Page 1

arXiv:0708.1551v1 [math.QA] 11 Aug 2007

Left-symmetric Bialgebras and An Analogue of the Classical

Yang-Baxter Equation

Chengming Bai1,2

1. Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, P.R. China

2. Department of Mathematics, Rutgers, The State University of New Jersey, Piscataway, NJ

08854, U.S.A.

Abstract

We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie

bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra

with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian

subalgebras. The latter is called a parak¨ ahler Lie algebra or a phase space of a Lie algebra in

mathematical physics. We introduce and study coboundary left-symmetric bialgebras and our

study leads to what we call “S-equation”, which is an analogue of the classical Yang-Baxter

equation. In a certain sense, the S-equation associated to a left-symmetric algebra reveals the

left-symmetry of the products. We show that a symmetric solution of the S-equation gives a

parak¨ ahler Lie algebra. We also show that such a solution corresponds to the symmetric part of a

certain operator called “O-operator”, whereas a skew-symmetric solution of the classical Yang-

Baxter equation corresponds to the skew-symmetric part of an O-operator. Thus a method

to construct symmetric solutions of the S-equation (hence parak¨ ahler Lie algebras) from O-

operators is provided. Moreover, by comparing left-symmetric bialgebras and Lie bialgebras, we

observe that there is a clear analogue between them and, in particular, parak¨ ahler Lie groups

correspond to Poisson-Lie groups in this sense.

Key Words

parak¨ ahler Lie algebra, left-symmetric algebra, left-symmetric bialgebra, S-

equation

Mathematics Subject Classification

17B, 53C, 81R

1

Page 2

1 Introduction

Left-symmetric algebras (or under other names like pre-Lie algebras, quasi-associative algebras,

Vinberg algebras and so on) are Lie-admissible algebras ((nonassociative) algebras whose com-

mutators are Lie algebras) whose left multiplication operators form a Lie algebra. They have

already been introduced by A. Cayley in 1896 as a kind of rooted tree algebras ([Ca]). They

also arose from the study of several topics in geometry and algebra in 1960s, such as convex

homogenous cones ([V]), affine manifolds and affine structures on Lie groups ([Ko],[Mat]), defor-

mation of associative algebras ([G]) and so on. In particular, a Lie algebra G with a compatible

left-symmetric algebra structure is the Lie algebra of a Lie group G with a left-invariant affine

structure, that is, there exists a left-invariant (locally) flat and torsion free connection ∇ in G.

The left-symmetric algebra structure corresponds to the connection ∇ given by XY = ∇XY

for X,Y ∈ G and (a geometric interpretation of) the left-symmetry is just the flatness of the

connection ∇ ([Me],[Ki]).

Furthermore, as it was pointed out in [CL] by Chapoton and Livernet, the left-symmetric

algebra “deserves more attention than it has been given”. It appears in many fields in math-

ematics and mathematical physics. In [Bu2], Burde gave a survey of certain different fields in

which left-symmetric algebras play an important role, such as vector fields, rooted tree algebras,

words in two letters, vertex algebras, operad theory, deformation complexes of algebras, convex

homogeneous cones, affine manifolds, left-invariant affine structures on Lie groups (see [Bu2]

and the references therein). Here are some more examples (partly overlap with some examples

in [Bu2]).

(a) Symplectic structures on Lie groups and Lie algebras. A symplectic Lie group is a Lie

group G with a left-invariant symplectic form ω+. One can define an affine structure on G by

([Ch])

ω+(∇x+y+,z+) = −ω+(y+,[x+,z+]) (1.1)

for any left-invariant vector fields x+,y+,z+and hence x+y+= ∇x+y+gives a left-symmetric

algebra. In fact, equation (1.1) is of great importance to the study of symplectic and K¨ ahler Lie

groups ([H],[LM],[DaM1-2],[MS],[KGM]).

(b) Complex and complex product structures on Lie groups and Lie algebras. From a real

left-symmetric algebra A, it is natural to define a Lie algebra structure on the vector space A⊕A

2

Page 3

(that is G(A) ⋉LG(A)) such that

J(x,y) = (−y,x), ∀ x,y ∈ A

(1.2)

is a complex structure on it. Moreover, there is a correspondence between left-symmetric algebras

and complex product structures on Lie algebras ([AS]), which plays an important role in the

theory of hypercomplex and hypersympletic manifolds ([Bar],[AD]).

(c) Vertex algebras. Vertex algebras are fundamental algebraic structures in conformal field

theory ([FLM],[FHL],[JL]). For any vertex algebra V ,

a ∗ b = a−1b, ∀ a,b ∈ V

(1.3)

defines a left-symmetric algebra. And a vertex algebra is equivalent to a left-symmetric algebra

and a Lie conformal algebra with some compatibility conditions ([BK]).

Vertex algebras are also closely related to a subclass of (finite-dimensional) left-symmetric

algebras, namely, Novikov algebras. Novikov algebras are left-symmetric algebras with com-

mutative right multiplication operators. They were introduced in connection with Hamiltonian

operators in the formal variational calculus ([GD]) and the following Poisson brackets of hydro-

dynamic type ([BN])

{u(x1),v(x2)} =

∂

∂x1((uv)(x1))x−1

1δ(x1

x2) + (uv + vu)(x1)

∂

∂x1x−1

1δ(x1

x2).

(1.4)

Furthermore, let A be a Novikov algebra and set A = A⊗C[t,t−1], where t is an indeterminate.

Then the bracket

[a ⊗ tm,b ⊗ tn] = (−mab + nba) ⊗ tm+n−1, ∀a,b ∈ A, m,n ∈ Z

(1.5)

defines a Lie algebra structure on A and this Lie algebra can be used to construct a vertex Lie

algebra and a vertex algebra ([Li]). Conversely, vertex algebras satisfying certain conditions must

correspond to some Novikov algebras (roughly speaking, such a vertex algebra V is generated

from V(2)which is a Novikov algebra, with some additional conditions) ([BKL]).

(d) Phase spaces of Lie algebras. The concept of phase space of a Lie algebra was introduced

by Kupershmidt in [Ku1] by replacing the underlying vector space with a Lie algebra and was

generalized in [Bai2]. In [Ku2], Kupershmidt pointed out that left-symmetric algebras appear

as an underlying structure of those Lie algebras that possess a phase space and thus they form

a natural category from the point of view of classical and quantum mechanics.

3

Page 4

(e) Left-symmetric algebras are closely related to certain integrable systems ([Bo1], [SS],[W]),

classical and quantum Yang-Baxter equation ([ES],[Ku3],[GS],[DiM]), combinatorics ([E]) and

so on. In particular, they play a crucial role in the Hopf algebraic approach of Connes and

Kreimer to renormalization of perturbative quantum field theory ([CK]).

In this paper, we study a structure, namely, parak¨ ahler structure, which appears in both ge-

ometry and mathematical physics, in terms of left-symmetric algebras. In geometry, a parak¨ ahler

manifold is a symplectic manifold with a pair of transversal Lagrangian foliations ([Lib]). A

parak¨ ahler Lie algebra G is the Lie algebra of a Lie group G with a G-invariant parak¨ ahler

structure ([Ka]). It is a symplectic Lie algebra with a decomposition into a direct sum of the

underlying vector spaces of two Lagrangian subalgebras. Some basic facts on the parak¨ ahler

structures on Lie groups and Lie algebras have been given in [Bai4]. On the other hand, a phase

space of a Lie algebra in mathematical physics ([Ku1-2], [Bai2]) is a parak¨ ahler Lie algebra. We

will show in this paper that conversely every parak¨ ahler Lie algebra is isomorphic to a phase

space of a Lie algebra.

We have obtained a structure theory of parak¨ ahler Lie algebras in terms of matched pairs

of Lie algebras (cf. Theorem 2.5) in [Bai2] and [Bai4]. This theory in fact gives a construction

of parak¨ ahler Lie algebras. However, except for some examples, it is still unclear when the

compatibility conditions appearing in the structure theory are satisfied.

The aim of this paper is to study further the structures of parak¨ ahler Lie algebras or phase

spaces of Lie algebras in terms of left-symmetric algebras and interpret the construction men-

tioned above using certain equivalent conditions which are much easier to use. Briefly speaking,

a parak¨ ahler Lie algebra is equivalent to a certain bialgebra structure, namely, a left-symmetric

bialgebra structure. From the point of view of phase spaces of Lie algebras, such a struc-

ture seems to be very similar to the Lie bialgebra structure given by Drinfeld ([D]). In fact,

left-symmetric bialgebras have many properties similar to those of Lie bialgebras. In particular,

there are so-called coboundary left-symmetric bialgebras which lead to an analogue (S-equation)

of the classical Yang-Baxter equation. In a certain sense, the S-equation in a left-symmetric

algebra reveals the left-symmetry of the products. A symmetric solution of the S-equation gives

a parak¨ ahler Lie algebra.

Furthermore, comparing left-symmetric bialgebras and Lie bialgebras in terms of several

different properties, we observe that there is a clear analogy between them and in particular,

parak¨ ahler Lie groups correspond to Poisson-Lie groups whose Lie algebras are Lie bialgebras in

4

Page 5

this sense. Since the classical Yang-Baxter equation can be regarded as a “classical limit” of the

quantum Yang-Baxter equation ([Be]), the analogy mentioned above, especially, the S-equation

corresponding to the classical Yang-Baxter equation found in this paper, suggests that there

might exist an analogue (“quantum S-equation” ) of the quantum Yang-Baxter equation. The

results in this paper are the beginning of a program to develop the theory of such analogues

of the quantum Yang-Baxter equation. We expect that our future study will be related to the

theory of quantum groups, tensor categories and vertex operator algebras.

We would like to point out that many structures (for example, see Theorem 3.8, Theorem

5.4 and so on) appearing in this paper exhibit features of both Lie algebras and left-symmetric

algebras, although the study of parak¨ ahler Lie algebras seems to be purely a topic in Lie algebras.

Indeed, the theory of Lie algebras alone is not enough here. Hence, unlike the theory of Lie

bialgebras which is purely Lie-algebra-theoretic, we need to combine the ideas and methods from

both the theory of Lie algebras and the theory of left-symmetric algebras.

The paper is organized as follows. In Section 2, we give some necessary definitions and

notations and basic results on left-symmetric algebras and parak¨ ahler Lie algebras. In Section

3, we study how to construct a left-symmetric algebra which is the direct sum of two left-

symmetric subalgebras. We observe that in the case of parak¨ ahler Lie algebras, matched pairs of

left-symmetric algebras are equivalent to the corresponding matched pairs of their sub-adjacent

Lie algebras, whereas it is not true in general. This also partly explains why left-symmetric

algebras appear in a problem which seems to be purely Lie-algebra-theoretic. In Section 4,

we introduce the notion of left-symmetric bialgebra which is precisely equivalent to the notion

of parak¨ ahler Lie algebra. In Section 5, we consider the special case that a certain 1-cocycle

appearing in a left-symmetric bialgebra is coboundary. A sufficient and necessary condition for

the existence of such a structure leads to certain explicit equations. In Section 6, we discuss only

the simplest cases in Section 5. We obtain an equation in the left-symmetric algebra, namely,

the S-equation, which is an analogue of the classical Yang-Baxter equation in a Lie algebra. We

also give some important properties of the S-equation. In Section 7, we compare left-symmetric

bialgebras and Lie bialgebras by recalling some facts on Lie bialgebras. We also consider the

case that a left-symmetric bialgebra is also a Lie bialgebra.

Throughout this paper, all algebras are finite-dimensional, although many results still hold

in the infinite-dimensional case.

5

Page 6

2 Preliminaries and basic results

Definition 2.1

Let A be a vector space over a field F with a bilinear product (x,y) → xy. A

is called a left-symmetric algebra if for any x,y,z ∈ A, the associator

(x,y,z) = (xy)z − x(yz) (2.1)

is symmetric in x,y, that is,

(x,y,z) = (y,x,z), or equivalently (xy)z − x(yz) = (yx)z − y(xz).

(2.2)

Left-symmetric algebras are Lie-admissible algebras (cf. [Me]).

Proposition 2.1

Let A be a left-symmetric algebra. For any x,y ∈ A, let Lx and Rx

denote the left and right multiplication operator respectively, that is, Lx(y) = xy, Rx(y) = yx.

Let L : A → gl(A) with x → Lxand R : A → gl(A) with x → Rx(for every x ∈ A) be two

linear maps. Then we have the following results:

(1) The commutator

[x,y] = xy − yx, ∀x,y ∈ A,

(2.3)

defines a Lie algebra G(A), which is called the sub-adjacent Lie algebra of A and A is also called

the compatible left-symmetric algebra structure on the Lie algebra G(A).

(2) L gives a regular representation of the Lie algebra G(A), that is,

[Lx,Ly] = L[x,y], ∀x,y ∈ A.

(2.4)

(3) The identity (2.2) is equivalent to the following equation

[Lx,Ry] = Rxy− RyRx, ∀x,y ∈ A.

(2.5)

Left-symmetric algebras can be obtained from some known algebraic structures (it can be

regarded as a “realization theory”). Recall that a Novikov algebra A is a left-symmetric algebra

satisfying RxRy= RyRxfor any x,y ∈ A.

Example 2.1

Let (A,·) be a commutative associative algebra and D be its derivation.

Then the new product

x ∗ay = x · Dy + a · x · y, ∀ x,y ∈ A

(2.6)

makes (A, ∗a) become a Novikov algebra for a = 0 by S. Gel’fand ([GD]), for a ∈ F by Filipov

([F]) and for a fixed element a ∈ A by Xu ([X]). In [BM2-3], we constructed a deformation

6

Page 7

theory of Novikov algebras. In particular, the two kinds of Novikov algebras given by Filipov

and Xu are the special deformations of the algebras (A, ∗) = (A, ∗0) given by S. Gel’fand.

Moreover, we proved that the Novikov algebras in dimension ≤ 3 ([BM1]) can be realized as

the algebras defined by S. Gel’fand and their compatible linear deformations. We conjectured

that this conclusion can be extended to higher dimensions. On the other hand, due to the

structures of free Novikov algebras, any Novikov algebra is a quotient of a subalgebra of an

(infinite-dimensional) algebra given by equation (2.6) for a = 0 ([DL]).

Example 2.2

Let (G,[,]) be a Lie algebra and R : G → G be a linear map satisfying the

operator form of classical Yang-Baxter equation ([Se] and the Remark after Proposition 7.5)

[R(x),R(y)] = R([R(x),y] + [x,R(y)]), ∀ x,y ∈ G.

(2.7)

Then

x ∗ y = [R(x),y], ∀ x,y ∈ G (2.8)

defines a left-symmetric algebra ([GS],[BM5]). Recall that the matrix form of the above R

(satisfying equation (2.7)) is a classical r-matrix. Hence the above construction of left-symmetric

algebras can be regarded as a Lie algebra “left-twisted” by a classical r-matrix, which gives an

algebraic interpretation of “left-symmetry” (comparing with the geometric interpretation given

in the Introduction) ([Bai3]).

