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Introduction to Co-Split Lie Algebras

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Introduction to Co-Split Lie Algebras Algebr Represent Theor DOI 10.1007/s10468-009-9183-0 Introduction to Co-split Lie Algebras Limeng Xia · Naihong Hu Received: 9 September 2006 / Accepted: 3 July 2007 © Springer Science + Business Media B.V. 2009 Abstract In this work, we introduce a new concept which is obtained by defining a new compatibility condition between Lie algebras and Lie coalgebras. With this terminology, we describe the interrelation between the Killing form and the adjoint representation in a new perspective. Keywords Co-split Lie algebra · Lie coalgebra · Adjoint action Mathematics Subject Classification (2000) 17B62 1 Introduction During the past decade, there have appeared a number of papers on the study of Lie bialgebras (see [1–3] and references therein, etc). It is well-known that a Lie bialgebra is a vector space endowed simultaneously with a Lie algebra structure and a Lie coalgebra structure, together with a certain compatibility condition, which was suggested by a study of Hamiltonian mechanics and Poisson Lie groups [3]. In the present work, we consider a new [Lie algebra]–[Lie coalgebra] structure, say, a co-split Lie algebra. Using this concept, we can easily study the Lie algebra structure on the dual space of a semi-simple Lie algebra from another point of view. Presented by Claus Ringel. L. Xia Faculty of Science, Jiangsu University, Zhenjiang 212013, Jiangsu Province, People’s Republic of China N. Hu (B) Department of Mathematics, East China Normal University, Minhang Campus, Dongchuan Road 500, Shanghai 200241, People’s Republic of China e-mail: [email protected] L. Xia, N. Hu This paper is arranged as follows: At first we recall some concepts and study the relations between Lie algebras and Lie coalgebras. Then we give the definition of a co-split Lie algebra. In Section 4, we prove that sln+1(C) is a co-split Lie algebra. Then we discuss the interrelation of the Killing form and the adjoint representation of sln+1(C). Finally, the results are proved to hold for all finite dimensional complex semi-simple Lie algebras. 2Basics In this section, we mainly recall the definitions of Lie algebras, Lie coalgebras and Lie bialgebras, and also their relationship. For more information, one can see [1–3] and references therein. A Lie algebra is a pair (L, [, ]),whereL is a linear space and [, ]:L × L −→ L is a bilinear map (in fact, it is a linear map from L ⊗ L to L) satisfying: (L1) [a,b]+[b,a]=0, (L2) [a,[b,c]]+[b,[c,a]]+[c,[a,b]]=0. For any spaces U, V, W, define maps τ : U ⊗ V −→ V ⊗ U u ⊗ v −→ v ⊗ u, ξ : U ⊗ V ⊗ W −→ V ⊗ W ⊗ U u ⊗ v ⊗ w −→ v ⊗ w ⊗ u. A Lie coalgebra is a pair (L,δ),whereL is a linear space and δ : L −→ L ⊗ L is a linear map satisfying: (Lc1) (1 + τ)◦ δ = 0, (Lc2) (1 + ξ + ξ 2) ◦ (1 ⊗ δ) ◦ δ = 0. A Lie bialgebra is a triple (L, [, ],δ)such that (Lb1) (L, [, ]) is a Lie algebra, (Lb2) (L,δ)is a Lie coalgebra, (Lb3) For any x, y ∈ L, δ([x, y]) = x · δ(y) − y · δ(x). The compatibility condition (Lb3) shows that δ is a derivation map. In the following lemmas, c is an arbitrary constant. Lemma 2.1 For any finite dimensional Lie algebra (L, [, ]), the dual space L∗ has a Lie coalgebra structure defined by ∗ ∗ δL∗ ( f ) = f1 ⊗ f2 : x ⊗ y −→ f1(x) f2(y) = cf ([x, y]). ( f ) Lemma 2.2 For any finite dimensional Lie coalgebra (L,δ), the dual space L∗ has a Lie algebra structure defined by ∗ ∗ ∗ ∗ f , g : x −→ cf (x1)g (x2), Introduction to Co-split Lie Algebras δ( ) = ⊗ . where x (x) x1 x2 These two lemmas are natural conclusions and easy to be verified. 