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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

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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

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(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.

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(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

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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.

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