Example 2.3

Let (A,·) be an associative algebra and R : A → A be a linear map satisfying

R(x) · R(y) + R(x · y) = R(R(x) · y + x · R(y)), ∀x,y ∈ A.

(2.9)

Then

x ∗ y = R(x) · y − y · R(x) − x · y, ∀x,y ∈ A

(2.10)

defines a left-symmetric algebra ([GS]). The above R is called a Rota-Baxter operator which

was introduced to solve analytic ([Bax]) and combinatorial problems ([R]) and attracts more

attention in many fields in mathematics and mathematical physics ([EGK] and the references

therein). It is also related to the “modified classical Yang-Baxter equation” ([Se]).

Example 2.4

Let V be a vector space over the complex field C with the ordinary scalar

product (,) and a be a fixed vector in V , then

u ∗ v = (u,v)a + (u,a)v, ∀ u,v ∈ V

(2.11)

7

Page 8

defines a left-symmetric algebra on V which gives the integrable (generalized) Burgers equation

([SS])

Ut= Uxx+ 2U ∗ Ux+ (U ∗ (U ∗ U)) − ((U ∗ U) ∗ U).

(2.12)

In [Bai1], we generalized the above construction to get left-symmetric algebras from linear

functions and in particular, we proved that the left-symmetric algebras given by equation (2.11)

are simple (without any ideals except zero and itself) (cf. [Bu1]).

On the other hand,

Definition 2.2

A Lie algebra G is called a symplectic Lie algebra if there is a nondegenerate

skew-symmetric 2-cocycle ω (the symplectic form) on G, that is,

ω([x,y],z) + ω([y,z],x) + ω([z,x],y) = 0, ∀x,y,z ∈ G.

(2.13)

We denote it by (G,ω). Let H be a subalgebra of a symplectic Lie algebra (G,ω) and H⊥=

{x ∈ G|ω(x,y) = 0, ∀ y ∈ H}. If H ⊂ H⊥, then H is called totally isotropic. If H = H⊥, then

H is called Lagrangian.

Theorem 2.2

([Ch]) Let (G,ω) be a symplectic Lie algebra. Then there exists a com-

patible left-symmetric algebra structure “∗′′on G given by

ω(x ∗ y,z) = −ω(y,[x,z]), ∀x,y,z ∈ G.

(2.14)

Throughout this paper, we mainly study the following symplectic Lie algebras.

Definition 2.3

Let (G,ω) be a symplectic Lie algebra. G is called a parak¨ ahler Lie algebra

if G is a direct sum of the underlying vector spaces of two Lagrangian subalgebras G+and G−. It

is denoted by (G,G+,G−,ω). Two parak¨ ahler Lie algebras (G1,G+

1,G−

1,ω1) and (G2,G+

2,G−

2,ω2)

are isomorphic if there exists a Lie algebra isomorphism ϕ : G1→ G2such that

ϕ(G+

1) = G+

2, ϕ(G−

1) = G−

2, ω1(x,y) = ϕ∗ω2(x,y) = ω2(ϕ(x),ϕ(y)), ∀x,y ∈ G1.

(2.15)

It is easy to see that a symplectic Lie algebra (G,ω) is a parak¨ ahler Lie algebra if and only

if G is a direct sum of the underlying vector spaces of two totally isotropic subalgebras.

Proposition 2.3

([Bai2])Let (G,G+,G−,ω) be a parak¨ ahler Lie algebra, then there exists

a left-symmetric algebra structure on G given by equation (2.14) such that G+and G−are

left-symmetric subalgebras. Moreover, two parak¨ ahler Lie algebras (Gi,G+

i,G−

i,ωi) (i = 1,2)

are isomorphic if and only if there exist an isomorphism of left-symmetric algebras satisfying

equation (2.15) which the compatible left-symmetric algebras are given by equation (2.14).

8

Page 9

Definition 2.4

Let G be a Lie algebra. If there is a Lie algebra structure on a direct sum

of the underlying vector spaces of G and its dual space G∗= Hom(G,F) such that G and G∗are

Lie subalgebras and the natural skew-symmetric bilinear form on G ⊕ G∗

ωp(x + a∗,y + b∗) = ?a∗,y? − ?x,b∗?, ∀x,y ∈ G,a∗,b∗∈ G∗,

(2.16)

is a 2-cocycle, where ?,? is the ordinary pair between G and G∗, then it is called a phase space

of the Lie algebra G ([Ku1],[Bai2]).

Obviously, a phase space of a Lie algebra is a parak¨ ahler Lie algebra. Furthermore, we have

Proposition 2.4

Every parak¨ ahler Lie algebra (G,G+,G−,ω) is isomorphic to a phase

space of G+.

Proof

Let ϕ : G−→ (G+)∗be a linear isomorphism given by ?ϕ(a),x? = ω(a,x) for

any x ∈ G+and a ∈ G−. Extending ϕ to be a linear isomorphism from G = G+⊕ G−to

Gp= G+⊕ (G+)∗with ϕ(x) = x for any x ∈ G+. Let the Lie algebra structure on Gpgiven by

[x,y] = [x,y], [a∗,b∗] = ϕ[ϕ−1(a∗),ϕ−1(b∗)], [x,a∗] = ϕ([x,ϕ−1(a∗)]), ∀x ∈ G+,a∗∈ (G+)∗.

Therefore ϕ is an isomorphism of Lie algebras. Furthermore, for any x,y ∈ G+,a,b ∈ G−, we

have

ω(x + a,y + b) = ω(a,y) + ω(x,b) = ?ϕ(a),y? − ?x,ϕ(b)? = ωp(ϕ(x + a),ϕ(y + b)).

Therefore ωpis a 2-cocycle in the above Lie algebra Gpand ϕ is an isomorphism of parak¨ ahler

Lie algebras.

?

Remark

It is obvious that, by symmetry, every parak¨ ahler Lie algebra (G,G+,G−,ω) is

also isomorphic to a phase space of G−.

On the other hand, recall that (G,H,ρ,µ) is a matched pair of Lie algebras if G and H are

Lie algebras and ρ : G → gl(H) and µ : H → gl(G) are representations satisfying

ρ(x)[a,b] − [ρ(x)a,b] − [a,ρ(x)b] + ρ(µ(a)x)b − ρ(µ(b)x)a = 0;(2.17)

µ(a)[x,y] − [µ(a)x,y] − [x,µ(a)y] + µ(ρ(x)a)y − µ(ρ(y)a)x = 0,

(2.18)

for any x,y ∈ G and a,b ∈ H. In this case, there exists a Lie algebra structure on the vector

space G ⊕ H given by

[x + a,y + b] = [x,y] + µ(a)y − µ(b)x + [a,b] + ρ(x)b − ρ(y)a, ∀x,y ∈ G,a,b ∈ H.

(2.19)

9

Page 10

We denote it by G ⊲⊳ρ

µ H or simply G ⊲⊳ H. Moreover, every Lie algebra which is a direct sum

of the underlying vector spaces of two subalgebras can be obtained from a matched pair of Lie

algebras ([Maj], [T]).

Theorem 2.5

([Bai2]) Let (A,·) be a left-symmetric algebra. Suppose there is another

left-symmetric algebra structure “◦′′on its dual space A∗. Let L∗

·and L∗

◦be the dual represen-

tation of the regular representation of G(A) and G(A∗) respectively, that is,

?L∗

·(x)a∗,y? = −?a∗,x · y?, ?L∗

◦(a∗)x,b∗? = −?x,a∗◦ b∗?, ∀x,y ∈ A, a∗,b∗∈ A∗.

(2.20)

Then there exists a parak¨ ahler Lie algebra structure on the vector space A ⊕ A∗such that

G(A) and G(A∗) are Lagrangian subalgebras associated to the symplectic form (2.16) if and only

if (G(A),G(A∗),L∗

·,L∗

◦) is a matched pair of Lie algebras. Furthermore, every parak¨ ahler Lie

algebra can be obtained from the above way.

3 Bimodules and matched pairs of left-symmetric algebras

Definition 3.1

Let A be a left-symmetric algebra and V be a vector space. Let S,T : A →

gl(V ) be two linear maps. V (or the pair (S,T), or (S,T,V )) is called a bimodule of A if

S(x)S(y)v − S(xy)v = S(y)S(x)v − S(yx)v, ∀x,y ∈ A,v ∈ V ; (3.1)

S(x)T(y)v − T(y)S(x)v = T(xy)v − T(y)T(x)v, ∀x,y ∈ A,v ∈ V.

(3.2)

According to [Sc], the following result is obvious.

Proposition 3.1

Let (A,·) be a left-symmetric algebra and V be a vector space. Let

S,T : A → gl(V ) be two linear maps. Then (S,T) is a bimodule of A if and only if the direct

sum A ⊕ V of vector spaces is turned into a left-symmetric algebra (the semidirect sum) by

defining multiplication in A ⊕ V by

(x1+ v1) ∗ (x2+ v2) = x1· x2+ (S(x1)v2+ T(x2)v1), ∀x1,x2∈,v1,v2∈ V

(3.3)

We denote it by A ⋉S,TV or simply A ⋉ V .

Lemma 3.2

Let (S,T,V ) be a bimodule of a left-symmetric algebra A. Then

(1) S : A → gl(V ) is a representation of the sub-adjacent Lie algebra G(A).

(2) ρ = S − T is a representation of the Lie algebra G(A).

(3) For any representation µ : G(A) → gl(G(A)) of the Lie algebra G(A), (µ,0) is a bimodule

of A.

10

Page 11

(4) The left-symmetric algebras A⋉S,TV and A⋉S−T,0V given by the bimodules (S,T) and

(S − T,0) respectively have the same sub-adjacent Lie algebra:

[x1+ v1,x2+ v2] = [x1,x2] + (S − T)(x1)v2− (S − T)(x2)v1, ∀x1,x2∈ G(A),v1,v2∈ V. (3.4)

It is the semidirect sum of a Lie algebra G(A) and its module (ρ = S − T,V ), we denote it by

G(A) ⋉S−TV .

Proof

(1) and (3) follow directly from Definition 3.1. For any x,y ∈ A, we have

[(S − T)(x),(S − T)(y)]=[S(x),S(y)] − [S(x),T(y)] − [T(x),S(y)] + [T(x),T(y)]

=

S([x,y]) − T(xy) + T(y)T(x) + T(yx) − T(x)T(y) + [T(x),T(y)]

=(S − T)([x,y]).

Thus ρ = S − T is a representation of the Lie algebra G(A). Hence (4) follows immediately. ?

Proposition 3.3

Let (A,·) be a left-symmetric algebra and (S,T,V ) be its bimodule. Let

S∗,T∗: A → gl(V∗) be the linear maps given by

?S∗(x)u∗,v? = −?S(x)v,u∗?, ?T∗(x)u∗,v? = −?T(x)v,u∗?, ∀x ∈ A,u∗∈ V∗,v ∈ V.

(3.5)

Then (S∗−T∗,−T∗,V∗) is a bimodule of A. Therefore there are two compatible left-symmetric

algebra structures A ⋉S∗,0V∗and A ⋉S∗−T∗,−T∗ V∗on the Lie algebra G(A) ⋉S∗ V∗.

Proof

Obviously, S∗− T∗is just the dual representation of the representation ρ = S − T

of the sub-adjacent Lie algebra G(A). For any x,y ∈ A, u∗∈ V∗, v ∈ V , we have

?[(S∗− T∗)(x),−T∗(y)]u∗,v? = ?(S∗− T∗)(x)u∗,−T(y)v? + ?T∗(y)u∗,(S − T)(x)v?

= ?u∗,(S − T)(x)T(y)v? − ?u∗,T(y)(S − T)(x)v? = ?u∗,(T(xy) − T(x)T(y))v?

= ?−T∗(xy)u∗− T∗(y)T∗(x)u∗,v?.

Therefore (S∗− T∗,−T∗,V∗) is a bimodule of A. The other conclusions follow from Lemma

3.2 directly.

?

However, (S∗,T∗,V∗) is not a bimodule of A in general. Similar to the above discussion, we

have the following conclusion:

Proposition 3.4

Let A be a left-symmetric algebra and (S,T,V ) be its bimodule. Then

the following conditions are equivalent:

(1) (S − T,−T) is a bimodule of A;

(2) (S∗,T∗) is a bimodule of A;

11

Page 12

(3) T(x)T(y) = T(y)T(x) for any x,y ∈ A.

Example 3.1

Let A be a left-symmetric algebra.Then (L,0), (ad,0) and (L,R) are

bimodules of A, where ad = L − R : A → gl(A) is the adjoint representation of the Lie algebra

G(A). On the other hand, (L∗,0), (ad∗,0) and (ad∗,−R∗) are bimodules of A, too, where ad∗

and L∗are the dual representations of the adjoint representation and the regular representation

of the sub-adjacent Lie algebra G(A) respectively, and R∗= −ad∗+ L∗.

Example 3.2

Let A be a left-symmetric algebra. Then (L∗,R∗) is a bimodule of A if and

only if (ad,−R) is a bimodule of A, if and only if A is a Novikov algebra.

Example 3.3

Let A be a left-symmetric algebra. Then there are two compatible left-

symmetric algebras A⋉L∗,0A∗and A⋉ad∗,−R∗ A∗on the Lie algebra G(A)⋉L∗ G∗(A), which are

given by the bimodules (L∗,0) and (ad∗,−R∗) respectively. This conclusion has already been

given in [Bai2].

Next, we study how to construct a left-symmetric algebra structure on a direct sum A ⊕ B

of the underlying vector spaces of two left-symmetric algebras A and B such that A and B are

subalgebras.

Theorem 3.5

Let (A,·) and (B,◦) be two left-symmetric algebras. Suppose that there

are linear maps lA,rA: A → gl(B) and lB,rB: B → gl(A) such that (lA,rA) is a bimodule of A

and (lB,rB) is a bimodule of B and they satisfy the following conditions:

rA(x)[a,b] = rA(lB(b)x)a − rA(lB(a)x)b + a ◦ (rA(x)b) − b ◦ (rA(x)a); (3.6)

lA(x)(a◦b) = −lA(lB(a)x−rB(a)x)b+(lA(x)a−rA(x)a)◦b+rA(rB(b)x)a+a◦(lA(x)b); (3.7)

rB(a)[x,y] = rB(lA(y)a)x − rB(lA(x)a)y + x · (rB(a)y) − y · (rB(a)x); (3.8)

lB(a)(x·y) = −lB(lA(x)a−rA(x)a)y +(lB(a)x−rB(a)x)·y +rB(rA(y)a)x+x·(lB(a)y), (3.9)

for any x,y ∈ A,a,b ∈ B. Then there is a left-symmetric algebra structure on the vector space

A ⊕ B given by

(x+a)∗(y+b) = (x·y+lB(a)y+rB(b)x)+(a◦b+lA(x)b+rA(y)a), ∀x,y ∈ A,a,b ∈ B. (3.10)

We denote this left-symmetric algebra by A ⊲⊳lA,rA

lB,rBB or simply A ⊲⊳ B. And (A,B,lA,rA,lB,rB)

satisfying the above conditions is called a matched pair of left-symmetric algebras. On the other

hand, every left-symmetric algebra which is a direct sum of the underlying vector spaces of two

subalgebras can be obtained from the above way.

