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Pseudo-Hessian Lie algebras and L-dendriform bialgebras
Journal of Algebra 400 (2014) 273–289
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Pseudo-Hessian Lie algebras and L-dendriform bialgebras Xiang Ni a , Chengming Bai b,∗ a b
Department of Mathematics, Caltech, Pasadena, CA 91125, USA Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, PR China
a r t i c l e
i n f o
Article history: Received 15 August 2010 Available online 24 December 2013 Communicated by Efim Zelmanov MSC: 17A30 17B05 17B60 Keywords: Lie algebra Pre-Lie algebra L-dendriform algebra Hessian structure
a b s t r a c t In this paper, we study a special class of pseudo-Hessian Lie algebras satisfying an additional condition that they are decomposed into a direct sum of underlying vector spaces of two Lagrangian subalgebras in terms of L-dendriform algebras which are the underlying algebraic structures. Such structures are equivalent to certain bialgebra structures, namely, Ldendriform bialgebras. Furthermore, we introduce and study the so-called triangular pseudo-Hessian Lie algebras and relate them to some semidirect product constructions from the “Lietype” operations of L-dendriform algebras. © 2013 Elsevier Inc. All rights reserved.
1. Introduction At first, we assume that the base field is taken to be R of real numbers. Definition 1.1. (See [6,7,11].) (a) A pre-Lie algebra is a vector space A equipped with a bilinear operation (x, y) → x·y satisfies * Corresponding author. E-mail addresses: [email protected] (X. Ni), [email protected] (C. Bai). 0021-8693/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jalgebra.2013.12.003
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(x · y) · z − x · (y · z) = (y · x) · z − y · (x · z),
∀x, y, z ∈ A.
(1)
(b) Suppose that (A, ·) is a pre-Lie algebra. Then the commutator [x, y] := x · y − y · x,
∀x, y ∈ A,
(2)
defines a Lie algebra g(A), which is called the sub-adjacent Lie algebra of A and A is called a compatible pre-Lie algebra structure on the Lie algebra g(A). Pre-Lie algebras are the algebraic structures corresponding to the (left-invariant) affine structures on Lie groups [7,11]. See a survey article [4] for the study of pre-Lie algebras. Definition 1.2. (See [8,10].) (a) Let (A, ·) be a pre-Lie algebra. A bilinear form B : A ⊗ A → R 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.
(3)
(b) A pre-Lie algebra (A, ·) is called Hessian (resp. pseudo-Hessian) if there is a symmetric and positive definite (resp. nondegenerate) 2-cocycle B. It is denoted by (A, ·, B). (c) A Lie algebra g is called Hessian (resp. pseudo-Hessian) if there is a compatible pre-Lie algebra structure · : g ⊗ g → g and a bilinear form B : g ⊗ g → R such that (g, ·, B) becomes a Hessian pre-Lie algebra (resp. pseudo-Hessian pre-Lie algebra). It is denoted by (g, B). Let (g, B) be a real Hessian Lie algebra. Then B induces a left-invariant Hessian metric on the corresponding connected real Lie group G, thus making it be a Hessian manifold. Recall that a Hessian manifold M is a flat affine manifold provided with a Hessian metric. Note that a Hessian metric on a smooth manifold M is a Riemannian metric g such that for each point p ∈ M there exists a C ∞ -function ϕ defined on a 2 ϕ neighborhood of p such that gij = ∂x∂i ∂x j [10]. Obviously, from an algebraic point of view, the base field can be extended to a field of characteristic different from 2, which is the field from which we take all the constants and over which we take all the algebras, vector spaces and linear maps, etc. in this paper. Furthermore, we study a special class of pseudo-Hessian Lie algebras satisfying an additional condition that they are decomposed into a direct sum of the underlying vector spaces of two Lagrangian subalgebras, which lead to the notion of double construction of pseudo-Hessian Lie algebra. On the other hand, a notion of L-dendriform algebra was introduced in [3] as the underlying algebraic structure of a pseudo-Hessian pre-Lie algebra or a pseudo-Hessian Lie algebra. This paper deals with further the connection between pseudo-Hessian Lie algebras and L-dendriform algebras. To be more precise, we interpret
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a double construction of pseudo-Hessian Lie algebra in terms of some bialgebra structure of an L-dendriform algebra, namely, an L-dendriform bialgebra. Note that another class of pseudo-Hessian Lie algebras related to the left-invariant flat pseudo-metrics on Lie groups [9] were studied in terms of L-dendriform algebras in [2]. Furthermore, we introduce and study so-called triangular pseudo-Hessian Lie algebras and relate them to some semidirect product constructions which are related to the “Lie-type” operations of L-dendriform algebras. The paper is organized as follows. In Section 2 we recall some necessary definitions and basic results on L-dendriform algebras given in [3]. We introduce a notion of double construction of pseudo-Hessian Lie algebra in Section 3 and interpret it in terms of some bialgebra structure of an L-dendriform algebra. In Section 4 we study a special subclass of pseudo-Hessian Lie algebras, which are called triangular pseudo-Hessian Lie algebras arising from the skew-symmetric solutions of the LD-equation introduced in [3]. We note that the “Lie-type” operations of L-dendriform algebras could be used to form some semidirect constructions in a natural way and we show that all the triangular pseudo-Hessian Lie algebras are isomorphic to them. Throughout this paper all algebras (may have more than one operation) and vector spaces are assumed to be finite-dimensional, although many results still hold in infinitedimensional case. 2. Preliminaries and basic results Definition 2.1. (See [3].) An L-dendriform algebra (A, ≺, ) is a vector space A equipped with two bilinear operations ≺, : A ⊗ A → A such that for any x, y, z ∈ A, (x ≺ y) ≺ z + y (x ≺ z) = x ≺ (y · z) + (y x) ≺ z,
(4)
(x · y) z + y (x z) = x (y z) + (y · x) z,
(5)
where x · y = x ≺ y + x y. Definition 2.2. (See [3].) Let (A, ≺, ) be an L-dendriform algebra. (a) Define x · y := x ≺ y + x y,
∀x, y ∈ A.
(6)
Then (A, ·) is a pre-Lie algebra and we denote it by (Ah , ·). It is called the associated horizontal pre-Lie algebra of (A, ≺, ) and (A, ≺, ) is called a compatible L-dendriform algebra structure on the pre-Lie algebra (Ah , ·). (b) Define x ◦ y := x y − y ≺ x,
∀x, y ∈ A.
(7)
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Then (A, ◦) is a pre-Lie algebra and we denote it by (Av , ◦). It is called the associated vertical pre-Lie algebra of (A, ≺, ) and (A, ≺, ) is called a compatible L-dendriform algebra structure on the pre-Lie algebra (Av , ◦). (c) The associated horizontal and vertical pre-Lie algebras (Ah , ·) and (Av , ◦) have the same sub-adjacent Lie algebra g(A) = g(Ah ) = g(Av ) given by [x, y] := x y + x ≺ y − y x − y ≺ x,
∀x, y ∈ A.
(8)
It is called the sub-adjacent Lie algebra of the L-dendriform algebra (A, ≺, ). Definition 2.3. (See [1].) Let (A, ·) be a pre-Lie algebra and V be a vector space. Let l, r : A → gl(V ) be two linear maps. V (or (V, l, r)) is called a bimodule of A if l(x)l(y) − l(x · y) = l(y)l(x) − l(y · x), l(x)r(y) − r(y)l(x) = r(x · y) − r(y)r(x),
∀x, y ∈ A.
(9)
In fact, (V, l, r) is a bimodule of a pre-Lie algebra (A, ·) if and only if the direct sum A ⊕ V of the underlying vector spaces of A and V is turned into a pre-Lie algebra (the semidirect product) by defining multiplication in A ⊕ V by (it is denoted by A l,r V ) (x1 + v1 ) ∗ (x2 + v2 ) := x1 · x2 + l(x1 )v2 + r(x2 )v1 ,
∀x1 , x2 ∈ A, v1 , v2 ∈ V. (10)
Definition 2.4. Let (A, ) be a vector space with a bilinear operation : A ⊗ A → A. We denote the left action and right action by L : A → gl(A) and R : A → gl(A) respectively, i.e., L (x)y = R (y)x = x y for any x, y ∈ A. We also define linear maps L∗ , R∗ : A → gl(A∗ ) by
L∗ (x)v ∗ , u = − v ∗ , L (x)u , ∗ R (x)v ∗ , u = − v ∗ , R (x)u , ∀x, u ∈ A, v ∗ ∈ A∗ .
(11)
∗ Proposition 2.5. (See [3].) Let (A, ≺, ) be an L-dendriform algebra. Then (A∗ , L∗◦ , −R≺ ) is a bimodule of Ah .
Theorem 2.6. (See [3].) Let (A, ·) be a pre-Lie algebra with a nondegenerate symmetric 2-cocycle B. Then there exists a compatible L-dendriform algebra structure on (A, ·) given by B(x ≺ y, z) = B(x, z · y), such that Ah = (A, ·).
