Extending Structures for Lie Conformal Algebras

Abstract

Let (g, [•, •], δ g ) be a fixed Lie bialgebra, E be a vector space containing g as a subspace and V be a complement of g in E. A natural problem is that how to classify all Lie bialgebraic structures on E such that (g, [•, •], δ g ) is a Lie sub-bialgebra up to an isomorphism of Lie bialgebras whose restriction on g is the identity map. This problem is called the extending structures problem. In this paper, we introduce a general co-product on E, called the unified co-product of (g, δ g ) by V . With this unified co-product and the unified product of (g, [•, •]) by V developed in [1], the unified bi-product of (g, [•, •], δ g ) by V is introduced. Moreover, we show that any E in the extending structures problem is isomorphic to a unified bi-product of (g, [•, •], δ g ) by V . Then an object HBI 2 g (V, g) is constructed to classify all E in the extending structures problem. Moreover, several special unified bi-products are also introduced. In particular, the unified bi-products when dimV = 1 are investigated in detail.

Representations of Some Associative Pseudoalgebras

“…After this, Y. Su determined the cohomology groups of a general linear conformal Lie algebra in [Su]. Further, Su and his coauthors study representations of various conformal Lie algebras (see [FHS,HS1,HS2,SY,SY1,SY2]).…”

Unified products for Malcev algebras

“…On the other hand, extending structures for Lie algebras, associative algebras and Leibniz algebras are studied by Agore and Militaru in [1,2,3,4,5,6]. The extending structures for left-symmetric algebras, associative and Lie conformal algebras has also been studied by Y. Hong and Y. Su in [7,8,9].…”

Extending Structures for Lie 2-Algebras

“…Meanwhile, crossed modules of Lie algebras can be identified with strict Lie 2-algebras. The extending structures problem for some algebra objects such as Lie algebras, Hopf algebras, Leibniz algebras, associative algebras, left-symmetric algebras and Lie conformal algebras have been studied in [6][7][8][9][10][11] respectively. Lie 2-algebras were first introduced by J. C. Baez and A. S. Crans in 2004.…”