12

Page 13

Proof

For any x,y,z ∈ A and a,b,c ∈ B, the associator in A ⊲⊳ B satisfies

(x+a,y+b,z+c) = (x,y,z)+(x,b,c)+(a,y,c)+(x,y,c)+(x,b,z)+(a,b,c)+(a,y,z)+(a,b,z).

Therefore, the equation (3.10) defines a left-symmetric algebra structure on A ⊕ B if and only

if the following equations are satisfied:

(x,y,z) = (y,x,z) ⇐⇒

A is a left − symmetric algebra;

(x,y,c) = (y,x,c) ⇐⇒

lA is a representation of G(A) and equation (3.8) holds;

(a,y,z) = (y,a,z) ⇐⇒ [(x,b,z) = (b,x,z) ⇐⇒] equation (3.9) holds and rA satisfies

lA(y)rA(z)a − rA(z)lA(y)a = rA(y · z)a − rA(z)rA(y)a;

(a,b,c) = (b,a,c) ⇐⇒

B is a left − symmetric algebra;

(a,b,z) = (b,a,z) ⇐⇒

lB is a representation of G(B) and equation (3.6) holds;

(x,b,c) = (b,x,c) ⇐⇒[(a,y,c) = (y,a,c) ⇐⇒] equation (3.7) holds and rB satisfies

lB(b)rB(c)x − rB(c)lB(b)x = rB(b ◦ c)x − rB(c)rB(b)x.

Hence A ⊲⊳ B is a left-symmetric algebra if and only if (lA,rA) is a bimodule of A and (lB,rB) is

a bimodule of B and equations (3.6-3.9) hold. On the other hand, if A and B are left-symmetric

subalgebras of a left-symmetric C such that C = A⊕B which is a direct sum of the underlying

vector spaces of A and B, then it is easy to show that the linear maps lA,rA: A → gl(B) and

lB,rB: B → gl(B) determined by

x ∗ a = lA(x)a + rB(a)x, a ∗ x = lB(a)x + rA(x)a, ∀x ∈ A,a ∈ B

satisfy equations (3.6-3.9). In addition, (lA,rA) is a bimodule of A and (lB,rB) is a bimodule

of B.

?

Remark

Obviously B is an ideal of A ⊲⊳ B if and only if lB= rB= 0. In this case, if in

addition, B is trivial (that is, all the products of B are zero), then A ⊲⊳lA,rA

0,0

B∼= A ⋉lA,rAB.

Furthermore, some other special cases of the above theorem have already been studied. For

example, Diatta and Medina studied the case that both A and B are left ideals of A ⊲⊳ B, that

is, rA= rB= 0 ([DiM]).

Corollary 3.6 ([DiM, Lemma 3.0.3])Let (A,·) and (B,◦) be two left-symmetric algebras.

Suppose there exist two Lie algebra representations ρ : G(A) → gl(B) and µ : G(B) → gl(A)

such that

ρ(x)(a ◦ b) = ρ(x)a ◦ b + a ◦ (ρ(x)b) − ρ(µ(a)x)b; (3.11)

13

Page 14

µ(a)(x · y) = µ(a)x · y + x · (µ(a)y) − µ(ρ(x)a)y,

(3.12)

for any x,y ∈ A,a,b ∈ B. A and B are called to be (ρ,µ)-linked. Then there is a left-symmetric

algebra structure on the vector space A ⊕ B given by

(x + a) ∗ (y + b) = (x · y + µ(a)y) + (a ◦ b + ρ(x)b),∀x,y ∈ A,a,b ∈ B.

(3.13)

Therefore, its sub-adjacent Lie algebra (which is the same Lie algebra given by the matched pair

of Lie algebras (G(A),G(B),ρ,µ)) is given by

[(x+a),(y+b)] = (x·y−y·x+µ(a)y−µ(b)x)+(a◦b−b◦a+ρ(x)b−ρ(y)a),∀x,y ∈ A,a,b ∈ B.

(3.14)

Corollary 3.7

Let (A,B,lA,rA,lB,rB) be a matched pair of left-symmetric algebras. Then

(G(A),G(B),lA− rA,lB− rB) is a matched pair of Lie algebras.

Proof

This conclusion can be proved by a direct computation or from the relation be-

tween the left-symmetric algebra A ⊲⊳ B and its sub-adjacent algebra Lie algebra. In fact, the

sub-adjacent Lie algebra G(A ⊲⊳ B) is just the Lie algebra obtained from the matched pair

(G(A),G(B),ρ,µ):

[x + a,y + b] = [x,y] + µ(a)y − µ(b)x + [a,b] + ρ(x)b − ρ(y)a, ∀x,y ∈ G(A),a,b ∈ G(B),

where ρ = lA− rA,µ = lB− rB.

?

Now we turns to the case of a parak¨ ahler Lie algebra: ρ = L∗

·and µ = L∗

◦.

Theorem 3.8

Let (A,·) be a left-symmetric algebra. Suppose there is another left-

symmetric algebra structure “◦′′on its dual space A∗. Then (G(A),G(A∗),L∗

·,L∗

◦) is a matched

pair of Lie algebras if and only if (A,A∗,ad∗

·,−R∗

·,ad∗

◦,−R∗

◦) is a matched pair of left-symmetric

algebras.

Proof

We only need to prove the “only if” part. A direct proof is given as follows. In

fact, in the case lA= ad∗

·,rA= −R∗

·, lB= lA∗ = ad∗

◦,rB= rA∗ = −R∗

◦, ρ = L∗

·and µ = L∗

◦, we

have

equation (2.17) ⇐⇒ equation (3.6) ⇐⇒ equation (3.9);

equation (2.18) ⇐⇒ equation (3.7) ⇐⇒ equation (3.8).

As an example (the proof of other equivalent relations is similar), we show how equation (2.17)

is equivalent to equation (3.6). In fact, it follows from

?−R∗

·(x)[a∗,b∗],y? = −?L∗(y)[a∗,b∗],x?;

14

Page 15

?−R∗

·(ad∗

◦(b∗)x)a∗,y? = ?[b∗,L∗

·(y)a∗],x?;

−?−R∗

·(ad∗

◦(a∗)x)b∗,y? = ?−[a∗,L∗

·(y)b∗],x?;

?a∗◦ (−R∗

·(x)b∗),y? = ?L∗

·(L∗

◦(a∗)y)(b∗),x?;

?−b∗◦ (−R∗

·(x)a∗),y? = ?−L∗

·(L∗

◦(b∗)y)(a∗),x?,

for any x,y ∈ A and a∗,b∗∈ A∗.

There is another proof. If (G(A),G(A∗),L∗

·,L∗

◦) is a matched pair of Lie algebras, then

G(A) ⊲⊳L∗

·

◦G(A∗) is a parak¨ ahler Lie algebra with the natural symplectic form (2.16). Hence

there exists a compatible left-symmetric algebra structure on G(A) ⊲⊳L∗

L∗

·

◦G(A∗) given by equation

L∗

(2.14). With a simple and direct computation, we know that A and A∗are its subalgebras and

the other products are given by

x ∗ a∗= ad∗

·(x)a∗− R∗

◦(a∗)(x), a∗∗ x = ad∗

◦(a∗)x − R∗

·(x)a∗, ∀x ∈ A,a∗∈ A∗.

Therefore (A,A∗,ad∗

·,−R∗

·,ad∗

◦,−R∗

◦) is a matched pair of left-symmetric algebras.

?

Remark

Unlike the case of semidirect sum in Example 3.3, it is not true that there is

a compatible left-symmetric structure on the Lie algebra G(A) ⊲⊳L∗

·

◦G(A∗) coming from the

L∗

bimodules (L∗

·,0) and (L∗

◦,0) since in this case, (A,A∗,L∗

·,0,L∗

◦,0) is not a matched pair of

left-symmetric algebras in general. In fact, from Corollary 3.6, (A,A∗,L∗

·,0,L∗

◦,0) is a matched

pair of left-symmetric algebras if and only if

L∗

·(x)(a∗◦ b∗) = −L∗

·(L∗

◦(a∗)x)b∗+ L∗

·(x)a∗◦ b∗+ a∗◦ L∗

·(x)b∗; (3.15)

L∗

◦(a∗)(x · y) = −L∗

◦(L∗

·(x)a∗)y + L∗

◦(a∗)x · y + x · L∗

◦(a∗)y,

(3.16)

for any x,y ∈ A and a∗,b∗∈ A∗which is a little stronger than equations (2.17-2.18) in the case

ρ = L∗

·and µ = L∗

◦.

4 Left-symmetric bialgebras

At first, we give some notations as follows.

Let G be a Lie algebra. For any two representations (ρ,V ) and (µ,W) of G, it is easy to

know that (ρ ⊗ 1 + 1 ⊗ µ,V ⊗ W) is also a representation of G, where

(ρ ⊗ 1 + 1 ⊗ µ)(x)(v ⊗ w) = (ρ(x) ⊗ 1 + 1 ⊗ µ(x))(v ⊗ w) = ρ(x)v ⊗ w + v ⊗ µ(x)w,

(4.1)

for any x ∈ G,v ∈ V,w ∈ W.

15

Page 16

For a Lie algebra G and a representation (ρ,V ) of G, recall that a 1-cocycle δ associated to

ρ (denoted by (ρ,δ)) is a linear map from G to V satisfying

δ([x,y]) = ρ(x)δ(y) − ρ(y)δ(x), ∀x,y ∈ G.

(4.2)

For a linear map φ : V1→ V2, we denote the dual (linear) map by φ∗: V∗

2→ V∗

1given by

?v,φ∗(u∗)? = ?φ(v),u∗?, ∀v ∈ V1,u∗∈ V2.

(4.3)

Theorem 4.1

Let (A,·) be a left-symmetric algebra whose product is given by a linear

map β∗: A⊗A → A. Suppose there is another left-symmetric algebra structure “◦′′on its dual

space A∗given by a linear map α∗: A∗⊗A∗→ A∗. Then (G(A),G(A∗),L∗

·,L∗

◦) is a matched pair

of Lie algebras if and only if α : A → A⊗A is a 1-cocycle of G(A) associated to L·⊗1+1⊗ad·

and β : A∗→ A∗⊗ A∗is a 1-cocycle of G(A∗) associated to L◦⊗ 1 + 1 ⊗ ad◦.

Proof

Let {e1,···,en} be a basis of A and {e∗

n ?

k=1

1,···,e∗

n} be its dual basis. Set ei· ej =

n ?

k=1

ck

ijekand e∗

i◦ e∗

j=

fk

ije∗

k. Therefore, we have

α(ek) =

n

?

i,j=1

fk

ijei⊗ej, β(e∗

k) =

n

?

i,j=1

ck

ije∗

i⊗e∗

j; L∗

·(ei)e∗

j= −

n

?

k=1

cj

ike∗

k, L∗

◦(e∗

i)ej= −

n

?

k=1

fj

ikek.

Hence the coefficient of em⊗ enin

α([ei,ej]) = [L·(ei) ⊗ 1 + 1 ⊗ ad·(ei)]α(ej) − [L·(ej) ⊗ 1 + 1 ⊗ ad·(ej)]α(ei)

gives the following relation (for any i,j,m,n)

n

?

k=1

(ck

ij− ck

ji)fk

mn=

n

?

k=1

[cm

ikfj

kn+ (cn

ik− cn

ki)fj

mk− cm

jkfi

kn− (cn

jk− cn

kj)fi

mk],

which is precisely the relation given by the coefficient of enin

−L∗

◦(e∗

m)[ei,ej] = L∗

◦(L∗

·(ei)e∗

m)ej− [ei,L∗

◦(e∗

m)ej] − L∗

◦(L∗

·(ej)e∗

m)ei− [L∗

◦(e∗

m)ei,ej].

Then α is a 1-cocycle of G(A) associated to L·⊗ 1 + 1 ⊗ ad·if and only if equation (2.18) holds

in the case ρ = L∗

·and µ = L∗

◦. Similarly (or by symmetry), β is a 1-cocycle of G(A∗) associated

to L◦⊗ 1 + 1 ⊗ ad◦if and only if equation (2.17) holds.

?

Definition 4.1

Let A be a vector space. A left-symmetric bialgebra structure on A is a

pair of linear maps (α,β) such that α : A → A ⊗ A,β : A∗→ A∗⊗ A∗and

(a) α∗: A∗⊗ A∗→ A∗is a left-symmetric algebra structure on A∗;

16

Page 17

(b) β∗: A ⊗ A → A is a left-symmetric algebra structure on A;

(c) α is a 1-cocycle of G(A) associated to L ⊗ 1 + 1 ⊗ ad with values in A ⊗ A;

(d) β is a 1-cocycle of G(A∗) associated to L ⊗ 1 + 1 ⊗ ad with values in A∗⊗ A∗.

We also denote this left-symmetric bialgebras by (A,A∗,α,β) or simply (A,A∗).

Proposition 4.2

Let (A,·) be a left-symmetric algebra and (A∗,◦) be a left-symmetric

algebra structure on its dual space A∗. Then the following conditions are equivalent:

(1) (G(A) ⊲⊳ G(A)∗,G(A),G(A∗),ωp) is a parak¨ ahler Lie algebra, where ωpis given by equa-

tion (2.16);

(2) (G(A),G(A∗),L∗

·,L∗

◦) is a matched pair of Lie algebras;

(3) (A,A∗,ad∗

·,−R∗

·,ad∗

◦,−R∗

◦) is a matched pair of left-symmetric algebras;

(4) (A,A∗) is a left-symmetric bialgebra.

Proof

The proofs are straightforward.

?

Definition 4.2

Let (A,A∗,αA,βA) and (B,B∗,αB,βB) be two left-symmetric bialgebras.

A homomorphism of left-symmetric bialgebras ϕ : A → B is a homomorphism of left-symmetric

algebras such that ϕ∗: B∗→ A∗is also a homomorphism of left-symmetric algebras, that is, ϕ

satisfies

(ϕ ⊗ ϕ)αA(x) = αB(ϕ(x)), (ϕ∗⊗ ϕ∗)βB(a∗) = βA(ϕ∗(a∗)),∀ x ∈ A,a∗∈ B∗.

(4.4)

An isomorphism of left-symmetric bialgebras is an invertible homomorphism of left-symmetric

bialgebras.

Proposition 4.3

Two parak¨ ahler Lie algebras are isomorphic if and only if their corre-

sponding left-symmetric bialgebras are isomorphic.

Proof

Let (G(A) ⊲⊳ G(A∗),G(A),G(A∗),ωp) and (G(B) ⊲⊳ G(B∗),G(B),G(B∗),ωp) be two

parak¨ ahler Lie algebras. Let {e1,···,en} be a basis of A and {e∗

1,···,e∗

n} be its dual basis.