B(x y, z) = B y, [z, x] ,
∀x, y, z ∈ A,
(12)
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3. Double constructions of pseudo-Hessian Lie algebras and L-dendriform bialgebras In this section we introduce a concept of a Manin triple of pre-Lie algebras associated to a nondegenerate symmetric 2-cocycle as our basic setup of a double construction of pseudo-Hessian structure on a Lie algebra. Then we interpret it as certain bialgebra structure of an L-dendriform algebra, namely, an L-dendriform bialgebra. First we recall a basic concept. Definition 3.1. Let V be a vector space and let B : V ⊗V → R be a bilinear form. Suppose that W is a subspace of V . Define W ⊥ := x ∈ V B(x, y) = 0, ∀y ∈ W . If W = W ⊥ , then W is called Lagrangian. Definition 3.2. A Manin triple of pre-Lie algebras associated to a nondegenerate symmetric 2-cocycle is a triple of pre-Lie algebras (A, A+ , A− ) together with a nondegenerate symmetric 2-cocycle B of A such that (a) A+ and A− are subalgebras of A. (b) A = A+ ⊕ A− as vector spaces. (c) A+ and A− are Lagrangian with respect to B. It is denoted by (A, A+ , A− , B). Proposition 3.3. Let (A, A+ , A− , B) be a Manin triple (A, A+ , A− ) of pre-Lie algebras associated to a nondegenerate symmetric 2-cocycle B. Then there exists an L-dendriform algebra structure (≺, ) on A given by Eq. (12) such that A+ and A− are subalgebras of (A, ≺, ) and A is the associated horizontal pre-Lie algebra. Proof. Let x, y, z ∈ A+ . Then B(x ≺ y, z) = B(x, z · y) = 0. Since A+ is a Lagrangian subalgebra of A, we have x ≺ y ∈ A+ . Similar arguments apply to and A− . So the conclusion holds. 2 Definition 3.4. Let A be a pre-Lie algebra. Suppose that there is a pre-Lie algebra structure on the dual space A∗ . If there is a pre-Lie algebra structure on the direct sum of the underlying vector spaces of A and A∗ such that A and A∗ are subalgebras and the symmetric bilinear form on A ⊕ A∗ given by Bp x + a∗ , y + b∗ := a∗ , y + x, b∗ ,
∀x, y ∈ A; a∗ , b∗ ∈ A∗ ,
(13)
is a 2-cocycle, where , is the canonical pairing between A and A∗ , then (A ⊕ A∗ , A, A∗ , Bp ) is called a standard Manin triple of pre-Lie algebras associated to a
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nondegenerate symmetric 2-cocycle. In this case, g(A ⊕ A∗ ) is a pseudo-Hessian Lie algebra and it is called a double construction of pseudo-Hessian Lie algebra. Obviously, a standard Manin triple of pre-Lie algebras associated to a nondegenerate symmetric 2-cocycle is a Manin triple of pre-Lie algebras associated to a nondegenerate symmetric 2-cocycle. Proposition 3.5. Every Manin triple of pre-Lie algebras (A, A+ , A− ) associated to a nondegenerate symmetric 2-cocycle B is isomorphic to a standard one. Proof. Since A− and (A+ )∗ are identified by the nondegenerate 2-cocycle B, we can transfer the pre-Lie algebra structure on A− to (A+ )∗ . Hence the pre-Lie algebra structure on A+ ⊕ A− can be transferred to A+ ⊕ (A+ )∗ . Therefore the conclusion follows. 2 Proposition-Definition 3.6. (See [1, Theorem 3.5].) Let (A1 , ·1 ) and (A2 , ·2 ) be two pre-Lie algebras. Suppose there are linear maps l1 , r1 : A1 → gl(A2 ) and l2 , r2 : A2 → gl(A1 ) such that (A2 , l1 , r1 ) is a bimodule of A1 and (A1 , l2 , r2 ) is a bimodule of A2 and they satisfy r1 (x) [a, b]2 = r1 l2 (b)x a − r1 l2 (a)x b + a ·2 r2 (x)b − b ·2 r1 (x)a , l1 (x)(a ·2 b) = −l1 l2 (a)x − r2 (a)x b + l1 (x)a − r1 (x)a ·2 b + r1 r2 (b)x a + a ·2 l1 (x)b , r2 (a) [x, y]1 = r2 l1 (y)a x − r2 l1 (x)a y + x ·1 r1 (a)y − y ·1 r2 (a)x , l2 (a)(x ·1 y) = −l2 l1 (x)a − r1 (x)a y + l2 (a)x − r2 (a)x ·1 y + r2 r1 (y)a x + x ·1 l2 (a)y ,
(14)
(15) (16)
(17)
where x, y ∈ A1 , a, b ∈ A2 and [,]i is the sub-adjacent Lie bracket of (A, ·i ) (i = 1, 2). Then there is a pre-Lie algebra structure “ ·” on the vector space A1 ⊕ A2 given by (x + a) · (y + b) := x ·1 y + l2 (a)y + r2 (b)x + a ·2 b + l1 (x)b + r1 (y)a , ∀x, y ∈ A1 , a, b ∈ A2 .
(18)
2 We denote this pre-Lie algebra by A1 ll21 ,r ,r1 A2 or simply A1 A2 . Moreover, (A1 , A2 , l1 , r1 , l2 , r2 ) satisfying the above conditions is called a matched pair of pre-Lie algebras. On the other hand, every pre-Lie algebra which is a direct sum of the underlying vector spaces of two subalgebras can be obtained from the above way.
Proposition 3.7. Let (A, ≺1 , 1 ) be an L-dendriform algebra. Suppose that there is an L-dendriform algebra structure (A∗ , ≺2 , 2 ) on the dual space A∗ . Then (Ah ⊕(A∗ )h , Ah , (A∗ )h , Bp ) is a standard Manin triple of pre-Lie algebras associated to a nondegenerate ∗ ∗ , L∗◦2 , −R≺ ) is a matched pair symmetric 2-cocycle if and only if (Ah , (A∗ )h , L∗◦1 , −R≺ 1 2 of pre-Lie algebras.
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∗ Proof. First note that by Proposition 2.5, (A∗ , L∗◦1 , −R≺ ) is a bimodule of Ah and 1 ∗ ∗ ∗ (A, L◦2 , −R≺2 ) is a bimodule of (A )h . ∗ ∗ , L∗◦2 , −R≺ ) is a matched pair of pre-Lie algebras, then it is If (A, A∗ , L∗◦1 , −R≺ 1 2 straightforward to show that the bilinear form Bp defined by Eq. (13) is a 2-cocycle L∗ ,−R∗
1 of the pre-Lie algebra A L◦∗◦1 ,−R≺ A∗ . Moreover, Ah and (A∗ )h are Lagrangian subal∗ ≺ 2
2
L∗ ,−R∗
1 gebras of A L◦∗◦1 ,−R≺ A∗ . Thus, (Ah ⊕ (A∗ )h , Ah , (A∗ )h , Bp ) is a standard Manin triple ∗ ≺2 2 of pre-Lie algebras associated to a nondegenerate symmetric 2-cocycle. Conversely, suppose that (Ah ⊕ (A∗ )h , Ah , (A∗ )h , Bp ) is a standard Manin triple of pre-Lie algebras associated to a nondegenerate symmetric 2-cocycle. By Proposition 3.3, Ah and (A∗ )h are subalgebras of Ah ⊕ (A∗ )h . Set
x ∗ a∗ = l·1 (x)a∗ + r·2 a∗ x,
a∗ ∗ x = l·2 a∗ x + r·1 (x)a∗ ,
∀x ∈ A, a∗ ∈ A∗ .
Then (A, A∗ , l1 , r1 , l2 , r2 ) is a matched pair of pre-Lie algebras. Note that ∗ (x)a∗ , y , r·1 (x)a∗ , y = a∗ · x, y = Bp y, a∗ · x = Bp y ≺1 x, a∗ = −R≺ 1 ∗ ∗ ∗
l·2 a x, b = a · x, b∗ = a∗ , x + x · a∗ , b∗ = −Bp b∗ , x, a∗ + Bp b∗ , x · a∗ = −Bp a∗ 2 b∗ , x + Bp b∗ ≺2 a∗ , x = b∗ , L∗◦2 a∗ x ,
∗ for any x, y ∈ A, a∗ , b∗ ∈ A∗ . Hence, r·1 = −R≺ , l·2 = L∗◦2 . Similarly, l·1 = L◦1 , r·2 = 1 ∗ −R≺2 . 2
Definition 3.8. (a) Let V be a vector space. Define the exchanging operator τ : V ⊗ V → V ⊗ V as τ (x ⊗ y) := y ⊗ x,
∀x, y ∈ V.
(19)
(b) Let V1 , V2 be two vector spaces and T : V1 → V2 be a linear map. Denote the dual (linear) map by T ∗ : V2∗ → V1∗ defined by
v1 , T ∗ v2∗ = T (v1 ), v2∗ ,
∀v1 ∈ V1 , v2∗ ∈ V2∗ .
(20)
Note the difference between T ∗ and the linear maps defined by Eq. (11). Proposition 3.9. Let (A, ≺, , α, β) be an L-dendriform algebra equipped with two linear maps α, β : A → A ⊗ A. Suppose that α∗ , β ∗ : A∗ ⊗ A∗ → A∗ induce an L-dendriform ∗ ∗ , L∗◦∗ , −R≺ ) is algebra structure on A∗ denoted by “≺∗ , ∗ ”. Then (Ah , (A∗ )h , L∗◦ , −R≺ ∗ a matched pair of pre-Lie algebra if and only if the following equations are satisfied (for any x, y ∈ A):
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(α + β − τ α − τ β)(x ≺ y) = L≺ (x) ⊗ 1 (α − τ β)(y) + 1 ⊗ L≺ (x) (−τ α + β)(y) + 1 ⊗ R≺ (y) (α + β)(x) − R≺ (y) ⊗ 1 (τ α + τ β)(x),
(α + β)(x ◦ y) = 1 ⊗ R◦ (y) β(x) − L≺ (y) ⊗ 1 α(x) + L (x) ⊗ 1 + 1 ⊗ L◦ (x) (α + β)(y), α [x, y] = −L· (y) ⊗ 1 − 1 ⊗ L◦ (y) α(x) + L· (x) ⊗ 1 + 1 ⊗ L◦ (x) α(y), (τ α − β)(x · y) = L (x) ⊗ 1 + 1 ⊗ L· (x) (τ α − β)(y) − 1 ⊗ R· (y) β(x) + R≺ (y) ⊗ 1 τ α(x).
(21)
(22)
(23)
(24)
Proof. By Proposition-Definition 3.6, we need to prove that Eqs. (14)–(17) are equivalent to Eqs. (21)–(24) respectively in the case that A 1 = Ah , l1 = L∗◦ ,
A2 = A∗ h ,
∗ r1 = −R≺ ,
l2 = L∗◦∗ ,
and ∗ r2 = −R≺ . ∗
As an example we give an explicit proof that Eq. (24) holds if and only if Eq. (17) holds. The other equivalences are similar. In fact, in this case, Eq. (17) becomes ∗ ∗ ∗ L∗◦∗ a∗ (x · y) = −L∗◦∗ L∗◦ (x)a∗ + R≺ (x)a∗ y + L∗◦∗ a∗ x + R≺ a x ·y ∗ ∗ ∗ ∗ ∗ ∗ + R≺∗ R≺ (y)a x + x · L◦∗ a y , ∀x, y ∈ A, a∗ ∈ A∗ . Let both sides of the above equation act on an arbitrary element b∗ ∈ A∗ . Then
x · y, −a∗ ◦∗ b∗ = y, L∗ (x)a∗ ◦∗ b∗ + x, a∗ ∗ R·∗ (y)b∗ ∗ − x, b∗ ≺∗ R≺ (y)a∗ + y, a∗ ◦∗ L∗· (x)b∗ .
It is equivalent to the following equation
(τ α − β)(x · y), a∗ ⊗ b∗ = L (x) ⊗ 1 + 1 ⊗ L· (x) (τ α − β)(y) − 1 ⊗ R· (y) β(x) + R≺ (y) ⊗ 1 τ α(x), a∗ ⊗ b∗ ,
which exactly gives Eq. (24).
2
Definition 3.10. (a) Let (A, α, β) be a vector space with two linear maps α, β : A → A⊗A. If α∗ , β ∗ : A∗ ⊗ A∗ → A∗ induce an L-dendriform algebra structure on A∗ , then we call (A, α, β) an L-dendriform coalgebra.