3WhatisaCo-splitLieAlgebra Definition 3.1 Suppose that (L, [·, ·]) is a Lie algebra and (L,δ) is a Lie coalgebra. Atriple(L, [·, ·],δ)is called a co-split Lie algebra if the compatibility condition [·, ·] ◦ δ = idL holds, and δ is called a co-splitting of L. If in the compatibility condition, idL is replaced by a non-degenerate diagonal matrix, then (L, [, ],δ) is called a weak co-split Lie algebra and δ is called a weak co-splitting . Remark 3.1 Obviously, a co-split Lie algebra L should satisfies [L, L]=L. Remark 3.2 If (L, [, ],δ) is a finite dimensional (weak) co-split Lie algebra, so is (L∗,δ∗, [, ]∗),where δ∗( f ⊗ g)(x) = ( f ⊗ g)δ(x), [, ]∗( f )(x ⊗ y) = f ([x, y]), for all x, y ∈ L and f, g ∈ L∗. This follows from the fact that V −→ V∗ is a con- travariant functor. 4 Co-split Lie Algebras of Type A Suppose that L is a complex simple Lie algebra of type An, then it can be realized as the special linear Lie algebra sln+1(C) with basis Ei, j, E j,i, Ei,i − E j, j | 1 ≤ i < j ≤ n + 1 . The Lie bracket is the commutator Ei, j, Ek,l = δ j,k Ei,l − δl,i Ek, j. Define a linear map δ : sln+1(C) −→ sln+1(C) ⊗ sln+1(C) as + 1 n 1 δ E , = E , ⊗ E , − E , ⊗ E , . i j 2n + 2 i k k j k j i k k=1 Proposition 4.1 δ is well-defined. L. Xia, N. Hu Proof Assume that i = j,then + 1 n 1 δ E , = E , ⊗ E , − E , ⊗ E , i j 2n + 2 i k k j k j i k k=1 1 = E , ⊗ E , − E , ⊗ E , 2n + 2 i k k j k j i k k =i, j 1 + E , − E , ⊗ E , − E , ⊗ E , − E , . 2n + 2 i i j j i j i j i i j j + 1 n 1 δ E , − E , = E , ⊗ E , − E , ⊗ E , i i j j 2n + 2 i k k i j k k j k=1 + 1 n 1 − E , ⊗ E , − E , ⊗ E , 2n + 2 k i i k k j j k k=1 1 = E , ⊗ E , − E , ⊗ E , 2n + 2 i k k i k i i k k =i 1 − E , ⊗ E , − E , ⊗ E , . 2n + 2 j k k j k j j k k = j Hence δ is well-defined. Theorem 4.1 (sln+1(C), δ) is a Lie coalgebra. Proof At first, it is clear that (1 + τ)◦ δ = 0. By a direct calculation, we have + 1 n 1 ((1 ⊗ δ) ◦ δ)(E , ) = (1 ⊗ δ) E , ⊗ E , − E , ⊗ E , i j 2n + 2 i k k j k j i k k=1 1 = E , ⊗ E , ⊗ E , − E , ⊗ E , ⊗ E , 4(n + 1)2 i k k l l j i k l j k l 1≤k,l≤n+1 1 − E , ⊗ E , ⊗ E , − E , ⊗ E , ⊗ E , 4(n + 1)2 k j i l l k k j l k i l 1≤k,l≤n+1 1 = E , ⊗ E , ⊗ E , − E , ⊗ E , ⊗ E , 4(n + 1)2 i k k l l j l j i k k l 1≤k,l≤n+1 1 − E , ⊗ E , ⊗ E , − E , ⊗ E , ⊗ E , . 4(n + 1)2 i l k j l k k j l k i l 1≤k,l≤n+1 Hence, 1 + ξ + ξ 2 ◦ (1 ⊗ δ) ◦ δ = 0, Introduction to Co-split Lie Algebras that is, δ satisfies the anti-symmetry property and the Jacobi identity. Then (sln+1 (C), δ) is a Lie coalgebra. Theorem 4.2 (sln+1(C), [·, ·],δ)is a co-split Lie algebra. Proof For i = j, it is easy to check that + 1 n 1 ([·, ·] ◦ δ) E , = [E , , E , ]−[E , , E , ] i j 2n + 2 i k k j k j i k k=1 = Ei, j, + 1 n 1 ([·, ·] ◦ δ) E , − E , = [E , , E , ]−[E , , E , ] i i j j 2n + 2 i k k i k i i k k=1 + 1 n 1 − [E , , E , ]−[E , , E , ] 2n + 2 j k k j k j j k k=1 = Ei,i − E j, j, that is, [·, ·] ◦ δ = id, also by Theorem 4.1. So, the theorem holds. 5 Dual Lie Algebras, Killing Form and Adjoint Representation In this section, we discuss the interrelation of the Killing form and the adjoint representation for the Lie algebra of type A within our new terminology. ∗ ∗ Theorem 5.1 ((sln) , −2nδ ) is a Lie algebra isomorphic to sln, the isomorphism is given by B : fi, j −→ E j,i, ∗ ∗ where { fi, j | 1 ≤ i, j ≤ n} forms a basis of (gln) ⊃ (sln) ,and fi, j(Ek,l) = δi,kδ j,l. Proof By definition, we have n ∗ −2nδ ( fi, j ⊗ fk,l)(Es,t) =− fi, j(Es,r) fk,l(Er,t) − fi, j(Er,t) fk,l(Es,r) r=1 =−δ j,k( fi, j(Es, j) f j,l(E j,t) + δi,l fi, j(Ei,t) fk,i(Es,i) =−(δ j,k fi,l − δi,l fk, j)(Es,t), ∗ ∗ then (sln) is a Lie algebra under bracket −2nδ ,andB is an isomorphism. −1 Define a bilinear form (, )B : sln × sln −→ C as (x, y)B = B (x)(y). L. Xia, N. Hu Theorem 5.2 (, )B is just a non-zero scalar of the Killing form. Proof This result is direct. Now we can consider the following maps: ⊗ −1 η 2nδ idsln B / / ∗ +3 sln sln ⊗ sln sln ⊗ (sln) End(sln) where η is an isomorphism, η(x ⊗ f )(y) = f (y)x. Theorem 5.3 For the adjoint representation ad : sln −→ End(sln), we have = η ◦ ( ⊗ −1) ◦ δ.
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