If ϕ : G(A) ⊲⊳ G(A∗) → G(B) ⊲⊳ G(B∗) is an isomorphism of parak¨ ahler Lie algebras, then

ϕ|A: A → B and ϕ|A∗ : A∗→ B∗are isomorphisms of left-symmetric algebras due to Proposition

2.3. Moreover, ϕ|A∗ = (ϕ|A)∗−1since

?ϕ|A∗(e∗

i),ϕ(ej)?=

ωp(ϕ|A∗(e∗

i),ϕ(ej)) = ωp(e∗

i,ej)

=

δij= ?e∗

i,ej? = ?ϕ∗(ϕ|A)∗−1(e∗

i),ej?

=?(ϕ|A)∗−1(e∗

i),ϕ(ej)?.

Hence (A,A∗) and (B,B∗) are isomorphic as left-symmetric bialgebras. On the other hand,

let (A,A∗) and (B,B∗) be two isomorphic left-symmetric bialgebras and ϕ′: A → B be an

17

Page 18

isomorphism of left-symmetric bialgebras. Set ϕ : A ⊕ A∗→ B ⊕ B∗be a linear map given by

ϕ(x) = ϕ′(x),ϕ(a∗) = (ϕ′∗)−1(a∗), ∀x ∈ A,a∗∈ A∗.

Then it is easy to know that ϕ is an isomorphism of the two parak¨ ahler Lie algebras (G(A) ⊲⊳

G(A∗),G(A),G(A∗),ωp) and (G(B) ⊲⊳ G(B∗),G(B),G(B∗),ωp).

?

Example 4.1

Let (A,A∗,α,β) be a left-symmetric bialgebra. Then its dual (A∗,A,β,α)

is also a left-symmetric bialgebra.

Example 4.2

Let A be a left-symmetric algebra. Let the left-symmetric algebra structure

on A∗be trivial, then in this case (A,A∗,0,β) is a left-symmetric bialgebra. And its correspond-

ing left-symmetric algebra is A⋉ad∗,−R∗ A∗. Moreover, its corresponding parak¨ ahler Lie algebra

is just the semidirect sum G(A)⋉L∗ G(A∗) with the symplectic form ωpgiven by equation (2.16).

Dually, let A be a trivial left-symmetric algebra, then the left-symmetric bialgebra structures

on A are in one-to-one correspondence with the left-symmetric algebra structure on A∗.

Example 4.3

Let (A,A∗) be a left-symmetric bialgebra. In next section, we will prove

that there exists a natural left-symmetric bialgebra structure on the direct sum A ⊕ A∗of the

underlying vector spaces of A and A∗.

5Coboundary left-symmetric bialgebras

In this section, we study the case that α is a 1-coboundary associated to L ⊗ 1 + 1 ⊗ ad.

Definition 5.1

A left-symmetric bialgebra (A,A∗,α,β) is called coboundary if α is a 1-

coboundary of G(A) associated to L ⊗ 1 + 1 ⊗ ad, that is, there exists a r ∈ A ⊗ A such that

α(x) = (Lx⊗ 1 + 1 ⊗ adx)r, ∀x ∈ A.

(5.1)

Let A be a left-symmetric algebra whose product is given by β∗: A⊗A → A and r ∈ A⊗A.

Suppose α : A → A ⊗ A is a 1-coboundary of G(A) associated to L ⊗ 1 + 1 ⊗ ad (given by

equation (5.1)). Then it is obvious that α is a 1-cocycle of G(A) associated to L ⊗ 1 + 1 ⊗ ad.

Therefore, (A,A∗,α,β) is a left-symmetric bialgebra if and only if the following two conditions

are satisfied:

(1) α∗: A∗⊗ A∗→ A∗defines a left-symmetric algebra structure on A∗.

(2) β is a 1-cocycle of G(A∗) associated to L ⊗ 1 + 1 ⊗ ad, where the left-symmetric algebra

structure of A∗is given by (1).

18

Page 19

For any vector space A, let σ : A ⊗ A → A ⊗ A be the linear map (exchanging operator)

satisfying σ(x ⊗ y) = y ⊗ x, for any x,y ∈ A. Let r =?

denote r12= r and r21= σ(r) =?

Proposition 5.1

Let (A,·) be a left-symmetric algebra whose product is given by β∗:

iai⊗ bi∈ A ⊗ A, where ai,bi∈ A. We

ibi⊗ ai.

A ⊗ A → A and r ∈ A ⊗ A. Suppose there exists a left-symmetric algebra structure “ ◦ ” on A∗

given by α∗: A∗⊗ A∗→ A∗, where α is given by equation (5.1). Then β : A∗→ A∗⊗ A∗is a

1-cocycle of G(A∗) associated to L ⊗ 1 + 1 ⊗ ad if and only if r satisfies

[P(x · y) − P(x)P(y)](r12− r21) = 0,∀x,y ∈ A,

(5.2)

where P(x) = Lx⊗ 1 + 1 ⊗ Lx.

Proof

Let {e1,···,en} be a basis of A and {e∗

1,···,e∗

n} be its dual basis. Set r =?

i,j

aijei⊗

ej. Suppose ei· ej=?

kck

ijekand e∗

i◦ e∗

j=?

kfk

ije∗

k. Since

α(ei) =

?

k,l

fi

klek⊗ el= (Lei⊗ 1 + 1 ⊗ adei)r,

we have (for any i,k,l)

fi

kl=

?

t

[atlck

it+ akt(cl

it− cl

ti)].

Therefore β is a 1-cocycle of G(A∗) associated to L◦⊗ 1 + 1 ⊗ ad◦if and only if

β[e∗

i,e∗

j] = (Le∗

i⊗ 1 + 1 ⊗ ade∗

i)β∗(e∗

j) − (Le∗

j⊗ 1 + 1 ⊗ ade∗

j)β∗(e∗

i),

where β(e∗

i) =?

k,lci

kle∗

k⊗ e∗

l. Let both sides of the above equation act on em⊗ enand after

rearranging the terms suitably, we have

(F1) + (F2) + (F3) + (F4) + (F5) + (F6) = 0,

where

(F1)=

?

t,l

(atl− alt)(cj

ntci

ml− ci

ntcj

ml);

(F2)=

?

t,l

(−altci

tncj

ml+ atlcj

mtci

ln− atlci

mtcj

ln+ altci

mlcj

tn);

(F3)=

?

t,l

(atj− ajt)(ci

ltcl

mn− cl

ntci

ml);

(F4)=

?

t,l

ajt[ci

tlcl

mn− cl

tnci

ml+ (cl

mt− cl

tm)ci

ln];

19

Page 20

(F5)=

?

t,l

(ati− ait)(cl

ntcj

ml− cj

ltcl

mn);

(F6)=

?

t,l

ait[−cj

tlcl

mn+ cl

tncj

ml− (cl

mt− cl

tm)cj

ln].

(F1) is the coefficient of ei⊗ ejin

(−Lem⊗ Len− Len⊗ Lem)

?

t,l

(atl− alt)et⊗ el= (−Lem⊗ Len− Len⊗ Lem](r12− r21);

(F2) = 0 by interchanging the indices t and l;

(F3) is the coefficient of ei⊗ ejin

(Lem·en⊗ 1 − LemLen⊗ 1)

?

t,l

(atl− alt)et⊗ el= [Lem·en⊗ 1 − LemLen⊗ 1](r12− r21);

(F4) = 0 since the term in the bracket is the coefficient of eiin (em,et,en) − (et,em,en) = 0.

(F5) is the coefficient of ei⊗ ejin

(1 ⊗ Lem·en− 1 ⊗ LemLen)

?

t,l

(atl− alt)et⊗ el= [1 ⊗ Lem·en− 1 ⊗ LemLen](r12− r21);

F(6) = 0 since the term in the bracket is the coefficient of ejin (et,em,en) − (em,et,en) = 0.

Therefore we have

[P(em· en) − P(em)P(en)](r12− r21) = 0.

Hence the conclusion holds.

?

For any linear map α : A → A ⊗ A, let Jα: A → A ⊗ A ⊗ A be a linear map given by

Jα(x) = (α⊗id)α(x)−(id⊗α)α(x)−(σ⊗id)(α⊗id)α(x)+(σ⊗id)(id⊗α)α(x),∀x ∈ A. (5.3)

Lemma 5.2

Let A be a vector space and α : A ⊗ A → A be a linear map. Then α∗:

A∗⊗ A∗→ A∗defines a left-symmetric algebra structure on A∗if and only if Jα= 0.

Proof

We denote the product in A∗by “◦′′. Then for any a∗,b∗∈ A∗, we have

a∗◦ b∗= α∗(a∗⊗ b∗).

Hence the associator satisfies (for any a∗,b∗,c∗∈ A∗)

(a∗,b∗,c∗)=[α∗(α∗⊗ id) − α∗(id ⊗ α∗)](a∗⊗ b∗⊗ c∗);

(b∗,a∗,c∗)=[α∗(α∗⊗ id) − α∗(id ⊗ α∗)](b∗⊗ a∗⊗ c∗)

=[α∗(α∗⊗ id) − α∗(id ⊗ α∗)](σ ⊗ id)(a∗⊗ b∗⊗ c∗).

20

Page 21

Therefore (a∗,b∗,c∗) = (b∗,a∗,c∗) for any a∗,b∗,c∗∈ A∗if and only if Jα= 0.

?

Proposition 5.3

Let (A,·) be a left-symmetric algebra. Let r =?

ai,bi∈ A. Define α : A → A ⊗ A by equation (5.1). Then

iai⊗bi∈ A⊗A, where

Jα(x) = Q(x)[[r,r]] +

?

j

[P(x · aj) − P(x)P(aj)](r12− r21) ⊗ bj,∀x ∈ A,

(5.4)

where

[[r,r]] = r13· r12− r23· r21+ [r23,r12] − [r13,r21] − [r13,r23],

(5.5)

and Q(x) = Lx⊗ 1 ⊗ 1 + 1 ⊗ Lx⊗ 1 + 1 ⊗ 1 ⊗ adx, P(x) = Lx⊗ 1 + 1 ⊗ Lxfor any x ∈ A.

Before proving this result, let us explain the notations. Let (A,·) be a left-symmetric algebra

and r =?

iai⊗ bi, set

r12=

?

i

ai⊗bi⊗1,r21=

?

i

bi⊗ai⊗1; r13=

?

i

ai⊗1⊗bi; r23=

?

i

1⊗ai⊗bi ∈ U(G(A)),

(5.6)

where U(G(A)) is the universal enveloping algebra of the sub-adjacent Lie algebra G(A). Set

r13· r12=

?

i,j

ai· aj⊗ bj⊗ bi; r23· r21=

?

i,j

bj⊗ ai· aj⊗ bi;

[r23,r12] = r23· r12− r12· r23=

?

ij

aj⊗ [ai,bj] ⊗ bi;

[r13,r21] = r13· r21− r21· r13=

?

ij

[ai,bj] ⊗ aj⊗ bi;

[r13,r23] = r13· r23− r23· r13=

?

ij

ai⊗ aj⊗ [bi,bj].

Proof

Let x ∈ A. After rearranging the terms suitably, we divide Jα(x) into three parts:

Jα(x) = (F1) + (F2) + (F3),

where

(F1)=

?

i,j

{(x · aj) · ai⊗ bi− bi⊗ (x · aj) · ai+ bi· aj⊗ x · ai− x · ai⊗ bi· aj

+ai⊗ [x · aj,bi− ai⊗ [x,bi] · aj+ [x,bi] · aj⊗ ai− [x · aj,bi] ⊗ ai} ⊗ bj;

(F2)=

?

i,j

{aj· ai⊗ bi+ ai⊗ [aj,bi] − bi⊗ aj· ai− [aj,bi] ⊗ ai} ⊗ [x,bj]

(F3)=

?

i,j

{−x · ai⊗ aj⊗ [bi,bj] − ai⊗ aj⊗ [[x,bi],bj] − ai⊗ x · aj⊗ [bi,bj]

+ai⊗ aj⊗ [[x,bj],bi]]}.

21

Page 22

On the other hand,

Q(x)(r13· r12) =

?

i,j

{x · (aj· ai) ⊗ bi⊗ bj+ aj· ai⊗ x · bi⊗ bj+ aj· ai⊗ bi⊗ [x,bj]};

Q(x)(−r23· r21) =

?

i,j

−{x · bi⊗ aj· ai⊗ bj+ bi⊗ x · (aj· ai) ⊗ bj+ bi⊗ aj· ai⊗ [x,bj]};

Q(x)([r23,r12]) =

?

i,j

{x · ai⊗ [aj,bi] ⊗ bj+ ai⊗ x · [aj,bi] ⊗ bj+ ai⊗ [aj,bi] ⊗ [x,bj]};

Q(x)(−[r13,r21]) =

?

i,j

−{x · [aj,bi] ⊗ ai⊗ bj+ [aj,bi] ⊗ x · ai⊗ bj+ [aj,bi] ⊗ ai⊗ [x,bj]};

Q(x)(−[r13,r23]) =

?

i,j

−{x · ai⊗ aj⊗ [bi,bj] + ai⊗ x · aj⊗ [bi,bj] + ai⊗ aj⊗ [x,[bi,bj]]};

?

j

[P(x · aj) − P(x)P(aj)](r12− r21) ⊗ bj=

?

i,j

{(x · aj) · ai⊗ bi+ ai⊗ (x · aj) · bi

−x · ai⊗ aj· bi− aj· ai⊗ x · bi− x · (aj· ai) ⊗ bi− ai⊗ x · (aj· bi)

−(x · aj) · bi⊗ ai− bi⊗ (x · aj) · ai+ x · bi⊗ aj· ai+ aj· bi⊗ x · ai

+x · (aj· bi) ⊗ ai+ bi⊗ x · (aj· ai)} ⊗ bj

After rearranging the terms suitably, the sum of the terms whose third component is bjin the

right hand side of equation (5.4) is

(F1′) = (F1a) + (F1b) + (F1c) + (F1d) + (F1e) + (F1f) + (F1g),

where

(F1a)=

?

i,j

{x · (aj· ai) + (x · aj) · ai− x · (aj· ai)} ⊗ bi⊗ bj=

?

i,j

(x · aj) · ai⊗ bi⊗ bj;

(F1b)=

?

i,j

bi⊗ {−x · (aj· ai) − (x · aj) · ai+ x · (aj· ai)} ⊗ bj=

?

i,j

−bi⊗ (x · aj) · ai⊗ bj;

(F1c)=

?

i,j

{−[aj,bi] ⊗ x · ai+ aj· bi⊗ x · ai} ⊗ bj=

?

i,j

bi· aj⊗ x · ai⊗ bj;

(F1d)=

?

i,j

{x · ai⊗ [aj,bi] − x · ai⊗ aj· bi} ⊗ bj=

?

i,j

−x · ai⊗ bi· aj⊗ bj;

(F1e)=

?

ij

{ai⊗ x · [aj,bi] + ai⊗ (x · aj) · bi− ai⊗ x · (aj· bi)} ⊗ bj

=

?

i,j

ai⊗ {−x · (bi· aj) + (x · aj) · bi} ⊗ bj=

?

i,j

ai⊗ {[x · aj,bi] − [x,bi] · aj} ⊗ bj;

(F1f)=

?

i,j

{−x · [aj,bi] ⊗ ai− (x · aj) · bi⊗ ai+ x · (aj· bi) ⊗ ai} ⊗ bj

=

?

i,j

{x · (bi· aj) − (x · aj) · bi} ⊗ ai⊗ bj=

?

i,j

{[x,bi] · aj− [x · aj,bi]} ⊗ ai⊗ bj;

22

Page 23

(F1g)=

?

i,j

{aj· ai⊗ x · bi− x · bi⊗ aj· ai− aj· ai⊗ x · bi+ x · bi⊗ aj· ai} ⊗ bj= 0.