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(b) Let (A, ≺, ) be an L-dendriform algebra. If there are two linear maps α, β : A → A⊗ A such that (A, α, β) is an L-dendriform coalgebra and α and β satisfy Eq. (21)–(24), then the set of the linear maps α, β is called an L-dendriform bialgebra structure on (A, ≺, ). We denote it by (A, ≺, , α, β). By Propositions 3.7 and 3.9 we have the following result which gives a structure theory of a double construction of pseudo-Hessian Lie algebra. Theorem 3.11. Let (A, ≺, ) be an L-dendriform algebra. Let α, β : A → A ⊗ A be two linear maps such that α∗ , β ∗ : A∗ ⊗ A∗ → A∗ induce an L-dendriform algebra structure on A∗ , that is, (A, α, β) is an L-dendriform coalgebra. Then the following conditions are equivalent: (a) (Ah ⊕ (A∗ )h , Ah , (A∗ )h , Bp ) is a standard Manin triple of pre-Lie algebras associated to a nondegenerate symmetric 2-cocycle, i.e., g(Ah ⊕ (A∗ )h ) is a double construction of pseudo-Hessian Lie algebra. (b) (A, ≺, , α, β) is an L-dendriform bialgebra. 4. Triangular pseudo-Hessian Lie algebras In this section we study some double constructions of pseudo-Hessian Lie algebras arising from some skew-symmetric tensors. We also relate them to some semidirect product constructions from the “Lie-type” operations of L-dendriform algebras. 4.1. Triangular pseudo-Hessian Lie algebras Lemma 4.1. Let (A, ≺, ) be an L-dendriform algebra and r ∈ A⊗A. Let α, β : A → A⊗A be two linear maps defined by α(x) := 1 ⊗ L◦ (x) + L· (x) ⊗ 1 r, (25) β(x) := −1 ⊗ ad(x) − L (x) ⊗ 1 r, (26) respectively, where x ∈ A. Then α and β satisfy Eqs. (21)–(24) if and only if the following equations are satisfied: R◦ (x ≺ y) ⊗ 1 + 1 ⊗ L≺ (x ≺ y) + L≺ (x) ⊗ 1 1 ⊗ L (y) + ad(y) ⊗ 1 − R≺ (y) ⊗ 1 R◦ (x) ⊗ 1 − 1 ⊗ L≺ (x) 1 ⊗ L· (y) + L (y) ⊗ 1 r + τ (r) = 0,
(27) L◦ (x · y) ⊗ 1 + 1 ⊗ L· (x · y) − L (x) ⊗ 1 L◦ (y) ⊗ 1 + 1 ⊗ L· (y) − 1 ⊗ L· (x) 1 ⊗ L· (y) + L (y) ⊗ 1 − R≺ (y) ⊗ 1 L◦ (x) ⊗ 1 r + τ (r) = 0, (28)
where x, y ∈ A.
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Proof. Obviously α and β satisfy Eqs. (22) and (23) automatically. Moreover, substituting Eqs. (25) and (26) into Eq. (21), we have
1 ⊗ R◦ (x ≺ y) + L≺ (x ≺ y) ⊗ 1 r − R◦ (x ≺ y) ⊗ 1 + 1 ⊗ L≺ (x ≺ y) τ (r) = L≺ (x) ⊗ 1 1 ⊗ L◦ (y) + L· (y) ⊗ 1 r + L≺ (x) ⊗ 1 ad(y) ⊗ 1 + 1 ⊗ L (y) τ (r) − 1 ⊗ L≺ (x) L◦ (y) ⊗ 1 + 1 ⊗ L· (y) τ (r) − 1 ⊗ L≺ (x) 1 ⊗ ad(y) + L (y) ⊗ 1 r + 1 ⊗ R≺ (y) 1 ⊗ R◦ (x) + L≺ (x) ⊗ 1 r − R≺ (y) ⊗ 1 R◦ (x) ⊗ 1 + 1 ⊗ L≺ (x) τ (r),
for any x, y ∈ A. It is equivalent to the following equation 1 ⊗ R◦ (x ≺ y) + L≺ (x ≺ y) ⊗ 1 + R◦ (x ≺ y) ⊗ 1 + 1 ⊗ L≺ (x ≺ y) r − R◦ (x ≺ y) ⊗ 1 + 1 ⊗ L≺ (x ≺ y) r + τ (r) = L≺ (x) ⊗ 1 −ad(y) ⊗ 1 − 1 ⊗ L (y) + 1 ⊗ L◦ (y) + L· (y) ⊗ 1 r + L≺ (x) ⊗ 1 ad(y) ⊗ 1 + 1 ⊗ L (y) r + τ (r) + 1 ⊗ L≺ (x) −1 ⊗ ad(y) − L (y) ⊗ 1 + L◦ (y) ⊗ 1 + 1 ⊗ L· (y) r − 1 ⊗ L≺ (x) L◦ (y) ⊗ 1 + 1 ⊗ L· (y) r + τ (r) + 1 ⊗ R≺ (y) 1 ⊗ R◦ (x) + L≺ (x) ⊗ 1 r + R≺ (y) ⊗ 1 R◦ (x) ⊗ 1 + 1 ⊗ L≺ (x) r − R≺ (y) ⊗ 1 R◦ (x) ⊗ 1 + 1 ⊗ L≺ (x) r + τ (r) . It is easy to show that the above equation holds if and only if Eq. (27) holds. So α and β satisfy Eq. (21) if and only if Eq. (27) holds. Similarly, α and β satisfy Eq. (24) if and only if Eq. (28) holds. 2 Lemma 4.2. Let V be a vector space and α, β : V → V ⊗ V be two linear maps. Then (A, α, β) is an L-dendriform coalgebra if and only if both R1 and R2 are zero, where Ri : V → V ⊗ V ⊗ V (i = 1, 2) are defined as follows: R1 (x) := (α ⊗ id)α(x) + (τ ⊗ id)(id ⊗ α)β(x) − id ⊗ (α + β) α(x) − (τ ⊗ id)(β ⊗ id)α(x), R2 (x) := (α + β) ⊗ id β(x) + (τ ⊗ id)(id ⊗ β)β(x) − (id ⊗ β)β(x) − (τ ⊗ id) (α + β) ⊗ id β(x), for any x ∈ V . Proof. It follows immediately from the definition of an L-dendriform algebra. 2
(29)
(30)
X. Ni, C. Bai / Journal of Algebra 400 (2014) 273–289
Definition 4.3. Let V be a vector space and r = r12 =
ai ⊗ bi ⊗ 1,
i
r13 =
i
r23 =
283
ai ⊗ bi ∈ V ⊗ V . Set
ai ⊗ 1 ⊗ bi ,
i
1 ⊗ ai ⊗ bi ,
i
r21 =
bi ⊗ ai ⊗ 1,
(31)
i
where 1 is a symbol playing a similar role of the unit. If in addition, there exists a binary operation : V ⊗ V → V on V , then the operation between two rs is in an obvious way. For example, r12 r13 =
ai aj ⊗ bi ⊗ bj ,
i,j
r23 r12 =
r13 r23 =
ai ⊗ aj ⊗ bi bj ,
i,j
aj ⊗ ai bj ⊗ bi .
(32)
i,j
Note that the above equation is independent of the existence of a unit. Lemma 4.4. Let (A, ≺, ) be an L-dendriform algebra and r = i ai ⊗bi ∈ A ⊗ A. Define α, β : A → A ⊗ A by Eqs. (25) and (26) respectively. Let ad be the adjoint action, i.e., ad(x)y = [x, y], for any x, y ∈ A. Then we have that R1 (x) = 1 ⊗ 1 ⊗ L◦ (x) Jr, rKQ1 + 1 ⊗ L (x) ⊗ 1 + L· (x) ⊗ 1 ⊗ 1 Jr, rKQ2 + P1 (x, aj ) r + τ (r) ⊗ bj ,
(33)
j
R2 (x) = 1 ⊗ 1 ⊗ ad(x) Jr, rKQ3 + 1 ⊗ L (x) ⊗ 1 + L (x) ⊗ 1 ⊗ 1 Jr, rKQ4 + P2 (x, aj ) r + τ (r) ⊗ bj ,
(34)
j
where Jr, rKQ1 := r23 ◦ r12 + r13 · r12 − r23 ◦ r13 + [r13 , r21 ] + r23 r21 ,
(35)
Jr, rKQ2 := r12 · r13 − r12 ≺ r23 − r23 ◦ r13 ,
(36)
Jr, rKQ3 := −r12 ◦ r23 − r13 ≺ r12 + r21 ◦ r13 + r23 ≺ r21 + [r23 , r13 ],
(37)
Jr, rKQ4 := [r23 , r13 ] − r12 r23 − r12 r13 , (38) P1 (x, y) := 1 ⊗ L (x · y) + ad(x · y) ⊗ 1 − R· (y) ⊗ 1 ad(x) ⊗ 1 + 1 ⊗ L (x) , (39) P2 (x, y) := 1 ⊗ L≺ (x y) + R◦ (x y) ⊗ 1 + R (y) ⊗ 1 1 ⊗ L (x) + ad(x) ⊗ 1 . (40)
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284
Proof. We give an explicit proof of Eq. (33) as an example. The proof of Eq. (34) is similar. After rearranging the terms suitably, we divide R1 into the following four parts: R1 (x) = J1 (x) + J2 (x) + J3 (x) + J4 (x), x ∈ A, where J1 (x) = ai ⊗ (x · aj ) ◦ bi − ai ⊗ (x ◦ bi ) · aj + ai ⊗ (x ◦ bi ) aj + (x · aj ) · ai ⊗ bi i,j
− x · ai ⊗ bi · aj + x · ai ⊗ bi aj + bi ⊗ (x · aj ) ai + [x · aj , bi ] ⊗ ai − [x, bi ] · aj ⊗ ai − bi · aj ⊗ x ai ⊗ bj , J2 (x) = ai ⊗ aj ◦ bi ⊗ x ◦ bj + aj · ai ⊗ bi ⊗ x ◦ bj i,j
+ [aj , bi ] ⊗ ai ⊗ x ◦ bj + bi ⊗ aj ai ⊗ x ◦ bj , J3 (x) = −ai ⊗ aj ⊗ [x, bj ] ◦ bi − aj ⊗ ai ⊗ (x ◦ bj ) ◦ bi + aj ⊗ ai ⊗ [x ◦ bj , bi ], i,j
J4 (x) =
−ai ⊗ x aj ⊗ bj ◦ bi − x · aj ⊗ ai ⊗ bj ◦ bi + x · aj ⊗ ai ⊗ [bj , bi ].
i,j
On the other hand, the right hand side of Eq. (33) is
ai ⊗ aj ◦ bi ⊗ x ◦ bj + aj · ai ⊗ bi ⊗ x ◦ bj − ai ⊗ aj ⊗ x ◦ (bj ◦ bi )
i,j
+ [aj , bi ] ⊗ ai ⊗ x ◦ bj + bi ⊗ aj ai ⊗ x ◦ bj + ai · aj ⊗ x bi ⊗ bj − ai ⊗ x (bi ≺ aj ) ⊗ bj − aj ⊗ (x ai ) ⊗ bi ◦ bj + x · (ai · aj ) ⊗ bi ⊗ bj − x · ai ⊗ bi ≺ aj ⊗ bj − x · aj ⊗ ai ⊗ bi ◦ bj + ai ⊗ (x · aj ) bi ⊗ bj + bi ⊗ (x · aj ) ai ⊗ bj + [x · aj , ai ] ⊗ bi ⊗ bj + [x · aj , bi ] ⊗ ai ⊗ bj − [x, ai ] · aj ⊗ bi ⊗ bj − [x, bi ] · aj ⊗ ai ⊗ bj − ai · aj ⊗ x bi ⊗ bj − bi · aj ⊗ x ai ⊗ bj . After rearranging the terms suitably, the sum of terms whose third component is bj in the right hand side of Eq. (33) is J˜1 (x) = Ja (x) + Jb (x) + Jc (x) + Jd (x) + Je (x) + Jf (x) + Jg (x) + Jh (x), Ja (x) = {ai · aj ⊗ x bi − ai · aj ⊗ x bi } ⊗ bj = 0, i,j
Jb (x) =
−ai ⊗ x (bi ≺ aj ) + ai ⊗ (x · aj ) bi ⊗ bj
i,j
=
ai ⊗ (x · aj ) ◦ bi − ai ⊗ (x ◦ bi ) · aj + ai ⊗ (x ◦ bi ) aj ⊗ bj ,
i,j
where
X. Ni, C. Bai / Journal of Algebra 400 (2014) 273–289
Jc (x) =
x · (ai · aj ) + [x · aj , ai ] − [x, ai ] · aj ⊗ bi ⊗ bj = (x · aj ) · ai ⊗ bi ⊗ bj ,
i,j
Jd (x) =
−x · ai ⊗ bi ≺ aj ⊗ bj ,
Je (x) =
i,j
Jf (x) =
[x · aj , bi ] ⊗ ai ⊗ bj , i,j
−[x, bi ] · aj ⊗ ai ⊗ bj ,
i,j
bi ⊗ (x · aj ) ai ⊗ bj ,
i,j
i,j
Jg (x) =
285
Jh (x) =
−bi · aj ⊗ x ai ⊗ bj .
i,j
So J˜1 (x) = J1 (x). Similarly, the sum of whose third component is x ◦ bj in the right hand side of Eq. (33) is J2 (x). Moreover, the sum of the other terms in the right hand side of Eq. (33) is J3 (x) + J4 (x). So the conclusion follows. 2 Definition 4.5. (See [3].) Let (A, ≺, ) be an L-dendriform algebra and r ∈ A ⊗ A. The following equation is called LD-equation in (A, ≺, ): r13 ◦ r23 + r12 · r23 − r12 ≺ r13 = 0.
