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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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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
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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)
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(∗∗) 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∗
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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.
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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