Therefore (F1′) = (F1). Obviously, the sum of the terms whose third component is [x,bj] in

the right hand side of equation (5.4) is just (F2). Moreover, the sum of the other terms in the

right hand side of equation (5.4) is Q(x)(−[r13,r23]), which precisely equals to (F3) by Jacobi

identity in the sub-adjacent Lie algebra G(A). Hence equation (5.4) holds.

?

With the discussion above together, we have the following result.

Theorem 5.4

Let A be a left-symmetric algebra and r ∈ A⊗A. Then the map α defined

by equation (5.1) induces a left-symmetric algebra structure on A∗such that (A,A∗) is a left-

symmetric bialgebra if and only if the following two conditions are satisfied:

(a) [P(x · y) − P(x)P(y)](r12− r21) = 0 for any x,y ∈ A;

(b) Q(x)[[r,r]] = 0,

where [[r,r]] is given by equation (5.5) and Q(x) = Lx⊗ 1 ⊗ 1 + 1 ⊗ Lx⊗ 1 + 1 ⊗ 1 ⊗ adx,

P(x) = Lx⊗ 1 + 1 ⊗ Lxfor any x ∈ A.

A direct application of Theorem 5.4 is given as follows (cf. Example 4.3).

Theorem 5.5

Let (A,A∗,α,β) be a left-symmetric bialgebra. Then there is a canonical

left-symmetric bialgebra structure on A⊕A∗such that both the inclusions i1: A → A⊕A∗and

i2: A∗→ A ⊕ A∗into the two summands are homomorphisms of left-symmetric bialgebras.

Proof

Let r ∈ A ⊗ A∗⊂ (A ⊕ A∗) ⊗ (A ⊕ A∗) correspond to the identity map id : A → A.

Let {e1,···,en} be a basis of A and {e∗

that the left-symmetric algebra structure “∗” on A ⊕ A∗is given by SD(A) = A ⊲⊳ad∗

1,···,e∗

n} be its dual basis. Then r =?

iei⊗e∗

i. Suppose

·,−R∗

ad∗

◦,−R∗

·

◦A∗.

Then by Theorem 3.5, we have

x ∗ y = x · y, a∗∗ b∗= a∗◦ b∗, x ∗ a∗= ad∗

·(x)a∗− R∗

◦(a∗)x, a∗∗ x = ad◦(a∗)x − R∗

·(x)a∗,

for any x,y ∈ A,a∗,b∗∈ A∗. Next we prove that r satisfies the two conditions in Theorem 5.4.

If so, then

αSD(u) = (Lu⊗ 1 + 1 ⊗ adu)(r), ∀u ∈ SD(A)

can induce a left-symmetric bialgebra structure on SD(A).

For any λ,µ ∈ SD(A), equation (5.2) is equivalent to

?

i

+ei⊗ (λ ∗ µ) ∗ e∗

{(λ ∗ µ) ∗ ei⊗ e∗

i− λ ∗ (µ ∗ ei) ⊗ e∗

i− (λ ∗ µ) ∗ e∗

i⊗ ei+ λ ∗ (µ ∗ e∗

i) ⊗ ei

i− ei⊗ λ ∗ (µ ∗ e∗

i) + e∗

i⊗ λ ∗ (µ ∗ ei) − e∗

i⊗ (λ ∗ µ) ∗ ei

−λ ∗ ei⊗ µ ∗ e∗

i+ µ ∗ e∗

i⊗ λ ∗ ei+ λ ∗ e∗

i⊗ µ ∗ ei− µ ∗ ei⊗ λ ∗ e∗

i} = 0

23

Page 24

We can prove the equation above in the following four cases: (I) λ,µ ∈ A; (II) λ,µ ∈ A∗; (III)

λ ∈ A,µ ∈ A∗and (IV) λ ∈ A∗,µ ∈ A. As an example, we give a detailed proof of the first case

(the proof of other cases is similar). Let λ = ek, µ = el, then we have

?

i

ei⊗ (ek∗ el) ∗ e∗

i

=

?

i

−[ek· el,ei] ⊗ e∗

i+

?

i,m

?e∗

m◦ e∗

i,ek· el?ei⊗ em;

?

i

−(ek∗ el) ∗ e∗

i⊗ ei

=

?

i

e∗

i⊗ [ek· el,ei] −

?

i,m

?e∗

m◦ e∗

i,ek· el?em⊗ ei;

?

i

−ei⊗ ek∗ (el∗ e∗

i)=

?

i

−[el,[ek,ei]] ⊗ e∗

i−

?

i,m

?e∗

m◦ e∗

i,el?ei⊗ ek· em

+

?

i,m,n

?[el,em],e∗

i??e∗

n◦ e∗

m,ek?ei⊗ en;

?

i

ek∗ (el∗ e∗

i) ⊗ ei

=

?

i

e∗

i⊗ [el,[ek,ei]] +

?

i,m

?e∗

m◦ e∗

i,el?ek· em⊗ ei

−

?

i,m,n

?[el,em],e∗

i??e∗

n◦ e∗

m,ek?en⊗ ei;

?

i

−ek∗ ei⊗ el∗ e∗

i

=

?

i

ek· [el,ei] ⊗ e∗

i−

?

i,m

?e∗

m◦ e∗

i,el?ek· ei⊗ em;

?

i

el∗ e∗

i⊗ ek∗ ei

=

?

i

−e∗

i⊗ ek· [el,ei] +

?

i,m

?e∗

m◦ e∗

i,el?em⊗ ek· ei;

?

i

−el∗ ei⊗ ek∗ e∗

i

=

?

i

el· [ek,ei] ⊗ e∗

i−

?

i,m

?e∗

m◦ e∗

i,ek?el· ei⊗ em;

?

i

ek∗ e∗

i⊗ el∗ ei

=

?

i

−e∗

i⊗ el· [ek,ei] +

?

i,m

?e∗

m◦ e∗

i,ek?em⊗ el· ei.

The sum of the terms which are in A ⊗ A∗or A∗⊗ A is zero since

(ek· el) · ei− ek· (el· ei) − [ek· el,ei] − [el,[ek,ei]] + ek· [el,ei] + el· [ek,ei]

= ei· (ek· el) − (ei· ek) · el− ek(ei· el) + (ek· ei) · el= 0.

The coefficient of ei⊗ emin the sum of the terms which are in A ⊗ A is

?e∗

m◦ e∗

i,ek· el? − ?e∗

i◦ e∗

m,ek· el?

−

?

n

i,ek· en??e∗

{?e∗

n◦ e∗

i,el??e∗

m,ek· en? + ?[el,en],e∗

i??e∗

m◦ e∗

n,ek?

?e∗

n◦ e∗

m,el? − ?e∗

m,[el,en]??e∗

i◦ e∗

n,ek?

−?e∗

i,ek· en??e∗

m◦ e∗

n,el? + ?e∗

m,ek· en??e∗

i◦ e∗

n,el?

−?e∗

i,el· en??e∗

m◦ e∗

n,ek? + ?e∗

m,el· en??e∗

i◦ e∗

n,ek?}

= ?−L∗

·(ek)[e∗

m,e∗

i],el? + ?L∗

·(ek)e∗

m◦ e∗

i,el? + ?−L∗

·(L∗

◦(e∗

m)ek)e∗

i,el?

24

Page 25

−?L∗

·(ek)e∗

i◦ e∗

m,el? + ?L∗

·(L∗

◦(e∗

i)ek)e∗

m,el?

+?e∗

m◦ L∗

·(ek)e∗

i,el? − ?e∗

i◦ L∗

·(ek)e∗

m,el? = 0

since (G(A),G(A∗),L∗

·,L∗

◦) is a matched pair of Lie algebras. Hence P(ek·el)−P(ek)P(el)(r12−

r21) = 0. Furthermore,

[[r,r]]=

?

i,j

−[ei,e∗

{ei· ej⊗ e∗

j⊗ e∗

i+ ej⊗ [ei,e∗

j] ⊗ e∗

i− e∗

j⊗ ei· ej⊗ e∗

i

j] ⊗ ej⊗ e∗

i− ei⊗ ej⊗ [e∗

i,e∗

j]}

Note that

?

i,j

{ei· ej⊗ e∗

j⊗ e∗

i+ ej⊗ L∗

·(ei)e∗

j⊗ e∗

i} = 0;

?

i,j

{−ej⊗ L∗

◦(e∗

j)ei⊗ e∗

i+ L∗

◦(e∗

j)ei⊗ ej⊗ e∗

i− ei⊗ ej⊗ [e∗

i,e∗

j]} = 0;

?

i,j

{−L∗

·(ei)e∗

j⊗ ej⊗ e∗

i− e∗

j⊗ ei· ej⊗ e∗

i} = 0.

Therefore [[r,r]] = 0. Hence SD(A) is a left-symmetric bialgebra.

For ei∈ A, we have

αSD(ei)=

?

j

{ei· ej⊗ e∗

j+ ej⊗ [ei,e∗

j]}

=

?

j,m

{ei· ej⊗ e∗

j− ?e∗

j,ei· em?ej⊗ e∗

m+ ?e∗

j◦ e∗

m,ei?ej⊗ em}

=

?

j,m

αA(ei)

?e∗

j◦ e∗

m,ei?ej⊗ em

=

Therefore the inclusion i1 : A → A ⊕ A∗is a homomorphism of left-symmetric bialgebras.

Similarly, the inclusion i2: A∗→ A ⊕ A∗is also a homomorphism of left-symmetric bialgebras

since αSD(e∗

i) = αA∗(e∗

i).

?

Definition 5.2

Let (A,A∗) be a left-symmetric bialgebra. With the left-symmetric bial-

gebra structure given in Theorem 5.5, A ⊕ A∗is called a symplectic double of A. We denote it

by SD(A).

Therefore, by Theorem 5.5, we have the following conclusion.

Proposition 5.6

Let (A,A∗) be a left-symmetric bialgebra. Then the symplectic double

SD(A) of A is a left-symmetric bialgebra and its sub-adjacent Lie algebra is a parak¨ ahler Lie

algebra with the symplectic form ωpgiven by equation (2.16).

25

Page 26

At the end of this section, we would like to point out that, unlike the symmetry of 1-cocycle

of G(A) and G(A∗) appearing in the definition of a left-symmetric bialgebra (A,A∗), it is not

necessary that β is also a 1-coboundary of G(A∗) for a coboundary left-symmetric bialgebra

(A,A∗,α,β), where α is given by equation (5.1).

6

S-equation and its properties

The simplest way to satisfy the two conditions of Theorem 5.4 is to assume that r is symmetric

and [[r,r]] = 0. Note that

[[r,r]] = r21· r13− r12· r23− [r13,r23] + r13· (r12− r21) + r23· (r12− r21).

(6.1)

Therefore, by Theorem 5.4, we have the following conclusion.

Proposition 6.1

Let A be a left-symmetric algebra and r ∈ A⊗A. Suppose r is symmetric.

Then the map α defined by equation (5.1) induces a left-symmetric algebra structure on A∗such

that (A,A∗) is a left-symmetric bialgebra if

[[r,r]] = −r12· r13+ r12· r23+ [r13,r23] = 0.

(6.2)

Definition 6.1

Let A be a left-symmetric algebra and r ∈ A ⊗ A. Then equation (6.2) is

called S-equation in A.

Remark 1

Let the products on {1,2,3} correspond to the products of A: for example, for

any i,j,k = 1,2,3, (i·j) ·k or i·(j ·k) corresponds to the (order) product (x·y)·z or x·(y ·z)

of A respectively, and so on. For any symmetric r ∈ A ⊗ A, set (for any rij,i < j,i,j = 1,2,3)

rij· rjl←→ j · (i · l), rij· ril←→ i · (j · l), rij· rlj←→ (i · l) · j, rij· rli←→ (j · l) · i.

Then the S-equation (6.2) in a left-symmetric algebra corresponds to the “left-symmetry” of

the products. It is similar to the relation between the classical Yang-Baxter equation in a Lie

algebra G and the Jacobi identity of G ([BD],[D],[Se]).

Remark 2

Let σ123,σ132: A ⊗ A ⊗ A → A ⊗ A ⊗ A be two linear maps satisfying

σ123(x ⊗ y ⊗ z) = z ⊗ x ⊗ y, σ132(x ⊗ y ⊗ z) = y ⊗ z ⊗ x, ∀ x,y,z ∈ A.

Suppose that r is symmetric. Then it is easy to know that under the action of σ123and σ132

respectively, the S-equation (6.2) turns to be

[r12,r13] − r23· (r12− r13) = 0;

26

Page 27

[r12,r23] − r13· (r12− r23) = 0,

respectively.

Let A be a vector space. For any r ∈ A ⊗ A, r can be regarded as a map from A∗to A in

the following way:

?u∗⊗ v∗,r? = ?u∗,r(v∗)?, ∀u∗,v∗∈ A∗.

(6.3)

Proposition 6.2

Let (A,·) be a left-symmetric and r ∈ A⊗ A be a symmetric solution of

S-equation in A. Then the left-symmetric algebra and its sub-adjacent Lie algebra structure on

the symplectic double SD(A) can be given from the products in A as follows:

(a) a∗∗ b∗= a∗◦ b∗= −R∗

·(r(b∗))a∗+ ad∗

·(r(a∗))b∗, for any a∗,b∗∈ A∗; (6.4)

(b) [a∗,b∗] = a∗◦ b∗− b∗◦ a∗= L∗

·(r(a∗))b∗− L∗

·(r(b∗))a∗, for any a∗,b∗∈ A∗; (6.5)

(c) x ∗ a∗= x · r(a∗) − r(ad∗

·(x)a∗) + ad∗

·(x)a∗, for any x ∈ A, a∗∈ A∗; (6.6)

(d) a∗∗ x = r(a∗) · x + r(R∗

·(x)a∗) − R∗

·(x)a∗, for any x ∈ A, a∗∈ A∗;(6.7)

(e) [x,a∗] = [x,r(a∗)] − r(L∗

·(x)a∗) + L∗

·(x)a∗, for any x ∈ A, a∗∈ A∗. (6.8)

Proof

Let {e1,···,en} be a basis of A and {e∗

1,···,e∗

n} be its dual basis. Suppose that

ei· ej=?