(41)
Let V be a vector space. Recall that for any r ∈ V ⊗ V , if r = −τ (r), then it is called skew-symmetric, where τ is the exchanging operator defined by Eq. (19). Lemma 4.6. Let (A, ≺, ) be an L-dendriform algebra and let r ∈ A ⊗ A be a skewsymmetric solution of LD-equation. Then we have Jr, rKQi = 0, where Jr, rKQi are defined by Eqs. (35)–(38), i = 1, 2, 3, 4. Proof. By Lemma 4.6 in [3], we know that if r is a skew-symmetric solution of LD-equation, then Jr, rKQ2 = 0 and the following two equations hold: r13 · r23 + r12 ◦ r23 + r13 ≺ r12 = 0.
(42)
r23 · r13 − r12 ◦ r13 − r23 ≺ r12 = 0.
(43)
Thus Jr, rKQ4 = 0. Also note that if r is skew-symmetric, then we have Jr, rKQ1 = Jr, rKQ2 = 0 and Jr, rKQ3 = Jr, rKQ4 = 0. So the conclusion holds. 2 Theorem 4.7. Let (A, ≺, ) be an L-dendriform algebra and let r ∈ A ⊗ A be a skewsymmetric solution of LD-equation. Define linear maps α, β : A → A ⊗ A by Eqs. (25) and (26) respectively. Then (A, ≺, , α, β) is an L-dendriform bialgebra. In this case, g(Ah ⊕ (A∗ )h ) becomes a double construction of pseudo-Hessian Lie algebra. Proof. By Lemmas 4.4 and 4.6, R1 and R2 , which are given by Eqs. (33) and (34) respectively, both are zero. So by Lemma 4.2, (A, α, β) is an L-dendriform coalgebra. On the other hand, since r is skew-symmetric, by Lemma 4.1, Eqs. (21)–(24) hold.
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So (A, ≺, , α, β) is an L-dendriform bialgebra. The last conclusion follows from Theorem 3.11. 2 Definition 4.8. Let (A, ≺, ) be an L-dendriform algebra and let r ∈ A ⊗ A be a skewsymmetric solution of LD-equation. Then the pseudo-Hessian Lie algebra g(Ah ⊕ (A∗ )h ) given by Theorem 4.7 is called triangular. The terminology “triangular” is motivated by quantum group theory [5]. 4.2. Lie-type operations of L-dendriform algebras and semidirect product constructions The second axiom defining an L-dendriform algebra (A, ≺, ), i.e., Eq. (5), can be re-written as [x, y] z = x (y z) − y (x z),
∀x, y, z ∈ A.
(44)
Eq. (44) means that L : g(A) → gl(A) is a representation of g(A). In this case we can form the semidirect product h := g(A) L∗ A∗ ,
(45)
which is the Lie algebra with the underlying vector space A ⊕ A∗ and the following Lie bracket:
x, a∗ , y, b∗ := [x, y], L∗ (x)b∗ − L∗ (y)a∗ ,
∀x, y ∈ A, a∗ , b∗ ∈ A∗ .
(46)
4.3. An isomorphic theorem Definition 4.9. Let V be a vector space. Then any r ∈ V ⊗ V can be identified as a linear map Tr : V ∗ → V in the following way:
u∗ ⊗ v ∗ , r = u∗ , Tr v ∗ ,
∀u∗ , v ∗ ∈ V ∗ .
(47)
Lemma 4.10. Let (A, ≺, ) be an L-dendriform algebra and r ∈ A⊗A be a skew-symmetric solution of LD-equation in (A, ≺, ). Let g := g(Ah ⊕ (A∗ )h ) be the corresponding triangular pseudo-Hessian Lie algebra. (a) The L-dendriform algebra structure on the dual space A∗ is given as follows: a∗ ≺ b∗ = R◦∗ Tr a∗ b∗ − R·∗ Tr b∗ a∗ , ∀a∗ , b∗ ∈ A∗ ; ∗ ∗ ∗ Tr b a , ∀a∗ , b∗ ∈ A∗ . a∗ b∗ = ad∗ Tr a∗ b∗ + R
(48) (49)
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(b) The following equations hold:
a∗ , b∗ = L∗ Tr a∗ b∗ − L∗ Tr b∗ a∗ , ∀a∗ , b∗ ∈ A∗ ;
∗ ∗
Tr a , Tr b = Tr a∗ , b∗ , ∀a∗ , b∗ ∈ A∗ .
(50) (51)
(c) The following equation in g holds:
Tr L∗ (x)a∗ − L∗ a∗ x = x, Tr a∗ ,
∀x ∈ A, a∗ ∈ A∗ .
(52)
Proof. (a) Let {e1 , . . . , en } be a basis of A and {e∗1 , . . . , e∗n } be the dual basis. Suppose that ei ≺ ej =
ckij ek ,
ei ej =
k
Then we have that aij = −aji and Tr (e∗i ) = e∗k ≺ e∗l =
s
=
dkij ek ,
r=
aij ei ⊗ ej .
i,j
k
k
aki ek . Thus, for any k, l, we have
e∗k ⊗ e∗l , 1 ⊗ L◦ (es ) + L· (es ) ⊗ 1 r e∗s
akt dlst − akt clts + ckst + dkst atl e∗s = R◦∗ Tr e∗k e∗l − R·∗ Tr e∗l e∗k .
s,t
So Eq. (48) holds. Eq. (49) is proved in a similar way. (b) By Eqs. (48) and (49), Eq. (50) holds. On the other hand, r is a solution of LD-equation if and only if, for any i, j, k, the following equation holds:
ail atk cjlt + djlt + ail ajt dklt − cktl − alj atk cilt = 0.
t,l
The left hand side of the above equation is precisely the coefficient of ej in ∗ ∗ ∗ −Tr e∗i · Tr e∗k + Tr L∗◦ Tr e∗i e∗k − Tr R≺ T r ek e i . Hence for any a∗ , b∗ ∈ A∗ we have that ∗ ∗
Tr a , Tr b = Tr L∗ Tr a∗ b∗ − Tr L∗ Tr b∗ a∗ . So by Eq. (50), Eq. (51) holds. (c) For any x ∈ A, a∗ , b∗ ∈ A∗ we have that
∗ ∗ ∗ L∗ b∗ x, a∗ = x, −b∗ a∗ = x, −ad∗ Tr b∗ a∗ − R Tr a b ∗ ∗ ∗ ∗ ∗
= Tr b , x , a + x Tr a , b = Tr b , x + Tr L∗ (x)b∗ , a∗ .
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Here the second equality follows from Eq. (49) and the last equality follows from the fact that r is skew-symmetric. So Eq. (52) holds. 2 At the end of this paper, we show that a triangular pseudo-Hessian Lie algebra is isomorphic (as Lie algebras) to the semidirect product h defined by Eq. (45). Theorem 4.11. Let (A, ≺, , α, β) be an L-dendriform bialgebra and r ∈ A ⊗ A be a skew-symmetric solution of LD-equation. Let g := g(Ah ⊕ (A∗ )h ) be the corresponding triangular pseudo-Hessian Lie algebra. Then g is isomorphic to the semidirect product h := g(A) L∗ A∗ defined by Eq. (45). Proof. By Proposition 3.7 and Theorem 4.7, the Lie bracket of g is given as follows:
x, a∗ , y, b∗ = [x, y] + L∗ a∗ y − L∗ b∗ x, a∗ , b∗ + L∗ (x)b∗ − L∗ (y)a∗ , ∀x, y ∈ A, a∗ , b∗ ∈ A∗ .
For any x ∈ A, a∗ ∈ A∗ , define a linear map θ : g → h by θ x, a∗ := Tr a∗ + x, a∗ . Then we have
θ x, a∗ , θ y, b∗ = Tr a∗ + x, a∗ , Tr b∗ + y, b∗
= Tr a∗ + x, Tr b∗ + y , L∗ Tr a∗ + x b∗ − L∗ Tr b∗ + y a∗
= Tr a∗ , b∗ + L∗ (x)b∗ − L∗ (y)a∗ + [x, y]
+ L∗ a∗ y − L∗ b∗ x, a∗ , b∗ + L∗ (x)b∗ − L∗ (y)a∗ = θ x, a∗ , y, b∗ . The last equality follows from Lemma 4.10(b) and (c). θ is obviously bijective. So the conclusion follows. 2 Acknowledgments This work is supported in part by NSFC (10921061, 11271202, 11221091), NKBRPC (2006CB 805905) and SRFDP (200800550015, 20120031110022). References [1] C. Bai, Left-symmetric bialgebras and an analogue of the classical Yang–Baxter equation, Commun. Contemp. Math. 10 (2008) 221–260. [2] C. Bai, D. Hou, Z. Chen, On a class of Lie groups with a left-invariant flat pseudo-metric, Monatsh. Math. 164 (2011) 243–269.
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[3] C. Bai, L. Liu, X. Ni, Some results on L-dendriform algebras, J. Geom. Phys. 60 (2010) 940–950. [4] D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Cent. Eur. J. Math. 4 (2006) 323–357. [5] V. Chari, A. Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1994. [6] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. Math. 78 (1963) 267–288. [7] J.-L. Koszul, Domaines bornés homogènes et orbites de groupes de transformation affines, Bull. Soc. Math. France 89 (1961) 515–533. [8] B.A. Kupershmidt, On the nature of the Virasoro algebra, J. Nonlinear Math. Phys. 6 (1999) 222–245. [9] J. Milnor, Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976) 293–329. [10] H. Shima, Homogeneous Hessian manifolds, Ann. Inst. Fourier (Grenoble) 30 (1980) 91–128. [11] E.B. Vinberg, Convex homogeneous cones, Trans. Moscow Math. Soc. 12 (1963) 340–403.
