Advances in Lie Groups and Applications in Applied SciencesView this Special Issue
Post-Lie Algebra Structures on the Lie Algebra
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.
Post-Lie algebras were introduced around 2007 by Vallette , who found the structure in a purely operadic manner as the Koszul dual of a commutative trialgebra. Moreover, Vallette  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 , 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  got the following classification theorem, which will be used in our proof.
Theorem 3 (see ). 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 .
(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).
Proposition 9. Suppose that is the matrix set of post-Lie algebra , then we have the following equations:
3. Classification of Post-Lie Algebra
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 .
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.
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).
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