Abstract and Applied Analysis

Volume 2013 (2013), Article ID 803540, 7 pages

http://dx.doi.org/10.1155/2013/803540

Research Article

Post-Lie Algebra Structures on the Lie Algebra

Yuqiu Sheng and Xiaomin Tang

Department of Mathematics, Heilongjiang University, Harbin 150080, China

Received 1 August 2013; Accepted 15 September 2013

Academic Editor: Teoman Özer

Copyright © 2013 Yuqiu Sheng and Xiaomin Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The post-Lie algebra is an enriched structure of the Lie algebra. We give a complete classification of post-Lie algebra structures on the Lie algebra up to isomorphism.

1. Introduction

Post-Lie algebras were introduced around 2007 by Vallette [1], who found the structure in a purely operadic manner as the Koszul dual of a commutative trialgebra. Moreover, Vallette [1] proves that post-Lie algebras have the important algebraic property of being Koszul. This property is shared by many other important algebras, such as Lie algebras, associative algebras, commutative algebras, pre-Lie algebras, LR-algebras, and dendriform algebras, see [2, 3]. Recently, many authors study some post-Lie algebras and post-Lie algebra structures [4–8]. We recall the definition of the post-Lie algebra (structure) as follows, see [1, 8].

Definition 1. A (left) post-Lie -algebra is a -vector space with two binary operations and which satisfy the following relations:

Let denote the Lie algebra defined by (1) and (2). Call a post-Lie algebra on .

Remark 2. Suppose is a Lie algebra. Two post-Lie algebra and on the Lie algebra are called isomorphic on the Lie algebra if there is an automorphism of the Lie algebra such that

One of the key problems in the study of post-Lie algebras is to find the post-Lie algebra structures on the (given) Lie algebras. In [8], the authors determined all isomorphic classes of post-Lie algebra structures on , the special linear Lie algebra of order . They use an important fact that the derivation of a semisimple Lie algebra is inner. But for the nonsemisimple Lie algebra, this fact dos not hold. So that we must find another way to study such problem for nonsemisimple Lie algebra. The purpose of this paper is to give a complete classification of post-Lie algebra structures on nonsemisimple Lie algebra , the general linear Lie algebra of order , up to isomorphism. Now, we recall the above two Lie algebras.

Denote

It is obvious that the previous four matrices form a -linear basis of and determine the Lie algebra through the Lie product

It is also well known that form a -linear basis of and determine the Lie algebra through the relations (7). The authors in [8] got the following classification theorem, which will be used in our proof.

Theorem 3 (see [8]). The following is a complete set of representatives for the isomorphic classes of post-Lie algebra on the Lie algebra .

(1); (2); (3), ;(4), , ;(5), , .

2. Equations from Post-Lie Algebras

From (4), we obtain and . Similarly, we have for any and . Thus, we get the following.

Lemma 4. is a post-Lie subalgebra of .

Proposition 5. Let and be post-Lie algebras on the Lie algebra , they are isomorphic through automorphism of the Lie algebra . Then, is a -subspace of , and is an isomorphism from to .

Proof. Suppose is the matrix of with respect to the basis ; that is,

From , we obtain . Moreover, means . Similarly, we have . Now, we see that has the form

From this we can easily get the conclusion.

Proposition 6. Let be post-Lie algebra on the Lie algebra .

(1) There exists a linear map such that

(2) There exist such that , . (3) There exist such that

Proof. (1) The conclusion is given by [8].

(2) For any , we have from (4) and (7). Thus, is in the center of , and so , where is a linear map from to . Let . The conclusion of is proved.

(3) Let , , and in (4), one can get the conclusion by a simple computation.

Definition 7. Suppose is the matrix of (from Proposition 6) with respect to the basis ; that is, . Denote

The matrix set is unique for a given . On the other hand, is defined uniquely by the matrix set . Because of their uniqueness, the matrix set is called the matrix set of the post-Lie algebra and is also denoted by .

Proposition 8. Suppose that and are the matrix sets of post-Lie algebras and , respectively. is an isomorphic map from to . Then,

(1)

 where , the group of all complex orthogonal matrices whose determinants are ;(2);(3);(4).

Proof. (1) The conclusion is given by Proposition 5 and [8].