Contents lists available at ScienceDirect
Journal of Algebra www.elsevier.com/locate/jalgebra
Pseudo-Hessian Lie algebras and L-dendriform bialgebras Xiang Ni a , Chengming Bai b,∗ a b
Department of Mathematics, Caltech, Pasadena, CA 91125, USA Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, PR China
a r t i c l e
i n f o
Article history: Received 15 August 2010 Available online 24 December 2013 Communicated by Efim Zelmanov MSC: 17A30 17B05 17B60 Keywords: Lie algebra Pre-Lie algebra L-dendriform algebra Hessian structure
a b s t r a c t In this paper, we study a special class of pseudo-Hessian Lie algebras satisfying an additional condition that they are decomposed into a direct sum of underlying vector spaces of two Lagrangian subalgebras in terms of L-dendriform algebras which are the underlying algebraic structures. Such structures are equivalent to certain bialgebra structures, namely, Ldendriform bialgebras. Furthermore, we introduce and study the so-called triangular pseudo-Hessian Lie algebras and relate them to some semidirect product constructions from the “Lietype” operations of L-dendriform algebras. © 2013 Elsevier Inc. All rights reserved.
1. Introduction At first, we assume that the base field is taken to be R of real numbers. Definition 1.1. (See [6,7,11].) (a) A pre-Lie algebra is a vector space A equipped with a bilinear operation (x, y) → x·y satisfies * Corresponding author. E-mail addresses: [email protected] (X. Ni), [email protected] (C. Bai). 0021-8693/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jalgebra.2013.12.003
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(x · y) · z − x · (y · z) = (y · x) · z − y · (x · z),
∀x, y, z ∈ A.
(1)
(b) Suppose that (A, ·) is a pre-Lie algebra. Then the commutator [x, y] := x · y − y · x,
∀x, y ∈ A,
(2)
defines a Lie algebra g(A), which is called the sub-adjacent Lie algebra of A and A is called a compatible pre-Lie algebra structure on the Lie algebra g(A). Pre-Lie algebras are the algebraic structures corresponding to the (left-invariant) affine structures on Lie groups [7,11]. See a survey article [4] for the study of pre-Lie algebras. Definition 1.2. (See [8,10].) (a) Let (A, ·) be a pre-Lie algebra. A bilinear form B : A ⊗ A → R 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.
(3)
(b) A pre-Lie algebra (A, ·) is called Hessian (resp. pseudo-Hessian) if there is a symmetric and positive definite (resp. nondegenerate) 2-cocycle B. It is denoted by (A, ·, B). (c) A Lie algebra g is called Hessian (resp. pseudo-Hessian) if there is a compatible pre-Lie algebra structure · : g ⊗ g → g and a bilinear form B : g ⊗ g → R such that (g, ·, B) becomes a Hessian pre-Lie algebra (resp. pseudo-Hessian pre-Lie algebra). It is denoted by (g, B). Let (g, B) be a real Hessian Lie algebra. Then B induces a left-invariant Hessian metric on the corresponding connected real Lie group G, thus making it be a Hessian manifold. Recall that a Hessian manifold M is a flat affine manifold provided with a Hessian metric. Note that a Hessian metric on a smooth manifold M is a Riemannian metric g such that for each point p ∈ M there exists a C ∞ -function ϕ defined on a 2 ϕ neighborhood of p such that gij = ∂x∂i ∂x j [10]. Obviously, from an algebraic point of view, the base field can be extended to a field of characteristic different from 2, which is the field from which we take all the constants and over which we take all the algebras, vector spaces and linear maps, etc. in this paper. Furthermore, we study a special class of pseudo-Hessian Lie algebras satisfying an additional condition that they are decomposed into a direct sum of the underlying vector spaces of two Lagrangian subalgebras, which lead to the notion of double construction of pseudo-Hessian Lie algebra. On the other hand, a notion of L-dendriform algebra was introduced in [3] as the underlying algebraic structure of a pseudo-Hessian pre-Lie algebra or a pseudo-Hessian Lie algebra. This paper deals with further the connection between pseudo-Hessian Lie algebras and L-dendriform algebras. To be more precise, we interpret
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a double construction of pseudo-Hessian Lie algebra in terms of some bialgebra structure of an L-dendriform algebra, namely, an L-dendriform bialgebra. Note that another class of pseudo-Hessian Lie algebras related to the left-invariant flat pseudo-metrics on Lie groups [9] were studied in terms of L-dendriform algebras in [2]. Furthermore, we introduce and study so-called triangular pseudo-Hessian Lie algebras and relate them to some semidirect product constructions which are related to the “Lie-type” operations of L-dendriform algebras. The paper is organized as follows. In Section 2 we recall some necessary definitions and basic results on L-dendriform algebras given in [3]. We introduce a notion of double construction of pseudo-Hessian Lie algebra in Section 3 and interpret it in terms of some bialgebra structure of an L-dendriform algebra. In Section 4 we study a special subclass of pseudo-Hessian Lie algebras, which are called triangular pseudo-Hessian Lie algebras arising from the skew-symmetric solutions of the LD-equation introduced in [3]. We note that the “Lie-type” operations of L-dendriform algebras could be used to form some semidirect constructions in a natural way and we show that all the triangular pseudo-Hessian Lie algebras are isomorphic to them. Throughout this paper all algebras (may have more than one operation) and vector spaces are assumed to be finite-dimensional, although many results still hold in infinitedimensional case. 2. Preliminaries and basic results Definition 2.1. (See [3].) An L-dendriform algebra (A, ≺, ) is a vector space A equipped with two bilinear operations ≺, : A ⊗ A → A such that for any x, y, z ∈ A, (x ≺ y) ≺ z + y (x ≺ z) = x ≺ (y · z) + (y x) ≺ z,
(4)
(x · y) z + y (x z) = x (y z) + (y · x) z,
(5)
where x · y = x ≺ y + x y. Definition 2.2. (See [3].) Let (A, ≺, ) be an L-dendriform algebra. (a) Define x · y := x ≺ y + x y,
∀x, y ∈ A.
(6)
Then (A, ·) is a pre-Lie algebra and we denote it by (Ah , ·). It is called the associated horizontal pre-Lie algebra of (A, ≺, ) and (A, ≺, ) is called a compatible L-dendriform algebra structure on the pre-Lie algebra (Ah , ·). (b) Define x ◦ y := x y − y ≺ x,
∀x, y ∈ A.
(7)
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Then (A, ◦) is a pre-Lie algebra and we denote it by (Av , ◦). It is called the associated vertical pre-Lie algebra of (A, ≺, ) and (A, ≺, ) is called a compatible L-dendriform algebra structure on the pre-Lie algebra (Av , ◦). (c) The associated horizontal and vertical pre-Lie algebras (Ah , ·) and (Av , ◦) have the same sub-adjacent Lie algebra g(A) = g(Ah ) = g(Av ) given by [x, y] := x y + x ≺ y − y x − y ≺ x,
∀x, y ∈ A.
(8)
It is called the sub-adjacent Lie algebra of the L-dendriform algebra (A, ≺, ). Definition 2.3. (See [1].) Let (A, ·) be a pre-Lie algebra and V be a vector space. Let l, r : A → gl(V ) be two linear maps. V (or (V, l, r)) is called a bimodule of A if l(x)l(y) − l(x · y) = l(y)l(x) − l(y · x), l(x)r(y) − r(y)l(x) = r(x · y) − r(y)r(x),
∀x, y ∈ A.
(9)
In fact, (V, l, r) is a bimodule of a pre-Lie algebra (A, ·) if and only if the direct sum A ⊕ V of the underlying vector spaces of A and V is turned into a pre-Lie algebra (the semidirect product) by defining multiplication in A ⊕ V by (it is denoted by A l,r V ) (x1 + v1 ) ∗ (x2 + v2 ) := x1 · x2 + l(x1 )v2 + r(x2 )v1 ,
∀x1 , x2 ∈ A, v1 , v2 ∈ V. (10)
Definition 2.4. Let (A, ) be a vector space with a bilinear operation : A ⊗ A → A. We denote the left action and right action by L : A → gl(A) and R : A → gl(A) respectively, i.e., L (x)y = R (y)x = x y for any x, y ∈ A. We also define linear maps L∗ , R∗ : A → gl(A∗ ) by
L∗ (x)v ∗ , u = − v ∗ , L (x)u , ∗ R (x)v ∗ , u = − v ∗ , R (x)u , ∀x, u ∈ A, v ∗ ∈ A∗ .
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∗ Proposition 2.5. (See [3].) Let (A, ≺, ) be an L-dendriform algebra. Then (A∗ , L∗◦ , −R≺ ) is a bimodule of Ah .
Theorem 2.6. (See [3].) Let (A, ·) be a pre-Lie algebra with a nondegenerate symmetric 2-cocycle B. Then there exists a compatible L-dendriform algebra structure on (A, ·) given by B(x ≺ y, z) = B(x, z · y), such that Ah = (A, ·).