5.1, we know that (for any k,l)

i,jck

ijekand r =?

i,jaijei⊗ ej, where aij= aji. Then from the proof of Proposition

e∗

k◦ e∗

l

=

?

i,t

{atlck

it+ akt(cl

it− cl

ti)}e∗

i=

?

i,t

{atl?ei· et,e∗

k? + akt?[ei,et],e∗

l?}e∗

i

=

?

i

−R∗

{?ei· r(e∗

l),e∗

k? + ?[ei,r(e∗

k)],e∗

l?}e∗

i

=

·(r(e∗

l))e∗

k+ ad∗

·(r(e∗

k))e∗

l.

Hence equation (6.4) holds. Therefore, we have

−R∗

◦(e∗

k)el

=

?

i

?

i

el· r(e∗

?−R∗

◦(e∗

k)el,e∗

i?ei=

?

i

?el,e∗

i◦ e∗

k?ei

=?el,−R∗

·(r(e∗

k))e∗

i+ ad∗

·(r(e∗

i))e∗

k?ei

=

k) −

?

i

?[r(e∗

i),el],e∗

k?ei

=

el· r(e∗

k) − r(ad∗

·(el)e∗

k).

Since el∗ e∗

k= ad∗

·(el)e∗

k− R∗

◦(e∗

k)el, we get equation (6.6). Similarly, we can prove equation

(6.7). Equations (6.5) and (6.8) then follow immediately.

?

27

Page 28

Definition 6.2

Let (A,·) be a left-symmetric algebra. A bilinear form B : A ⊗ A → F is

called a 2-cocycle of A if

B(x · y,z) − B(x,y · z) = B(y · x,z) − B(y,x · z),∀x,y,z ∈ A.

(6.9)

In fact, the above definition is precisely the definition of a 2-cocycle of a left-symmetric

algebra into the trivial bimodule F ([SW]). It is equivalent to the following central extension:

there exists a left-symmetric algebra structure on A ⊕ Fc given by

x ∗ y = x · y + B(x,y)c, x ∗ c = c ∗ x = c ∗ c = 0, ∀x,y ∈ A,

(6.10)

if and only if B is a 2-cocycle of A. Furthermore, for any 2-cocycle B of a left-symmetric algebra

A, it is easy to know that ([Ku2])

ω(x,y) = B(x,y) − B(y,x), ∀x,y ∈ A,

(6.11)

is a 2-cocycle of the sub-adjacent Lie algebra G(A).

Theorem 6.3

Let A be a left-symmetric algebra and r ∈ A ⊗ A. Suppose that r is

symmetric and nondegenerate. Then r is a solution of S-equation in A if and only if the inverse

of the isomorphism A∗→ A induced by r, regarded as a bilinear form B on A, is a 2-cocycle of

A. That is, B(x,y) = ?r−1x,y? for any x,y ∈ A.

Proof

Let r =?

i?v∗,ai?bi=?

iai⊗bi. Since r is symmetric, we have?

i?v∗,bi?aifor any v∗∈ A∗. Since r is nondegenerate, for any x,y,z ∈ A,

there exist u∗,v∗,w∗∈ A∗such that x = r(u∗),y = r(v∗),z = r(w∗). Therefore

iai⊗bi=?

ibi⊗ai. Therefore

r(v∗) =?

B(x · y,z)=?r(u∗) · r(v∗),w∗? =

?

i,j

?u∗,bi??v∗,bj??w∗,ai· aj?

=?u∗⊗ v∗⊗ w∗,r13· r23?;

−B(x,y · z)= −?u∗,r(v∗) · r(w∗)? =

?

i,j

−?v∗,bi??w∗,bj??u∗,ai· aj?

=?u∗⊗ v∗⊗ w∗,−r12· r13?;

−B(y · x,z)= −?r(v∗) · r(u∗),w∗? = −

?

i,j

?v∗,bi??u∗,bj??w∗,ai· aj?

=?u∗⊗ v∗⊗ w∗,−r23· r13?;

B(y,x · z)=?v∗,r(u∗) · r(w∗)? =

?

i,j

?u∗,bi??w∗,bj??v∗,ai· aj?

=?u∗⊗ v∗⊗ w∗,r12· r23?.

28

Page 29

Therefore B is a (symmetric) 2-cocycle of A if and only if ?u∗⊗ v∗⊗ w∗,[[r,r]]? = 0 for any

u∗,v∗,w∗∈ A∗, if and only if [[r,r]]=0.

?

Corollary 6.4

Let (A,·) be a left-symmetric and r ∈ A⊗A be a nondegenerate symmetric

solution of S-equation in A. Suppose the left-symmetric algebra structure “◦” on A∗is induced

by r from equation (6.4). Then we have

a∗◦ b∗= r−1(r(a∗) · r(b∗)), ∀a∗,b∗∈ A∗.

(6.12)

Therefore r : A∗→ A is an isomorphism of left-symmetric algebras.

Proof

For any x,y ∈ A, set B(x,y) = ?r−1(x),y?. Then B is a 2-cocycle of A. For any

a∗,b∗∈ A∗and x ∈ A, from Proposition 6.2 and the above theorem, we know that

?a∗◦ b∗,x?=?x · r(b∗),a∗? − ?[r(a∗),x],b∗?

=B(x · r(b∗),r(a∗)) − B([r(a∗),x],r(b∗)) = B(x,r(a∗) · r(b∗))

=?r−1(r(a∗) · r(b∗)),x?.

Hence equation (6.12) holds and therefore r is an isomorphism of left-symmetric algebras.

?

Definition 6.3

A left-symmetric algebra A over the real field R is called Hessian if there

exists a symmetric and positive definite 2-cocycle B of A, that is, there exists a 2-cocycle B of

A which is an inner product.

In geometry, a Hessian manifold M is a flat affine manifold provided with a Hessian metric g,

that is, g is a Riemannian metric such that for any each point p ∈ M there exists a C∞-function

ϕ defined on a neighborhood of p such that gij =

∂2ϕ

∂xi∂xj. A Hessian left-symmetric algebra

corresponds to an affine Lie group G with a G-invariant Hessian metric ([Sh]).

Hence, we have the following conclusion.

Proposition 6.5

Let (A,B) be a Hessian left-symmetric algebra with the inner product

B. Then there exists a basis {e1,···,en} such that B(ei,ej) = δij. Suppose ei· ej=?

Then

kck

ijek.

ck

ij− ck

ji+ cj

ik− ci

jk= 0, ∀i,j,k.

(6.13)

Under this basis, the corresponding symmetric solution of S-equation in A is given by

r =

n

?

i

ei⊗ ei.

(6.14)

29

Page 30

Moreover, let {e∗

1,···,e∗

n} be the dual basis of {e1,···,en}. Then the left-symmetric algebra

and its sub-adjacent Lie algebra structures on the symplectic double SD(A) are given by

e∗

i∗ e∗

j= r(ei· ej) =

n

?

k

ck

ije∗

k; [e∗

i,e∗

j] = r([ei,ej]) =

n

?

k

(ck

ij− ck

ji)e∗

k;

ei∗e∗

j=

n

?

k

[ci

kjek+(cj

ki−cj

ik)e∗

k]; e∗

j∗ei=

n

?

k

[(ck

ji−ck

ij)ek+cj

kie∗

k]; [ei,e∗

j] =

n

?

k

(ci

jkek−cj

ike∗

k).

(6.15)

On the other hand, if equation (6.14) is a symmetric solution of S-equation in a left-symmetric

algebra A, then the structure constants {ck

ij} associated to the basis {e1,···,en} satisfy equation

(6.13).

Example 6.1

For an arbitrary nonassociative algebra (A,·), there is an “invariant” bilinear

form B defined as a trace form ([Sc]):

B(x · y,z) = B(x,y · z), or equivalently,B(Lx(y),z) = B(x,Ry(z)), ∀x,y,z ∈ A.

(6.16)

Obviously, for a left-symmetric algebra A, a trace form is a 2-cocycle of A. Hence a nonde-

generate symmetric trace form of a left-symmetric algebra A can give a symmetric solution of

S-equation in A. For example, F can be regarded as a 1-dimensional associative algebra with

a basis {e} satisfying e · e = e. Then r = e ⊗ e as a solution of S-equation in F corresponds to

the trace form B(e,e) = 1. Moreover, a left-symmetric algebra with a nondegenerate symmetric

trace form must be associative and hence it is a Frobenius algebra ([Bo2]).

Example 6.2

Let (A,·) be a left-symmetric algebra. There is another “invariant” bilinear

form B in the following sense ([BM4]):

B(x · y,z) = −B(y,x · z), or equivalently,B(Lx(y),z) = −B(y,Lx(z)), ∀x,y,z ∈ A.

(6.17)

The existence of such a nondegenerate bilinear form on A is equivalent to the fact that L is

isomorphic to L∗as representations of G(A). Obviously, such a nondegenerate symmetric bilinear

form on a left-symmetric algebra A is also a 2-cocycle of A. In fact, we have constructed some

examples of parak¨ ahler Lie algebras from these bilinear forms in [Bai2]. Here, we explain why

such a bilinear form appears in our study. As it was pointed in [BM4], such bilinear forms are

degenerate on most of left-symmetric algebras.

Next we turn to the general symmetric solutions of S-equation.

30

Page 31

Theorem 6.6

Let (A,·) be a left-symmetric algebra and r ∈ A ⊗ A be symmetric. Then

r is a solution of S-equation in A if and only if r satisfies

[r(a∗),r(b∗)] = r(L∗

·(r(a∗))b∗− L∗

·(r(b∗))a∗), ∀a∗,b∗∈ A∗.

(6.18)

Proof

Let {e1,···,en} be a basis of A and {e∗

1,···,e∗

n} be its dual basis. Suppose that

ei·ej=?

of S-equation in A if and only if (for any i,j,k)

kck

ijekand r =?

i,jaijei⊗ej, aij= aji. Hence r(e∗

i) =?

kaikek. Then r is a solution

?

t,l

{−ci

tlatjalk+ cj

tlaitalk+ (ck

tl− ck

lt)aitalj} = 0.

The left-hand side of the above equation is precisely the coefficient of ekin

[r(e∗

i),r(e∗

j)] − r(L∗

·(r(e∗

i))e∗

j− L∗

·(r(e∗

j))e∗

i).

Therefore the conclusion follows.

?

Definition 6.4

([Ku3]) Let G be a Lie algebra and ρ : G → gl(V ) be its representation.

A linear map T : V → G is called an O-operator associated to ρ if T satisfies

[T(u),T(v)] = T(ρ(T(u))v − ρ(T(v))u),∀u,v ∈ V.

(6.19)

Obviously, for a left-symmetric algebra A, the identity map id is an O-operator associated

to the regular representation of the sub-adjacent Lie algebra G(A). On the other hand,

Lemma 6.7

([Bai3])Let G be a Lie algebra and ρ : G → gl(V ) be its representation. Let

T : V → G be an O-operator associated to ρ. Then the product

u ◦ v = ρ(T(u))v, ∀u,v ∈ V

(6.20)

defines a left-symmetric algebra structure on V .Therefore V is a Lie algebra as the sub-

adjacent Lie algebra of this left-symmetric algebra and T is a homomorphism of Lie algebras.

Furthermore, T(V ) = {T(v)|v ∈ V } ⊂ G is a Lie subalgebra of G and there is an induced

left-symmetric algebra structure on T(V ) given by

T(u) · T(v) = T(u ◦ v) = T(ρ(T(u))v), ∀u,v ∈ V.

(6.21)

Moreover, its sub-adjacent Lie algebra structure is just the Lie subalgebra structure of G and T

is a homomorphism of left-symmetric algebras.

Therefore we have the following result.

31

Page 32

Corollary 6.8

Let (A,·) be a left-symmetric algebra. Let r ∈ A ⊗ A be a symmetric

solution of S-equation in A. Then r is an O-operator associated to L∗

·. Therefore there is a

left-symmetric algebra structure on A∗given by

a∗◦′b∗= L∗

·(r(a∗))b∗, ∀a∗,b∗∈ A∗.

(6.22)

It has the same sub-adjacent Lie algebra of the left-symmetric algebra in A∗given by equation

(6.4), which is induced by r in the sense of coboundary left-symmetric bialgebras. If r is

nondegenerate, then there is a new compatible left-symmetric algebra structure on G(A) given

by

x ·′y = r(L∗

·(x)r−1y), ∀x,y ∈ A,

(6.23)

which is just the left-symmetric algebra structure given by

B(x ·′y,z) = −B(y,x · z),∀x,y,z ∈ A,

(6.24)

where B is the symmetric nondegenerate 2-cocycle of A induced by r−1, that is, for any x,y ∈ A,

B(x,y) = ?r−1(x),y?.

Remark

With the above conditions and the left-symmetric algebra structure on A∗given

by equation (6.22), (A,A∗) is a left-symmetric bialgebra if and only if the following two conditions

hold: (for any x,y ∈ A and a∗,b∗∈ A∗)

L∗

·[x · r(a∗) − r(L∗

·(x)a∗)]b∗= L∗

·[x · r(b∗) − r(L∗

·(x)b∗)]a∗; (6.25)

[x · r(a∗) − r(L∗

·(x)a∗)] · y = [y · r(a∗) − r(L∗

·(y)a∗)] · x.

(6.26)

Theorem 6.9

Let G be a Lie algebra. Let ρ : G → gl(V ) be a representation of G and

ρ∗: G → gl(V∗) be its dual representation. Suppose that T : V → G is an O-operator associated

to ρ. Then

r = T + T21

(6.27)

is a symmetric solution of S-equation in T(V ) ⋉ρ∗,0V∗, where T(V ) ⊂ G is a left-symmetric

algebra given by equation (6.21) and (ρ∗,0) is its bimodule since its sub-adjacent Lie algebra

G(T(V )) is a Lie subalgebra of G, and T can be identified as an element in T(V ) ⊗ V∗⊂

(T(V ) ⋉ρ∗,0V∗) ⊗ (T(V ) ⋉ρ∗,0V∗).

Proof

Let {e1,···,en} be a basis of G. Let {v1,···,vm} be a basis of V and {v∗

n ?

k=1

1,···,v∗

m}

be its dual basis. Set T(vi) =

aikek,i = 1,···,m. Then, we have

T =

m

?

i=1

T(vi) ⊗ v∗

i=

m

?

i=1

n

?

k=1

aikek⊗ v∗

i∈ T(V ) ⊗ V∗⊂ (T(V ) ⋉ρ∗,0V∗) ⊗ (T(V ) ⋉ρ∗ V∗).