(2) It is given by [8].

(3) Let and . Note that

So, we have by that , that is, .

(4) Since

we deduce by that , the proof is completed.

Proposition 9. Suppose that is the matrix set of post-Lie algebra , then we have the following equations:

Proof. We consider (3) and (10). Let and , and , respectively, we get (17); Let , , and , respectively, we get (18); let and , , , , , and , respectively, we get (19).

3. Classification of Post-Lie Algebra

Lemma 10 (see [9]). Suppose that is a complex skewsymmetric matrix, there exists such that

Based on Definition 7, we get the main result in this paper as follows.

Theorem 11. The following is a complete set of matrix sets of representatives for the isomorphic classes of post-Lie algebra on the Lie algebra

Proof. Case  1. . By (18), we obtain ; that is, . By Lemma 10 we have

Case  1.1. Consider

If , let

then

In view of Proposition 8, we can suppose that

Case  1.2. Consider . If , let

then

Thus, we can suppose that or .

Case  2. Consider . By Lemma 10 we have

By (18), we obtain . In a similar way with the proof of Case 1, we can suppose that or .

Case  3. Consider

By (18), we obtain . Then, by (19), we get

Therefore, by (17), we can have

Hence, . If , then . In a similar way with the proof of Case 1, we can suppose that . Else, if , then .

Case  4. Consider

By (18), we obtain

Case  4.1. When , . By (17), we obtain . Then, by (19), we get .

If , then .

Therefore,

In a similar way with the proof of Case 1, we can suppose that or .

Case  4.2. When , . By (19), we obtain

This, together with (17), implies that .

If , then . Hence, in a similar way with the proof of Case 1, we can suppose that

If and , then

If and , then . Hence,

Let

then

Thus, we can suppose that.

Case  5. Consider

By (18), we obtain . Then, by (19), we get

From (45) and (46), we have .

Case  5.1. When , . From (45), we have . Then by (44), (47), and (48) we obtain . Therefore, by (17), we get . So, . In a similar way with the proof of Case 1, we can suppose that

Case  5.2. When , . From (44), we have . Then, by (45) and (46), we get and ; that is, .

If , then .

Else, if , from (44) we obtain that ; that is, ; thus, . Therefore, by (45) and (46), we conclude that .

Therefore, we can get by (17). So,

The sufficiency of Theorem 11 is obvious from the proof of the necessity.

Acknowledgments

This work is supported in part by NSFC (11171294), The Natural Science Foundation of Heilongjiang Province of China (A201013), and the fund of Heilongjiang Education Committee (12531483).

References

B. Vallette, “Homology of generalized partition posets,” Journal of Pure and Applied Algebra, vol. 208, no. 2, pp. 699–725, 2007.

D. Burde, “Left-symmetric algebras, or pre-Lie algebras in geometry and physics,” Central European Journal of Mathematics, vol. 4, no. 3, pp. 323–357, 2006.

D. Burde, K. Dekimpe, and S. Deschamps, “LR-algebras,” in New Developments in Lie Theory and Geometry, vol. 491 of Contemporary Mathematics, pp. 125–140, American Mathematical Society, Providence, RI, USA, 2009.

D. Burde and K. Dekimpe, “Post-Lie algebra structures and generalized derivations of semisimple Lie algebras,” Moscow Mathematical Journal, vol. 13, no. 1, pp. 1–18, 2013.

D. Burde, K. Dekimpe, and K. Vercammen, “Affine actions on Lie groups and post-Lie algebra structures,” Linear Algebra and its Applications, vol. 437, no. 5, pp. 1250–1263, 2012.

J.-L. Loday, “Generalized bialgebras and triples of operads,” Astérisque, vol. 320, 2008.

H. Z. Munthe-Kaas and A. Lundervold, “On post-lie algebras, lie—butcher series and moving frames,” Foundations of Computational Mathematics, vol. 13, no. 4, pp. 583–613, 2013.

Y. Pan, Q. Liu, C. Bai, and L. Guo, “PostLie algebra structures on the Lie algebra ,” Electronic Journal of Linear Algebra, vol. 23, pp. 180–197, 2012.

F. R. Gantmacher, The Theory of Matrices, vol. 2, Chelsea Publishing, New York, NY, USA, 1959.