B(x y, z) = B y, [z, x] ,
∀x, y, z ∈ A,
(12)
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3. Double constructions of pseudo-Hessian Lie algebras and L-dendriform bialgebras In this section we introduce a concept of a Manin triple of pre-Lie algebras associated to a nondegenerate symmetric 2-cocycle as our basic setup of a double construction of pseudo-Hessian structure on a Lie algebra. Then we interpret it as certain bialgebra structure of an L-dendriform algebra, namely, an L-dendriform bialgebra. First we recall a basic concept. Definition 3.1. Let V be a vector space and let B : V ⊗V → R be a bilinear form. Suppose that W is a subspace of V . Define W ⊥ := x ∈ V B(x, y) = 0, ∀y ∈ W . If W = W ⊥ , then W is called Lagrangian. Definition 3.2. A Manin triple of pre-Lie algebras associated to a nondegenerate symmetric 2-cocycle is a triple of pre-Lie algebras (A, A+ , A− ) together with a nondegenerate symmetric 2-cocycle B of A such that (a) A+ and A− are subalgebras of A. (b) A = A+ ⊕ A− as vector spaces. (c) A+ and A− are Lagrangian with respect to B. It is denoted by (A, A+ , A− , B). Proposition 3.3. Let (A, A+ , A− , B) be a Manin triple (A, A+ , A− ) of pre-Lie algebras associated to a nondegenerate symmetric 2-cocycle B. Then there exists an L-dendriform algebra structure (≺, ) on A given by Eq. (12) such that A+ and A− are subalgebras of (A, ≺, ) and A is the associated horizontal pre-Lie algebra. Proof. Let x, y, z ∈ A+ . Then B(x ≺ y, z) = B(x, z · y) = 0. Since A+ is a Lagrangian subalgebra of A, we have x ≺ y ∈ A+ . Similar arguments apply to and A− . So the conclusion holds. 2 Definition 3.4. Let A be a pre-Lie algebra. Suppose that there is a pre-Lie algebra structure on the dual space A∗ . If there is a pre-Lie algebra structure on the direct sum of the underlying vector spaces of A and A∗ such that A and A∗ are subalgebras and the symmetric bilinear form on A ⊕ A∗ given by Bp x + a∗ , y + b∗ := a∗ , y + x, b∗ ,
∀x, y ∈ A; a∗ , b∗ ∈ A∗ ,
(13)
is a 2-cocycle, where , is the canonical pairing between A and A∗ , then (A ⊕ A∗ , A, A∗ , Bp ) is called a standard Manin triple of pre-Lie algebras associated to a
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nondegenerate symmetric 2-cocycle. In this case, g(A ⊕ A∗ ) is a pseudo-Hessian Lie algebra and it is called a double construction of pseudo-Hessian Lie algebra. Obviously, a standard Manin triple of pre-Lie algebras associated to a nondegenerate symmetric 2-cocycle is a Manin triple of pre-Lie algebras associated to a nondegenerate symmetric 2-cocycle. Proposition 3.5. Every Manin triple of pre-Lie algebras (A, A+ , A− ) associated to a nondegenerate symmetric 2-cocycle B is isomorphic to a standard one. Proof. Since A− and (A+ )∗ are identified by the nondegenerate 2-cocycle B, we can transfer the pre-Lie algebra structure on A− to (A+ )∗ . Hence the pre-Lie algebra structure on A+ ⊕ A− can be transferred to A+ ⊕ (A+ )∗ . Therefore the conclusion follows. 2 Proposition-Definition 3.6. (See [1, Theorem 3.5].) Let (A1 , ·1 ) and (A2 , ·2 ) be two pre-Lie algebras. Suppose there are linear maps l1 , r1 : A1 → gl(A2 ) and l2 , r2 : A2 → gl(A1 ) such that (A2 , l1 , r1 ) is a bimodule of A1 and (A1 , l2 , r2 ) is a bimodule of A2 and they satisfy r1 (x) [a, b]2 = r1 l2 (b)x a − r1 l2 (a)x b + a ·2 r2 (x)b − b ·2 r1 (x)a , l1 (x)(a ·2 b) = −l1 l2 (a)x − r2 (a)x b + l1 (x)a − r1 (x)a ·2 b + r1 r2 (b)x a + a ·2 l1 (x)b , r2 (a) [x, y]1 = r2 l1 (y)a x − r2 l1 (x)a y + x ·1 r1 (a)y − y ·1 r2 (a)x , l2 (a)(x ·1 y) = −l2 l1 (x)a − r1 (x)a y + l2 (a)x − r2 (a)x ·1 y + r2 r1 (y)a x + x ·1 l2 (a)y ,
(14)
(15) (16)
(17)
where x, y ∈ A1 , a, b ∈ A2 and [,]i is the sub-adjacent Lie bracket of (A, ·i ) (i = 1, 2). Then there is a pre-Lie algebra structure “ ·” on the vector space A1 ⊕ A2 given by (x + a) · (y + b) := x ·1 y + l2 (a)y + r2 (b)x + a ·2 b + l1 (x)b + r1 (y)a , ∀x, y ∈ A1 , a, b ∈ A2 .
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2 We denote this pre-Lie algebra by A1 ll21 ,r ,r1 A2 or simply A1 A2 . Moreover, (A1 , A2 , l1 , r1 , l2 , r2 ) satisfying the above conditions is called a matched pair of pre-Lie algebras. On the other hand, every pre-Lie algebra which is a direct sum of the underlying vector spaces of two subalgebras can be obtained from the above way.
Proposition 3.7. Let (A, ≺1 , 1 ) be an L-dendriform algebra. Suppose that there is an L-dendriform algebra structure (A∗ , ≺2 , 2 ) on the dual space A∗ . Then (Ah ⊕(A∗ )h , Ah , (A∗ )h , Bp ) is a standard Manin triple of pre-Lie algebras associated to a nondegenerate ∗ ∗ , L∗◦2 , −R≺ ) is a matched pair symmetric 2-cocycle if and only if (Ah , (A∗ )h , L∗◦1 , −R≺ 1 2 of pre-Lie algebras.
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∗ Proof. First note that by Proposition 2.5, (A∗ , L∗◦1 , −R≺ ) is a bimodule of Ah and 1 ∗ ∗ ∗ (A, L◦2 , −R≺2 ) is a bimodule of (A )h . ∗ ∗ , L∗◦2 , −R≺ ) is a matched pair of pre-Lie algebras, then it is If (A, A∗ , L∗◦1 , −R≺ 1 2 straightforward to show that the bilinear form Bp defined by Eq. (13) is a 2-cocycle L∗ ,−R∗
1 of the pre-Lie algebra A L◦∗◦1 ,−R≺ A∗ . Moreover, Ah and (A∗ )h are Lagrangian subal∗ ≺ 2
2
L∗ ,−R∗
1 gebras of A L◦∗◦1 ,−R≺ A∗ . Thus, (Ah ⊕ (A∗ )h , Ah , (A∗ )h , Bp ) is a standard Manin triple ∗ ≺2 2 of pre-Lie algebras associated to a nondegenerate symmetric 2-cocycle. Conversely, suppose that (Ah ⊕ (A∗ )h , Ah , (A∗ )h , Bp ) is a standard Manin triple of pre-Lie algebras associated to a nondegenerate symmetric 2-cocycle. By Proposition 3.3, Ah and (A∗ )h are subalgebras of Ah ⊕ (A∗ )h . Set
x ∗ a∗ = l·1 (x)a∗ + r·2 a∗ x,
a∗ ∗ x = l·2 a∗ x + r·1 (x)a∗ ,
∀x ∈ A, a∗ ∈ A∗ .
Then (A, A∗ , l1 , r1 , l2 , r2 ) is a matched pair of pre-Lie algebras. Note that ∗ (x)a∗ , y , r·1 (x)a∗ , y = a∗ · x, y = Bp y, a∗ · x = Bp y ≺1 x, a∗ = −R≺ 1 ∗ ∗ ∗
l·2 a x, b = a · x, b∗ = a∗ , x + x · a∗ , b∗ = −Bp b∗ , x, a∗ + Bp b∗ , x · a∗ = −Bp a∗ 2 b∗ , x + Bp b∗ ≺2 a∗ , x = b∗ , L∗◦2 a∗ x ,
∗ for any x, y ∈ A, a∗ , b∗ ∈ A∗ . Hence, r·1 = −R≺ , l·2 = L∗◦2 . Similarly, l·1 = L◦1 , r·2 = 1 ∗ −R≺2 . 2
Definition 3.8. (a) Let V be a vector space. Define the exchanging operator τ : V ⊗ V → V ⊗ V as τ (x ⊗ y) := y ⊗ x,
∀x, y ∈ V.
(19)
(b) Let V1 , V2 be two vector spaces and T : V1 → V2 be a linear map. Denote the dual (linear) map by T ∗ : V2∗ → V1∗ defined by
v1 , T ∗ v2∗ = T (v1 ), v2∗ ,
∀v1 ∈ V1 , v2∗ ∈ V2∗ .
(20)
Note the difference between T ∗ and the linear maps defined by Eq. (11). Proposition 3.9. Let (A, ≺, , α, β) be an L-dendriform algebra equipped with two linear maps α, β : A → A ⊗ A. Suppose that α∗ , β ∗ : A∗ ⊗ A∗ → A∗ induce an L-dendriform ∗ ∗ , L∗◦∗ , −R≺ ) is algebra structure on A∗ denoted by “≺∗ , ∗ ”. Then (Ah , (A∗ )h , L∗◦ , −R≺ ∗ a matched pair of pre-Lie algebra if and only if the following equations are satisfied (for any x, y ∈ A):
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(α + β − τ α − τ β)(x ≺ y) = L≺ (x) ⊗ 1 (α − τ β)(y) + 1 ⊗ L≺ (x) (−τ α + β)(y) + 1 ⊗ R≺ (y) (α + β)(x) − R≺ (y) ⊗ 1 (τ α + τ β)(x),
(α + β)(x ◦ y) = 1 ⊗ R◦ (y) β(x) − L≺ (y) ⊗ 1 α(x) + L (x) ⊗ 1 + 1 ⊗ L◦ (x) (α + β)(y), α [x, y] = −L· (y) ⊗ 1 − 1 ⊗ L◦ (y) α(x) + L· (x) ⊗ 1 + 1 ⊗ L◦ (x) α(y), (τ α − β)(x · y) = L (x) ⊗ 1 + 1 ⊗ L· (x) (τ α − β)(y) − 1 ⊗ R· (y) β(x) + R≺ (y) ⊗ 1 τ α(x).
(21)
(22)
(23)
(24)
Proof. By Proposition-Definition 3.6, we need to prove that Eqs. (14)–(17) are equivalent to Eqs. (21)–(24) respectively in the case that A 1 = Ah , l1 = L∗◦ ,
A2 = A∗ h ,
∗ r1 = −R≺ ,
l2 = L∗◦∗ ,
and ∗ r2 = −R≺ . ∗
As an example we give an explicit proof that Eq. (24) holds if and only if Eq. (17) holds. The other equivalences are similar. In fact, in this case, Eq. (17) becomes ∗ ∗ ∗ L∗◦∗ a∗ (x · y) = −L∗◦∗ L∗◦ (x)a∗ + R≺ (x)a∗ y + L∗◦∗ a∗ x + R≺ a x ·y ∗ ∗ ∗ ∗ ∗ ∗ + R≺∗ R≺ (y)a x + x · L◦∗ a y , ∀x, y ∈ A, a∗ ∈ A∗ . Let both sides of the above equation act on an arbitrary element b∗ ∈ A∗ . Then
x · y, −a∗ ◦∗ b∗ = y, L∗ (x)a∗ ◦∗ b∗ + x, a∗ ∗ R·∗ (y)b∗ ∗ − x, b∗ ≺∗ R≺ (y)a∗ + y, a∗ ◦∗ L∗· (x)b∗ .
It is equivalent to the following equation
(τ α − β)(x · y), a∗ ⊗ b∗ = L (x) ⊗ 1 + 1 ⊗ L· (x) (τ α − β)(y) − 1 ⊗ R· (y) β(x) + R≺ (y) ⊗ 1 τ α(x), a∗ ⊗ b∗ ,
which exactly gives Eq. (24).
2
Definition 3.10. (a) Let (A, α, β) be a vector space with two linear maps α, β : A → A⊗A. If α∗ , β ∗ : A∗ ⊗ A∗ → A∗ induce an L-dendriform algebra structure on A∗ , then we call (A, α, β) an L-dendriform coalgebra.