32

Page 33

Therefore we have

−r12· r13

=−

m

?

i,j=1

m

?

i,j=1

m

?

i,j=1

+v∗

{T(vi) · T(vj) ⊗ v∗

i⊗ v∗

j+ v∗

i⊗ v∗

j⊗ T(ρ(T(vj))vi)};

r12· r23

=

{v∗

i⊗ T(vi) · T(vj) ⊗ v∗

j− v∗

i⊗ v∗

j⊗ T(ρ(T(vi))vj)};

[r13,r23]= {−v∗

i⊗ T(ρ(T(vi))vj) ⊗ v∗

j+ T(ρ(T(vj))vi) ⊗ v∗

j⊗ v∗

i

i⊗ v∗

j⊗ [T(vi),T(vj)]}.

Since T is an O-operator associated to ρ and T(u) · T(v) = T(ρ(T(u))v)) for any u,v ∈ V , we

know that r is a symmetric solution of S-equation in T(V ) ⋉ρ∗,0V∗.

?

Remark

The above theorem also provides a method to construct the symmetric solutions

of S-equation (hence parak¨ ahler Lie algebras). We would like to point out that this method

starts from Lie algebras and only involves the representation theory of Lie algebras (to find the

O-operators), which avoids involving the nonassociativity of left-symmetric algebras.

Corollary 6.10

Let (A,·) be a left-symmetric algebra. Then

r =

n

?

i=1

(ei⊗ e∗

i+ e∗

i⊗ ei) (6.28)

is a symmetric solution of S-equation in A ⋉L∗,0A∗, where {e1,···,en} is a basis of A and

{e∗

1,···,e∗

A ⋉L∗,0A∗is given by

n} is its dual basis. Moreover, r is nondegenerate and the induced 2-cocycle B of

B(x + a∗,y + b∗) = ?r−1(x + a∗),y + b∗? = ?x,b∗? + ?a∗,y?, ∀x,y ∈ A,a∗,b∗∈ A∗.

(6.29)

Proof

Let V = A, ρ = L and T = id in Theorem 6.9. Then the conclusion follows

immediately.

?

Remark

Comparing with Theorem 5.5, we know that (the non-symmetric) T =

n ?

i=1

ei⊗e∗

i

induces a left-symmetric bialgebra structure on A ⋉ad∗,−R∗ A∗, whereas the above (symmetric)

r = T + T21induces a left-symmetric bialgebra structure on A ⋉L∗,0A∗.

7 Comparison between Lie bialgebras and left-symmetric bial-

gebras

In this section, we first recall some facts on Lie bialgebras from Chapters 1-2 in [CP] by Chari

and Pressley. In fact, a Lie bialgebra is the Lie algebra G of a Poisson-Lie group G equipped

33

Page 34

with additional structures induced from the Poisson structure on G and a Poisson-Lie group is

a Lie group with a Poisson structure compatible with the group operation in a certain sense.

Poisson-Lie groups play an important role in symplectic geometry and quantum group theory

([D],[KM],[LW]).

Definition 7.1

Let G be a Lie algebra. A Lie bialgebra structure on G is a skew-symmetric

linear map δG: G → G⊗G such that δ∗

G: G∗⊗G∗→ G∗is a Lie bracket on G∗and δ is a 1-cocycle

of G associated to ad ⊗ 1 + 1 ⊗ ad with values in G ⊗ G. We denote it by (G,G∗) or (G,δG).

Let G be a Lie algebra. A bilinear form B( , ) on G is called invariant if

B([x,y],z) = B(x,[y,z]), ∀x,y,z ∈ G.

(7.1)

Definition 7.2

A Manin triple is a triple of Lie algebras (P,P+,P−) together with a

non-degenerate symmetric invariant bilinear form B( , ) on P such that

(1) P+and P−are Lie subalgebras of P;

(2) P = P+⊕ P−as vector spaces;

(3) B(P+,P+) = B(P−,P−) = 0.

Two Manin triple (P1,P1,+,P1,−) and (P2,P2,+,P2,−) with the bilinear forms B1( , ) and B( , )

respectively are isomorphic if there exists an isomorphism of Lie algebras ϕ : P1→ P2such that

ϕ(P1,+) = P2,+, ϕ(P1,−) = P2,−, B1(x,y) = B2(ϕ(x),ϕ(y)), ∀x,y ∈ P1.

(7.2)

In particular, if there is a Lie algebra structure G ⊲⊳ G∗on G ⊕ G∗such that G and G∗are

Lie subalgebras and the natural symmetric bilinear form (also see equation (6.29))

(x + a∗,y + b∗) = ?a∗,y? + ?x,b∗?, ∀x,y ∈ G,a∗,b∗∈ G∗,

(7.3)

is invariant, then (G ⊲⊳ G∗,G,G∗) is a (standard) Manin triple. It is known that every Manin

triple is isomorphic to such a standard Manin triple.

Theorem 7.1

Let (G,[ , ]G) be a Lie algebra and (G∗,[ , ]G∗) be a Lie algebra structure

on its dual space G∗. Then the following conditions are equivalent:

(1) (G ⊲⊳ G∗,G,G∗) is a standard Manin triple with the bilinear form (7.3);

(2) (G,G∗,ad∗

G,ad∗

G∗) is a matched pair of Lie algebras;

(3) (G,G∗) is a Lie bialgebra.

Proposition 7.2

Let (G,G∗) be a Lie bialgebra. Then there is a canonical Lie bialgebra

structure on G ⊕ G∗such that the inclusions i1: G → G ⊕ G∗and i2: G∗→ G ⊕ G∗into the two

34

Page 35

summands are homomorphisms of Lie bialgebras. Such a structure is called a classical (Drinfeld)

double of G.

Definition 7.3

A Lie bialgebra (G,δ) is called coboundary if there exists a r ∈ G ⊗G such

that

δ(x) = [x ⊗ 1 + 1 ⊗ x,r], ∀x ∈ G.

(7.4)

Theorem 7.3

Let G be a Lie algebra and r ∈ G ⊗ G. Then the map δ : G → G ⊗ G

defined by equation (7.4) induces a Lie bialgebra structure on G if and only if the following two

conditions are satisfied:

(a) [x ⊗ 1 + 1 ⊗ x, r12+ r21] = 0 for any x ∈ G;

(b) [x ⊗ 1 ⊗ 1 + 1 ⊗ x ⊗ 1 + 1 ⊗ 1 ⊗ x, [r12,r13] + [r12,r23] + [r13,r23]] = 0 for any x ∈ G,

where the notations r12,r21,r13,r23are given as the notations after Proposition 5.3 for a Lie

algebra.

Corollary 7.4

Let G be a Lie algebra and r ∈ G⊗G. If r is skew-symmetric and r satisfies

[r12,r13] + [r12,r23] + [r13,r23] = 0,

(7.5)

then the map δ : G → G ⊗ G defined by equation (7.4) induces a Lie bialgebra structure on G.

Definition 7.4

Let G be a Lie algebra and r ∈ G ⊗ G. Equation (7.5) is called classical

Yang-Baxter equation (CYBE) in G.

The facts above are from Chapters 1-2 in [CP]. Next we recall some results from the literature.

Proposition 7.5

Let G be a Lie algebra and r ∈ G ⊗ G.

(1) [D] Suppose r is skew-symmetric and nondegenerate. Then r is a solution of CYBE in

G if and only if the isomorphism G∗→ G induced by r, regarded as a bilinear form on G is a

2-cocycle of G. Therefore under this situation, G is a symplectic Lie algebra.

(2) ([Ku3])Suppose r is skew-symmetric. Then r is a solution of CYBE in G if and only if

r is an O-operator associated to ad∗, that is, r satisfies

[r(a∗),r(b∗)] = r(ad∗(r(a∗))b∗− ad∗(r(b∗))a∗), ∀a∗,b∗∈ G∗.

(7.6)

Remark

Let G be a Lie algebra equipped with a nondegenerate symmetric invariant bi-

linear form. Let r ∈ G ⊗ G. Then r can be identified as a linear map from G to G. If r is

skew-symmetric, then r is a solution of CYBE in G if and only if r satisfies equation (2.7), that

is, the operator form of CYBE.

35

Page 36

Proposition 7.6 ([Bai3]) Let G be a Lie algebra and ρ : G → gl(V ) be a representation

of G. Let ρ∗: G → gl(V∗) be the dual representation of ρ and T : V → G be a linear map. Then

r = T − T21

(7.7)

is a skew-symmetric solution of CYBE in G ⋉ρ∗ V∗if and only if T is an O-operator associated

to ρ.

Therefore, roughly speaking, a symmetric solution of the S-equation corresponds to the

symmetric part of an O-operator, whereas a skew-symmetric solution of the classical Yang-

Baxter equation corresponds to the skew-symmetric part of an O-operator.

Proposition 7.7 ([Bai3])Let (A,·) be a left-symmetric algebra. Then

r =

n

?

i=1

(ei⊗ e∗

i− e∗

i⊗ ei) (7.7)

is a skew-symmetric solution of CYBE in G(A) ⋉L∗ G(A)∗, where {e1,···,en} is a basis of A

and {e∗

1,···,e∗

n} is its dual basis. Moreover, r is nondegenerate and the induced 2-cocycle B of

G(A) ⋉L∗ G(A)∗given by

B(x + a∗,y + b∗) = ?r−1(x + a∗),y + b∗?, ∀x,y ∈ A,a∗,b∗∈ A∗,

(7.8)

is precisely equation (2.16).

The facts above and the results in the previous sections allow us to compare left-symmetric

bialgebras and Lie bialgebras in terms of the following properties: structures on the correspond-

ing Lie groups, 1-cocycles of Lie algebras, matched pairs of Lie algebras, Lie algebra structures

on the direct sum of the Lie algebras in the matched pairs, bilinear forms on the direct sum of

the Lie algebras in the matched pairs, double structures on the direct sum of the Lie algebras

in the matched pairs, algebraic equations associated to coboundary cases, nondegenerate solu-

tions and related geometric interpretation, O-operators and constructions from left-symmetric

algebras. We list them in Table 1. From this table, we observe that there is a clear analogy

between them and in particular, parak¨ ahler Lie groups correspond to Poisson-Lie groups whose

Lie algebras are Lie bialgebras in this sense.

Moreover, since classical Yang-Baxter equations can be regarded as “classical limits” of quan-

tum Yang-Baxter equations ([Be]), we believe that there should exist an analogue (“quantum

S-equations” ) of the quantum Yang-Baxter equations. We use a question mark in Table 1 to

denote these still-to-be-found “quantum S-equations”.

36

Page 37

Table 1: Comparison between Lie bialgebras and left-symmetric bialgebras

Algebras Lie bialgebras

Corresponding Lie groupsPoisson-Lie groups

1-cocycles of Lie algebras1 ⊗ ad + ad ⊗ 1

Matched pairs of Lie algebras(G,G∗,ad∗

Lie algebra structures onManin triples

the direct sum of the Lie

algebras in the matched pairs

Bilinear forms onsymmetric

the direct sum of the Lie?x + a∗,y + b∗?

algebras in the matched pairs= ?x,b∗? + ?a∗,y?

invariant

Double structures onDrinfeld Doubles

the direct sum of the Lie

algebras in the matched pairs

Algebraic equations associatedskew-symmetric solutions

to coboundary casesCYBEs in Lie algebras

Left-symmetric bialgebras

parak¨ ahler Lie groups

L ⊗ 1 + 1 ⊗ ad

(G(A),G(A∗),L∗

phase spaces

G,ad∗

G∗)

A,L∗

A∗)

skew-symmetric

?x + a∗,y + b∗?

= −?x,b∗? + ?a∗,y?

2-cocycles

symplectic doubles

symmetric solutions

S-equations in left-symmetric

algebras

2-cocycles of left-symmetric

algebras

Hessian structures

associated to L∗

symmetric parts

n ?

i=1

induced bilinear forms

?x + a∗,y + b∗?

= ?x,b∗? + ?a∗,y?

?

Nondegenerate solutions2-cocycles of Lie algebras

symplectic structures

associated to ad∗

skew-symmetric parts

n ?

i=1

induced bilinear forms

?x + a∗,y + b∗?

= −?x,b∗? + ?a∗,y?

Quantum Yang-Baxter

equations

O-operators

Constructions from

r =

(ei⊗ e∗

i− e∗

i⊗ ei)

r =

(ei⊗ e∗

i+ e∗

i⊗ ei)

left-symmetric algebras

Quantum equations

At the end of this section, we consider the case that a left-symmetric bialgebra is also a Lie

bialgebra.

Theorem 7.7

Let (A,A∗,α,β) be a left-symmetric bialgebra. Then (G(A),G(A∗)) is a Lie

bialgebra if and only if

?R∗

·(x)a∗,R∗

◦(b∗)y? + ?R∗

·(x)b∗,R∗

◦(a∗)y? = ?R∗

·(y)b∗,R∗

◦(a∗)x? + ?R∗

·(y)a∗,R∗

◦(b∗)x?,

(7.9)

for any x,y ∈ A∗,a∗,b∗∈ A∗.

Proof

Denote the left-symmetric product on A by “·′′and the left-symmetric product on

A∗by “◦′′. Then (G(A),G(A∗)) is a Lie bialgebra if and only if for any x,y ∈ A, a∗,b∗∈ A∗,

(∗) ad∗

·(x)[a∗,b∗] − [ad∗

·(x)a∗,b∗] − [a∗,ad∗

·(x)b∗] + ad∗

·(ad∗

◦(a∗)x)b∗− ad∗

·(ad∗

◦(b∗)x)a∗= 0;

37

Page 38

(∗∗) ad∗

◦(a∗)[x,y] − [ad∗

◦(a∗)x,y] − [x,ad∗

◦(a∗)y] + ad∗

◦(ad∗

·(x)a∗)y − ad∗

◦(ad∗

·(y)a∗)x = 0.

Since ad∗

·= L∗

·− R∗

·and (G(A),G(A∗),L∗

·,L∗

◦) is a matched pair of Lie algebras, the equation

(∗) is reduced to

−R∗

·(x)[a∗,b∗] + [R∗

·(x)a∗,b∗] + [a∗,R∗

·(x)b∗] − R∗

·(ad∗

◦(a∗)x)b∗− L∗

·(R∗

◦(a∗)x)b∗

+R∗

·(ad∗

◦(b∗)x)a∗+ L∗

·(R∗

◦(b∗)x)a∗= 0.

Let the equation above act on y ∈ A and note that ?R∗

·(x)a∗,y? = ?L∗

·(y)a∗,x? for any x,y ∈ A

and a∗∈ A∗, we obtain

−L∗

·(y)[a∗,b∗] + L∗

·(ad∗(b∗)y)a∗− L∗

·(ad∗(a∗)y)b∗+ [a,L∗

·(y)b∗] + R∗

·(y)b∗◦ a∗

−[b∗,L∗

·(y)a∗] − R∗

·(y)a∗◦ b∗= 0.

Using the condition that (G(A),G(A∗),L∗

·,L∗

◦) is a matched pair of Lie algebras again, we know

that

−L∗

·(R∗

◦(b∗)y)a∗+ L∗

·(R∗

◦(a∗)y)b∗+ R∗

·(y)b∗◦ a∗− R∗

·(y)a∗◦ b∗= 0,

which gives equation (7.9) by acting on x ∈ A. Similarly, from equation (∗∗), we can get

−L∗

◦(R∗

·(y)b∗)x + L∗

◦(R∗

·(x)b∗)y + R∗

◦(b∗)y · x − R∗

◦(b∗)x · y = 0,

for any x,y ∈ A and b∗∈ A∗, which gives the same equation (7.9) by acting on a∗∈ A∗.