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(b) Let (A, ≺, ) be an L-dendriform algebra. If there are two linear maps α, β : A → A⊗ A such that (A, α, β) is an L-dendriform coalgebra and α and β satisfy Eq. (21)–(24), then the set of the linear maps α, β is called an L-dendriform bialgebra structure on (A, ≺, ). We denote it by (A, ≺, , α, β). By Propositions 3.7 and 3.9 we have the following result which gives a structure theory of a double construction of pseudo-Hessian Lie algebra. Theorem 3.11. Let (A, ≺, ) be an L-dendriform algebra. Let α, β : A → A ⊗ A be two linear maps such that α∗ , β ∗ : A∗ ⊗ A∗ → A∗ induce an L-dendriform algebra structure on A∗ , that is, (A, α, β) is an L-dendriform coalgebra. Then the following conditions are equivalent: (a) (Ah ⊕ (A∗ )h , Ah , (A∗ )h , Bp ) is a standard Manin triple of pre-Lie algebras associated to a nondegenerate symmetric 2-cocycle, i.e., g(Ah ⊕ (A∗ )h ) is a double construction of pseudo-Hessian Lie algebra. (b) (A, ≺, , α, β) is an L-dendriform bialgebra. 4. Triangular pseudo-Hessian Lie algebras In this section we study some double constructions of pseudo-Hessian Lie algebras arising from some skew-symmetric tensors. We also relate them to some semidirect product constructions from the “Lie-type” operations of L-dendriform algebras. 4.1. Triangular pseudo-Hessian Lie algebras Lemma 4.1. Let (A, ≺, ) be an L-dendriform algebra and r ∈ A⊗A. Let α, β : A → A⊗A be two linear maps defined by α(x) := 1 ⊗ L◦ (x) + L· (x) ⊗ 1 r, (25) β(x) := −1 ⊗ ad(x) − L (x) ⊗ 1 r, (26) respectively, where x ∈ A. Then α and β satisfy Eqs. (21)–(24) if and only if the following equations are satisfied: R◦ (x ≺ y) ⊗ 1 + 1 ⊗ L≺ (x ≺ y) + L≺ (x) ⊗ 1 1 ⊗ L (y) + ad(y) ⊗ 1 − R≺ (y) ⊗ 1 R◦ (x) ⊗ 1 − 1 ⊗ L≺ (x) 1 ⊗ L· (y) + L (y) ⊗ 1 r + τ (r) = 0,
(27) L◦ (x · y) ⊗ 1 + 1 ⊗ L· (x · y) − L (x) ⊗ 1 L◦ (y) ⊗ 1 + 1 ⊗ L· (y) − 1 ⊗ L· (x) 1 ⊗ L· (y) + L (y) ⊗ 1 − R≺ (y) ⊗ 1 L◦ (x) ⊗ 1 r + τ (r) = 0, (28)
where x, y ∈ A.
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Proof. Obviously α and β satisfy Eqs. (22) and (23) automatically. Moreover, substituting Eqs. (25) and (26) into Eq. (21), we have
1 ⊗ R◦ (x ≺ y) + L≺ (x ≺ y) ⊗ 1 r − R◦ (x ≺ y) ⊗ 1 + 1 ⊗ L≺ (x ≺ y) τ (r) = L≺ (x) ⊗ 1 1 ⊗ L◦ (y) + L· (y) ⊗ 1 r + L≺ (x) ⊗ 1 ad(y) ⊗ 1 + 1 ⊗ L (y) τ (r) − 1 ⊗ L≺ (x) L◦ (y) ⊗ 1 + 1 ⊗ L· (y) τ (r) − 1 ⊗ L≺ (x) 1 ⊗ ad(y) + L (y) ⊗ 1 r + 1 ⊗ R≺ (y) 1 ⊗ R◦ (x) + L≺ (x) ⊗ 1 r − R≺ (y) ⊗ 1 R◦ (x) ⊗ 1 + 1 ⊗ L≺ (x) τ (r),
for any x, y ∈ A. It is equivalent to the following equation 1 ⊗ R◦ (x ≺ y) + L≺ (x ≺ y) ⊗ 1 + R◦ (x ≺ y) ⊗ 1 + 1 ⊗ L≺ (x ≺ y) r − R◦ (x ≺ y) ⊗ 1 + 1 ⊗ L≺ (x ≺ y) r + τ (r) = L≺ (x) ⊗ 1 −ad(y) ⊗ 1 − 1 ⊗ L (y) + 1 ⊗ L◦ (y) + L· (y) ⊗ 1 r + L≺ (x) ⊗ 1 ad(y) ⊗ 1 + 1 ⊗ L (y) r + τ (r) + 1 ⊗ L≺ (x) −1 ⊗ ad(y) − L (y) ⊗ 1 + L◦ (y) ⊗ 1 + 1 ⊗ L· (y) r − 1 ⊗ L≺ (x) L◦ (y) ⊗ 1 + 1 ⊗ L· (y) r + τ (r) + 1 ⊗ R≺ (y) 1 ⊗ R◦ (x) + L≺ (x) ⊗ 1 r + R≺ (y) ⊗ 1 R◦ (x) ⊗ 1 + 1 ⊗ L≺ (x) r − R≺ (y) ⊗ 1 R◦ (x) ⊗ 1 + 1 ⊗ L≺ (x) r + τ (r) . It is easy to show that the above equation holds if and only if Eq. (27) holds. So α and β satisfy Eq. (21) if and only if Eq. (27) holds. Similarly, α and β satisfy Eq. (24) if and only if Eq. (28) holds. 2 Lemma 4.2. Let V be a vector space and α, β : V → V ⊗ V be two linear maps. Then (A, α, β) is an L-dendriform coalgebra if and only if both R1 and R2 are zero, where Ri : V → V ⊗ V ⊗ V (i = 1, 2) are defined as follows: R1 (x) := (α ⊗ id)α(x) + (τ ⊗ id)(id ⊗ α)β(x) − id ⊗ (α + β) α(x) − (τ ⊗ id)(β ⊗ id)α(x), R2 (x) := (α + β) ⊗ id β(x) + (τ ⊗ id)(id ⊗ β)β(x) − (id ⊗ β)β(x) − (τ ⊗ id) (α + β) ⊗ id β(x), for any x ∈ V . Proof. It follows immediately from the definition of an L-dendriform algebra. 2
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Definition 4.3. Let V be a vector space and r = r12 =
ai ⊗ bi ⊗ 1,
i
r13 =
i
r23 =
283
ai ⊗ bi ∈ V ⊗ V . Set
ai ⊗ 1 ⊗ bi ,
i
1 ⊗ ai ⊗ bi ,
i
r21 =
bi ⊗ ai ⊗ 1,
(31)
i
where 1 is a symbol playing a similar role of the unit. If in addition, there exists a binary operation : V ⊗ V → V on V , then the operation between two rs is in an obvious way. For example, r12 r13 =
ai aj ⊗ bi ⊗ bj ,
i,j
r23 r12 =
r13 r23 =
ai ⊗ aj ⊗ bi bj ,
i,j
aj ⊗ ai bj ⊗ bi .
(32)
i,j
Note that the above equation is independent of the existence of a unit. Lemma 4.4. Let (A, ≺, ) be an L-dendriform algebra and r = i ai ⊗bi ∈ A ⊗ A. Define α, β : A → A ⊗ A by Eqs. (25) and (26) respectively. Let ad be the adjoint action, i.e., ad(x)y = [x, y], for any x, y ∈ A. Then we have that R1 (x) = 1 ⊗ 1 ⊗ L◦ (x) Jr, rKQ1 + 1 ⊗ L (x) ⊗ 1 + L· (x) ⊗ 1 ⊗ 1 Jr, rKQ2 + P1 (x, aj ) r + τ (r) ⊗ bj ,
(33)
j
R2 (x) = 1 ⊗ 1 ⊗ ad(x) Jr, rKQ3 + 1 ⊗ L (x) ⊗ 1 + L (x) ⊗ 1 ⊗ 1 Jr, rKQ4 + P2 (x, aj ) r + τ (r) ⊗ bj ,
(34)
j
where Jr, rKQ1 := r23 ◦ r12 + r13 · r12 − r23 ◦ r13 + [r13 , r21 ] + r23 r21 ,
(35)
Jr, rKQ2 := r12 · r13 − r12 ≺ r23 − r23 ◦ r13 ,
(36)
Jr, rKQ3 := −r12 ◦ r23 − r13 ≺ r12 + r21 ◦ r13 + r23 ≺ r21 + [r23 , r13 ],
(37)
Jr, rKQ4 := [r23 , r13 ] − r12 r23 − r12 r13 , (38) P1 (x, y) := 1 ⊗ L (x · y) + ad(x · y) ⊗ 1 − R· (y) ⊗ 1 ad(x) ⊗ 1 + 1 ⊗ L (x) , (39) P2 (x, y) := 1 ⊗ L≺ (x y) + R◦ (x y) ⊗ 1 + R (y) ⊗ 1 1 ⊗ L (x) + ad(x) ⊗ 1 . (40)
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Proof. We give an explicit proof of Eq. (33) as an example. The proof of Eq. (34) is similar. After rearranging the terms suitably, we divide R1 into the following four parts: R1 (x) = J1 (x) + J2 (x) + J3 (x) + J4 (x), x ∈ A, where J1 (x) = ai ⊗ (x · aj ) ◦ bi − ai ⊗ (x ◦ bi ) · aj + ai ⊗ (x ◦ bi ) aj + (x · aj ) · ai ⊗ bi i,j
− x · ai ⊗ bi · aj + x · ai ⊗ bi aj + bi ⊗ (x · aj ) ai + [x · aj , bi ] ⊗ ai − [x, bi ] · aj ⊗ ai − bi · aj ⊗ x ai ⊗ bj , J2 (x) = ai ⊗ aj ◦ bi ⊗ x ◦ bj + aj · ai ⊗ bi ⊗ x ◦ bj i,j
+ [aj , bi ] ⊗ ai ⊗ x ◦ bj + bi ⊗ aj ai ⊗ x ◦ bj , J3 (x) = −ai ⊗ aj ⊗ [x, bj ] ◦ bi − aj ⊗ ai ⊗ (x ◦ bj ) ◦ bi + aj ⊗ ai ⊗ [x ◦ bj , bi ], i,j
J4 (x) =
−ai ⊗ x aj ⊗ bj ◦ bi − x · aj ⊗ ai ⊗ bj ◦ bi + x · aj ⊗ ai ⊗ [bj , bi ].
i,j
On the other hand, the right hand side of Eq. (33) is
ai ⊗ aj ◦ bi ⊗ x ◦ bj + aj · ai ⊗ bi ⊗ x ◦ bj − ai ⊗ aj ⊗ x ◦ (bj ◦ bi )
i,j
+ [aj , bi ] ⊗ ai ⊗ x ◦ bj + bi ⊗ aj ai ⊗ x ◦ bj + ai · aj ⊗ x bi ⊗ bj − ai ⊗ x (bi ≺ aj ) ⊗ bj − aj ⊗ (x ai ) ⊗ bi ◦ bj + x · (ai · aj ) ⊗ bi ⊗ bj − x · ai ⊗ bi ≺ aj ⊗ bj − x · aj ⊗ ai ⊗ bi ◦ bj + ai ⊗ (x · aj ) bi ⊗ bj + bi ⊗ (x · aj ) ai ⊗ bj + [x · aj , ai ] ⊗ bi ⊗ bj + [x · aj , bi ] ⊗ ai ⊗ bj − [x, ai ] · aj ⊗ bi ⊗ bj − [x, bi ] · aj ⊗ ai ⊗ bj − ai · aj ⊗ x bi ⊗ bj − bi · aj ⊗ x ai ⊗ bj . After rearranging the terms suitably, the sum of terms whose third component is bj in the right hand side of Eq. (33) is J˜1 (x) = Ja (x) + Jb (x) + Jc (x) + Jd (x) + Je (x) + Jf (x) + Jg (x) + Jh (x), Ja (x) = {ai · aj ⊗ x bi − ai · aj ⊗ x bi } ⊗ bj = 0, i,j
Jb (x) =
−ai ⊗ x (bi ≺ aj ) + ai ⊗ (x · aj ) bi ⊗ bj
i,j
=
ai ⊗ (x · aj ) ◦ bi − ai ⊗ (x ◦ bi ) · aj + ai ⊗ (x ◦ bi ) aj ⊗ bj ,
i,j
where
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Jc (x) =
x · (ai · aj ) + [x · aj , ai ] − [x, ai ] · aj ⊗ bi ⊗ bj = (x · aj ) · ai ⊗ bi ⊗ bj ,
i,j
Jd (x) =
−x · ai ⊗ bi ≺ aj ⊗ bj ,
Je (x) =
i,j
Jf (x) =
[x · aj , bi ] ⊗ ai ⊗ bj , i,j
−[x, bi ] · aj ⊗ ai ⊗ bj ,
i,j
bi ⊗ (x · aj ) ai ⊗ bj ,
i,j
i,j
Jg (x) =
285
Jh (x) =
−bi · aj ⊗ x ai ⊗ bj .
i,j
So J˜1 (x) = J1 (x). Similarly, the sum of whose third component is x ◦ bj in the right hand side of Eq. (33) is J2 (x). Moreover, the sum of the other terms in the right hand side of Eq. (33) is J3 (x) + J4 (x). So the conclusion follows. 2 Definition 4.5. (See [3].) Let (A, ≺, ) be an L-dendriform algebra and r ∈ A ⊗ A. The following equation is called LD-equation in (A, ≺, ): r13 ◦ r23 + r12 · r23 − r12 ≺ r13 = 0.
(41)
Let V be a vector space. Recall that for any r ∈ V ⊗ V , if r = −τ (r), then it is called skew-symmetric, where τ is the exchanging operator defined by Eq. (19). Lemma 4.6. Let (A, ≺, ) be an L-dendriform algebra and let r ∈ A ⊗ A be a skewsymmetric solution of LD-equation. Then we have Jr, rKQi = 0, where Jr, rKQi are defined by Eqs. (35)–(38), i = 1, 2, 3, 4. Proof. By Lemma 4.6 in [3], we know that if r is a skew-symmetric solution of LD-equation, then Jr, rKQ2 = 0 and the following two equations hold: r13 · r23 + r12 ◦ r23 + r13 ≺ r12 = 0.
(42)
r23 · r13 − r12 ◦ r13 − r23 ≺ r12 = 0.
(43)
Thus Jr, rKQ4 = 0. Also note that if r is skew-symmetric, then we have Jr, rKQ1 = Jr, rKQ2 = 0 and Jr, rKQ3 = Jr, rKQ4 = 0. So the conclusion holds. 2 Theorem 4.7. Let (A, ≺, ) be an L-dendriform algebra and let r ∈ A ⊗ A be a skewsymmetric solution of LD-equation. Define linear maps α, β : A → A ⊗ A by Eqs. (25) and (26) respectively. Then (A, ≺, , α, β) is an L-dendriform bialgebra. In this case, g(Ah ⊕ (A∗ )h ) becomes a double construction of pseudo-Hessian Lie algebra. Proof. By Lemmas 4.4 and 4.6, R1 and R2 , which are given by Eqs. (33) and (34) respectively, both are zero. So by Lemma 4.2, (A, α, β) is an L-dendriform coalgebra. On the other hand, since r is skew-symmetric, by Lemma 4.1, Eqs. (21)–(24) hold.
286
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So (A, ≺, , α, β) is an L-dendriform bialgebra. The last conclusion follows from Theorem 3.11. 2 Definition 4.8. Let (A, ≺, ) be an L-dendriform algebra and let r ∈ A ⊗ A be a skewsymmetric solution of LD-equation. Then the pseudo-Hessian Lie algebra g(Ah ⊕ (A∗ )h ) given by Theorem 4.7 is called triangular. The terminology “triangular” is motivated by quantum group theory [5]. 4.2. Lie-type operations of L-dendriform algebras and semidirect product constructions The second axiom defining an L-dendriform algebra (A, ≺, ), i.e., Eq. (5), can be re-written as [x, y] z = x (y z) − y (x z),
∀x, y, z ∈ A.
(44)
Eq. (44) means that L : g(A) → gl(A) is a representation of g(A). In this case we can form the semidirect product h := g(A) L∗ A∗ ,
(45)
which is the Lie algebra with the underlying vector space A ⊕ A∗ and the following Lie bracket:
x, a∗ , y, b∗ := [x, y], L∗ (x)b∗ − L∗ (y)a∗ ,
∀x, y ∈ A, a∗ , b∗ ∈ A∗ .
(46)
4.3. An isomorphic theorem Definition 4.9. Let V be a vector space. Then any r ∈ V ⊗ V can be identified as a linear map Tr : V ∗ → V in the following way:
u∗ ⊗ v ∗ , r = u∗ , Tr v ∗ ,
∀u∗ , v ∗ ∈ V ∗ .
(47)
Lemma 4.10. Let (A, ≺, ) be an L-dendriform algebra and r ∈ A⊗A be a skew-symmetric solution of LD-equation in (A, ≺, ). Let g := g(Ah ⊕ (A∗ )h ) be the corresponding triangular pseudo-Hessian Lie algebra. (a) The L-dendriform algebra structure on the dual space A∗ is given as follows: a∗ ≺ b∗ = R◦∗ Tr a∗ b∗ − R·∗ Tr b∗ a∗ , ∀a∗ , b∗ ∈ A∗ ; ∗ ∗ ∗ Tr b a , ∀a∗ , b∗ ∈ A∗ . a∗ b∗ = ad∗ Tr a∗ b∗ + R
(48) (49)
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287
(b) The following equations hold:
a∗ , b∗ = L∗ Tr a∗ b∗ − L∗ Tr b∗ a∗ , ∀a∗ , b∗ ∈ A∗ ;
∗ ∗
Tr a , Tr b = Tr a∗ , b∗ , ∀a∗ , b∗ ∈ A∗ .
(50) (51)
(c) The following equation in g holds:
Tr L∗ (x)a∗ − L∗ a∗ x = x, Tr a∗ ,
∀x ∈ A, a∗ ∈ A∗ .
(52)
Proof. (a) Let {e1 , . . . , en } be a basis of A and {e∗1 , . . . , e∗n } be the dual basis. Suppose that ei ≺ ej =
ckij ek ,
ei ej =
k
Then we have that aij = −aji and Tr (e∗i ) = e∗k ≺ e∗l =
s
=
dkij ek ,
r=
aij ei ⊗ ej .
i,j
k
k
aki ek . Thus, for any k, l, we have
e∗k ⊗ e∗l , 1 ⊗ L◦ (es ) + L· (es ) ⊗ 1 r e∗s
akt dlst − akt clts + ckst + dkst atl e∗s = R◦∗ Tr e∗k e∗l − R·∗ Tr e∗l e∗k .
s,t
So Eq. (48) holds. Eq. (49) is proved in a similar way. (b) By Eqs. (48) and (49), Eq. (50) holds. On the other hand, r is a solution of LD-equation if and only if, for any i, j, k, the following equation holds:
ail atk cjlt + djlt + ail ajt dklt − cktl − alj atk cilt = 0.
t,l
The left hand side of the above equation is precisely the coefficient of ej in ∗ ∗ ∗ −Tr e∗i · Tr e∗k + Tr L∗◦ Tr e∗i e∗k − Tr R≺ T r ek e i . Hence for any a∗ , b∗ ∈ A∗ we have that ∗ ∗
Tr a , Tr b = Tr L∗ Tr a∗ b∗ − Tr L∗ Tr b∗ a∗ . So by Eq. (50), Eq. (51) holds. (c) For any x ∈ A, a∗ , b∗ ∈ A∗ we have that
∗ ∗ ∗ L∗ b∗ x, a∗ = x, −b∗ a∗ = x, −ad∗ Tr b∗ a∗ − R Tr a b ∗ ∗ ∗ ∗ ∗
= Tr b , x , a + x Tr a , b = Tr b , x + Tr L∗ (x)b∗ , a∗ .
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288
Here the second equality follows from Eq. (49) and the last equality follows from the fact that r is skew-symmetric. So Eq. (52) holds. 2 At the end of this paper, we show that a triangular pseudo-Hessian Lie algebra is isomorphic (as Lie algebras) to the semidirect product h defined by Eq. (45). Theorem 4.11. Let (A, ≺, , α, β) be an L-dendriform bialgebra and r ∈ A ⊗ A be a skew-symmetric solution of LD-equation. Let g := g(Ah ⊕ (A∗ )h ) be the corresponding triangular pseudo-Hessian Lie algebra. Then g is isomorphic to the semidirect product h := g(A) L∗ A∗ defined by Eq. (45). Proof. By Proposition 3.7 and Theorem 4.7, the Lie bracket of g is given as follows:
x, a∗ , y, b∗ = [x, y] + L∗ a∗ y − L∗ b∗ x, a∗ , b∗ + L∗ (x)b∗ − L∗ (y)a∗ , ∀x, y ∈ A, a∗ , b∗ ∈ A∗ .
For any x ∈ A, a∗ ∈ A∗ , define a linear map θ : g → h by θ x, a∗ := Tr a∗ + x, a∗ . Then we have
θ x, a∗ , θ y, b∗ = Tr a∗ + x, a∗ , Tr b∗ + y, b∗
= Tr a∗ + x, Tr b∗ + y , L∗ Tr a∗ + x b∗ − L∗ Tr b∗ + y a∗
= Tr a∗ , b∗ + L∗ (x)b∗ − L∗ (y)a∗ + [x, y]
+ L∗ a∗ y − L∗ b∗ x, a∗ , b∗ + L∗ (x)b∗ − L∗ (y)a∗ = θ x, a∗ , y, b∗ . The last equality follows from Lemma 4.10(b) and (c). θ is obviously bijective. So the conclusion follows. 2 Acknowledgments This work is supported in part by NSFC (10921061, 11271202, 11221091), NKBRPC (2006CB 805905) and SRFDP (200800550015, 20120031110022). References [1] C. Bai, Left-symmetric bialgebras and an analogue of the classical Yang–Baxter equation, Commun. Contemp. Math. 10 (2008) 221–260. [2] C. Bai, D. Hou, Z. Chen, On a class of Lie groups with a left-invariant flat pseudo-metric, Monatsh. Math. 164 (2011) 243–269.
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[3] C. Bai, L. Liu, X. Ni, Some results on L-dendriform algebras, J. Geom. Phys. 60 (2010) 940–950. [4] D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Cent. Eur. J. Math. 4 (2006) 323–357. [5] V. Chari, A. Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1994. [6] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. Math. 78 (1963) 267–288. [7] J.-L. Koszul, Domaines bornés homogènes et orbites de groupes de transformation affines, Bull. Soc. Math. France 89 (1961) 515–533. [8] B.A. Kupershmidt, On the nature of the Virasoro algebra, J. Nonlinear Math. Phys. 6 (1999) 222–245. [9] J. Milnor, Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976) 293–329. [10] H. Shima, Homogeneous Hessian manifolds, Ann. Inst. Fourier (Grenoble) 30 (1980) 91–128. [11] E.B. Vinberg, Convex homogeneous cones, Trans. Moscow Math. Soc. 12 (1963) 340–403.