?

Corollary 7.8

Let (A,·) be a left-symmetric and r ∈ A ⊗ A be a symmetric solution of

S-equation in A. Suppose the left-symmetric algebra structure on A∗is induced by r from

equation (6.4). Then there exists a Lie bialgebra structure (G(A),G(A∗)) if and only if

−r(L∗

·(x)R∗

·(y)a∗) + [x,r(R∗

·(y)a∗] − (y · r(a∗) − r(ad∗

·(y)a∗)) · x

= −r(L∗

·(y)R∗

·(x)a∗) + [y,r(R∗

·(x)a∗] − (x · r(a∗) − r(ad∗

·(x)a∗)) · y,

(7.10)

for any x,y ∈ A and a∗∈ A∗.

Corollary 7.9

Let (A,A∗,α,β) be a left-symmetric bialgebra. If equation (7.9) is satisfied,

then there are two Lie algebra structures G(A) ⊲⊳L∗

·

◦G(A∗) and G(A) ⊲⊳ad∗

L∗

·

◦G(A∗) on the direct

ad∗

sum A ⊕ A∗of the underlying vector spaces of A and A∗such that both G(A) and G(A∗) are

Lie subalgebras and the bilinear form given by equation (2.16) is a 2-cocycle of G(A) ⊲⊳L∗

·

◦G(A∗)

L∗

and the bilinear form (7.3) is invariant on G(A) ⊲⊳ad∗

·

◦G(A∗).

ad∗

38

Page 39

Remark

Obviously the two Lie algebras above are not isomorphic in general. On the other

hand, if a Lie bialgebra (G,G∗) whose Lie algebra structure on G∗is induced by a non-degenerate

classical r-matrix ([D], [Se]), then both G and G∗are symplectic Lie algebras. Therefore there

is a compatible left-symmetric algebra structure on G and G∗respectively.

Example 7.1

Let (G,ω) be a symplectic Lie algebra. Then there is a Lie bialgebra whose

Lie algebra structure in G∗is given by a non-degenerate classical r-matrix as follows (cf. [DiM]).

δ(x) = [x ⊗ 1 + 1 ⊗ x,r], ∀x ∈ G,

(7.11)

where r : G∗→ G is given by ω(x,y) = ?r−1(x),y?. On the other hand, there exists a left-

symmetric algebra structure “·” on G given by equation (2.14), that is, ω(x·y,z) = −ω(y,[x,z])

for any x,y,z ∈ G. Moreover, there exists a compatible left-symmetric algebra structure on the

Lie algebra G∗given by

a∗◦ b∗= r−1(r(a∗) · r(b∗)),∀a∗,b∗∈ G∗.

(7.12)

Furthermore, it is easy to know that

L∗

·(x)a∗= r−1[x,r(a∗)], R∗

·(x)a∗= −r−1(r(a∗)·x)), L∗

◦(a∗)x = [r(a∗),x], R∗

◦(a∗)x = −x·r(a∗),

(7.13)

for any x ∈ A,a∗∈ A∗. Therefore according to Theorem 2.5, (G,G∗) (as left-symmetric algebras)

is a left-symmetric bialgebra if and only if G is 2-step nilpotent, that is, [[x,y],z] = 0 for any

x,y,z ∈ G. In this case, we can know that it is equivalent to [x,y] · z = 0 for any x,y,z ∈ G.

Therefore, equation (7.9) holds naturally.

Acknowledgments

The author thanks Professors P. Etingof, I.M. Gel’fand, B.A. Kupershmidt, and C. Woodward

for important suggestion and great encouragement. He is very grateful of referees’ important

suggestion. He also thanks Professors J. Lepowsky, Y.-Z. Huang and H.S. Li for the hospitality

extended to him during his stay at Rutgers, The State University of New Jersey and for valuable

discussions. This work was supported in part by S.S. Chern Foundation for Mathematical

Research, the National Natural Science Foundation of China (10571091, 10621101), NKBRPC

(2006CB805905), Program for New Century Excellent Talents in University and K.C. Wong

Education Foundation.

39

Page 40

References

[AD] A. Andrada, I.G. Dotti, Double products and hypersymplectic structures on R4n, arXiv:

math.DG/0401294.

[AS] A. Andrada, S. Salamon, Complex product structure on Lie algebras, Forum Math. 17

(2005) 261-295.

[Bai1] C.M. Bai, Left-symmetric algebras from linear functions, J. Algebra 281 (2004)651-665.

[Bai2] C.M. Bai, A further study on non-abelian phase spaces: Left-symmetric algebraic ap-

proach and related geometry, Rev. Math. Phys. 18 (2006), 545-564.

[Bai3] C.M. Bai, A unified algebraic approach to the classical Yang-Baxter equation, to appear

in J. Phys. A: Math. Theo. (2007).

[Bai4] C.M. Bai, Parak¨ ahler structures on Lie groups, preprint.

[BKL] C.M. Bai, L. Kong, H.S. Li, Novikov algebras and vertex algebras, preprint.

[BM1] C.M. Bai, D.J. Meng, The classification of Novikov algebras in low dimensions, J. Phys.

A: Math. Gen. 34 (2001) 1581-1594.

[BM2] C.M. Bai, D.J. Meng, On the realization of transitive Novikov algebras, J. Phys. A:

Math. Gen. 34 (2001) 3363-3372.

[BM3] C.M. Bai, D.J. Meng, The realizations of non-transitive Novikov algebras, J. Phys. A:

Math. Gen. 34 (2001) 6435-6442.

[BM4] C.M. Bai, D.J. Meng, Bilinear forms on Novikov algebras, Int. J. Theor. Phys. 41 (2002)

495-502.

[BM5] C.M. Bai, D.J. Meng, A Lie algebraic approach to Novikov algebras, J. Geom. Phys. 45

(2003) 218-230.

[BK] B. Bakalov, V. Kac, Field algebras, Int. Math. Res. Not. (2003) 123-159.

[BN] A.A. Balinskii, S.P. Novikov, Poisson brackets of hydrodynamic type, Frobenius algebras

and Lie algebras, Soviet Math. Dokl. 32 (1985) 228-231.

[Bar] M.L. Barbeis, Hypercomplex structures on four-dimensional Lie groups, Proc. Amer.

Math. Soc. 125 (1997) 1043-1054.

[Bax] G. Baxter, An analytic problem whose solution follows from a simple algebraic identity,

Pacific J. Math. 10 (1960) 731-742.

[Be] A.A. Belavin, Dynamical symmetry of integrable quantum systems, Nucl. Phys. B 180

(1981) 189-200.

40

Page 41

[BD] A.A. Belavin, V.G. Drinfeld, Solutions of classical Yang-Baxter equation for simple Lie

algebras, Funct. Anal. Appl. 16 (1982) 159-180.

[Bo1] M. Bordemann, Generalized Lax pairs, the modified classical Yang-Baxter equation, and

affine geometry of Lie groups, Comm. Math. Phys. 135 (1990) 201-216.

[Bo2] M. Bordemann, Nondegenerate invariant bilinear forms on nonassociative algebras, Acta

Math. Univ. Comen. LXVI (1997) 151-201.

[Bu1] D. Burde, Simple left-symmetric algebras with solvable Lie algebra, Manuscripta Math.

95 (1998) 397-411.

[Bu2] D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Cent.

Eur. J. Math. 4 (2006) 323-357.

[Ca] A. Cayley, On the theory of analytic forms called trees, Collected Mathematical Papers of

Arthur Cayley, Cambridge Univ. Press, Vol. 3 (1890) 242-246.

[CL] F. Chapoton, M. Livernet, Pre-Lie algebras and the rooted trees operad, Int. Math. Res.

Not. (2001) 395-408.

[CP] V. Chari, A. Pressley, A guide to quantum groups, Cambridge University Press, Cambridge

(1994).

[Ch] B.Y. Chu, Symplectic homogeneous spaces, Trans. Amer. Math. Soc. 197 (1974) 145-159.

[CK] A. Connes, D. Kreimer, Hopf algebras, renormalization and noncommutative geometry,

Comm. Math. Phys. 199 (1998) 203-242.

[DaM1] J.M. Dardie, A. Medina, Double extension symplectique d’un groupe de Lie symplec-

tique, Adv. Math. 117 (1996) 208-227.

[DaM2] J.M. Dardie, A. Medina, Alg` ebres de Lie k¨ ahl´ eriennes et double extension, J. Algebra

185 (1996) 774-795.

[DiM] A. Diatta, A. Medina, Classical Yang-Baxter equation and left-invariant affine geometry

on Lie groups, arXiv:math.DG/0203198.

[D] V. Drinfeld, Hamiltonian structure on the Lie groups, Lie bialgebras and the geometric

sense of the classical Yang-Baxter equations, Soviet Math. Dokl. 27 (1983) 68-71.

[DL] A. Dzhumadil’daev, C. Lofwall, Trees, free right-symmetric algebras, free Novikov algebras

and identities, Homology, Homotopy and applications 4 (2002) 165-190.

[E] K. Ebrahimi-Fard, Loday-type algebras and the Rota-Baxter relation, Lett. Math. Phys.

61 (2002) 139-147.

41

Page 42

[EGK] K. Ebrahimi-Fard, L. Guo, D. Kreimer, Integrable renormalization 1: The ladder case,

J. Math. Phys, 45 (2004) 3758-3769.

[ES] P. Etingof, A. Soloviev, Quantization of geometric classical r-matrix, Math. Res. Lett. 6

(1999) 223-228.

[FHL] I. Frenkel, Y.Z. Huang, J. Lepowsky, On axiomatic approaches to vertex operator algebras

and modules, Memoris. Amer. Math. Soc. 104 (1993).

[FLM] I. Frenkel, J. Lepowsky, A. Meurman, Vertex operator algebras and the Monster, Pure

and Appl. Math. 134, Academic Press, Boston (1988).

[F] V.T. Filipov, A class of simple nonassociative algebras, Mat. Zametki 45 (1989) 101-105.

[GD] I.M. Gel’fand, I. Ya. Dorfman, Hamiltonian operators and algebraic structures related to

them, Funct. Anal. Appl. 13 (1979) 248-262.

[G] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. Math. 78 (1963)

267-288.

[GS] I.Z. Golubschik, V.V. Sokolov, Generalized operator Yang-Baxter equations, integrable

ODEs and nonassociative algebras, J. Nonlinear Math. Phys., 7 (2000) 184-197.

[H] J.I. Hano, On kaehlerian homogeneous spaces of unimodular Lie groups, Amer. J. Math.

79 (1957) 885-900.

[JL] J. Lepowsky, H.S. Li, Introduction to vertex operator algebras and their representations,

Progress in Mathematics 227, Birk¨ auser, Boston (2004).

[Ka] S. Kaneyuki, Homogeneous symplectic manifolds and dipolarizations in Lie algebras,

Tokyo J. Math. 15 (1992) 313-325.

[KGM] Y. Khakimdjanov, M. Goze and A. Medina, Symplectic or contact structures on Lie

groups, arXiv:math.DG/0205290.

[Ki] H. Kim, Complete left-invariant affine structures on nilpotent Lie groups, J. Diff. Geom.

24 (1986) 373-394.

[KM] V. Kosmann-Schwarzbach, F. Magri, Poisson-Lie groups and complete integrability, I.

Drinfeld algebras, dual extensions and their canonical representations, Ann. Inst. Henri

Poincar´ e Phys. Th´ eor 49 (1988) 35-81.

[Ko] J.-L. Koszul, Domaines born´ es homog` enes et orbites de groupes de transformations affines,

Bull. Soc. Math. France 89 (1961) 515-533.

[Ku1] B.A. Kupershmidt, Non-abelian phase spaces, J. Phys. A: Math. Gen. 27 (1994) 2801-

2810.

42

Page 43

[Ku2] B.A. Kupershmidt, On the nature of the Virasoro algebra, J. Nonlinear Math. Phy. 6

(1999) 222-245.

[Ku3] B.A. Kupershmidt, What a classical r-matrix really is, J. Nonlinear Math. Phy. 6 (1999)

448-488.

[Li] H.S. Li, Vertex algebras and vertex Poisson algebras, Comm. Contemp. Math. 6 (2004)

61-110.

[Lib] P. Libermann, Sur le probl` eme d’´ equivalence de certaines structures infinit´ esimals, Ann.

Mat. Pura Appl. 36 (1954) 27-120.

[LM] A. Lichnerowicz, A. Medina, On Lie groups with left invariant symplectic or kahlerian

structures, Lett. Math. Phys. 16 (1988) 225-235.

[LW] J.H. Lu, A. Weinstein, Poisson-Lie groups, dressing transformations and Bruhat decom-

positions, J. Diff. Geom 31 (1990) 501-526.

[Maj] S. Majid, Matched pairs of Lie groups associated to solutions of the Yang-Baxter equa-

tions, Pacific J. Math. 141 (1990) 311-332.

[Mat] Y. Matsushima, Affine structures on complex mainfolds, Osaka J. Math. 5 (1968) 215-222.

[MS] D. Mcduff, D. Salamon, Introduction to symplectic topology, Clarendon Press, Oxford,

(1998).

[Me] A. Medina, Flat left-invariant connections adapted to the automorphism structure of a

Lie group, J. Diff. Geom. 16 (1981) 445-474.

[R] G.-C. Rota, Baxter operators, an introduction, In: “Gian-Carlo Rota on Combinatorics,

Introductory papers and commentaries”, Joseph P.S. Kung, Editor, Birkh¨ auser, Boston,

1995.

[SW] A.A. Sagle, R.E. Walde, Introduction to Lie groups and Lie algebras, Academic Press,

New York (1973).

[Sc] R. Schafer, An introduction to nonassociative algebras, Dover Publications Inc., New York

(1995).

[Se]M.A. Semonov-Tian-Shansky, What is a classical R-matrix? Funct. Anal. Appl. 17 (1983)

259-272.

[Sh] H. Shima, Homogeneous Hessian manifolds, Ann. Inst. Fourier 30 (1980) 91-128.

[SS] S.I. Svinolupov, V.V. Sokolov, Vector-matrix generalizations of classical integrable equa-

tions, Theoret. and Math. Phys. 100 (1994) 959-962.

43

Page 44

[T]M. Takeuchi, Matched pairs of groups and bimash products of Hopf algebras, Comm.

Algebra 9 (1981) 841-842.

[V] E.B. Vinberg, Convex homogeneous cones, Transl. of Moscow Math. Soc. No. 12 (1963)

340-403.

[W] A. Winterhalder, Linear Nijenhuis-tensors and the construction of integrable systems,

arXiv: physics/9709008.

[X] X. Xu, On simple Novikov algebras and their irreducible modules, J. Algebra 185 (1996)

905-934.

44