Extending Structures and Classifying Complements for Left-Symmetric Algebras

Abstract

Let A be a left-symmetric (resp. Novikov) algebra, E be a vector space containing A as a subspace and V be a complement of A in E. The extending structures problem which asks for the classification of all leftsymmetric (resp. Novikov) algebra structures on E such that A is a subalgebra of E is studied. In this paper, the definition of the unified product of left-symmetric (resp. Novikov) algebras is introduced. It is shown that there exists a left-symmetric (resp. Novikov) algebra structure on E such that A is a subalgebra of E if and only if E is isomorphic to a unified product of A and V . Two cohomological type objects H 2 A (V, A) and H 2 (V, A) are constructed to give a theoretical answer to the extending structures problem. Furthermore, given an extension A ⊂ E of left-symmetric (resp. Novikov) algebras, another cohomological type object is constructed to classify all complements of A in E. Several special examples are provided in details. 1 2 YANYONG HONGThis problem is called extending structures problem. The extending structures problems for groups, associative algebras, Hopf algebras, Lie algebras, and Leibniz algebras have been studied in [4,3,6,5,7] respectively. Since left-symmetric algebras and Novikov algebras are important algebraic structures in mathematics and mathematical physics, it is interesting and meaningful to study the extending structures problems for left-symmetric algebras and Novikov algebras. Certainly, this problem is very difficult. When A = {0}, this problem is equivalent to classify all left-symmetric algebras and Novikov algebras on an arbitrary vector space E. Of course, when the dimension of E is large, it is impossible to classify all algebraic structures. In fact, the classifications of left-symmetric algebras in dimension 2 and dimension 3 over C were obtained in [11,10], and the classifications of Novikov algebras in dimension 2 and dimension 3 over R and C were obtained in [12,16]. But, as far as we know, the complete classifications of left-symmetric algebras (resp. Novikov algebras) whose dimension are larger than 3 have not been obtained so far. There are only some papers on the classification of some special left-symmetric algebras and Novikov algebras of dimensions 4 and 5, such as [18,23,24,16]. Therefore, we will assume that A = {0} below. In fact, the extending structures problem generalizes some classic algebraic problems. For example, the extension of the left-symmetric algebra A by an abelian left-symmetric algebra C is characterized by the second cohomology group H 2 (C, A) (see [19]); the problem which asks for describing and classifying the extensions of the left-symmetric algebra A by another left-symmetric algebra B can be answered by bicrossed products of A and B (see [9]) and this problem is called the factorization problem. In this paper, our aim is to give a theoretical interpretation of the extending structures problem for left-symmetric (resp. Novikov) algebras using some cohomological type objects and provide some detailed answers to it in certain ...

Classifying complements for conformal algebras

“…Structure theory and representation theory of Lie conformal algebras and associative conformal algebras have been studied in a series of papers (see [5]- [8], [17]- [18], [20]- [21] and so on). In particular, the C[∂]-split extending structure problems of Lie conformal algebras and associative conformal algebras were investigated in [12] and [10] respectively. The tool for studying C[∂]-split extending structure problem is unified product, which includes crossed product, bicrossed product and so on.…”

“…In this paper, following the works in [12] and [10], we plan to study the following structure problem of Lie conformal algebras and associative conformal algebras:…”

“…From the point of view of algebra, this problem is natural and significant. Similar problems for classical algebraic objects such as associative algebras, Hopf algebras, Lie algebras, Leibniz algebras and left-symmetric algebras have been investigated in [1,2,3,10]. According to the theory of bicrossed product developed in [12] and [10], it is easy to see that E in the classifying complements problem is just a bicrossed product of R and Q associated to a matched pair.…”

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.…”

Extending structures for associative conformal algebras

“…Set E = A ⊕ Q where the direct sum is the sum of C[∂]-modules. Describe and classify all associative conformal algebra structures on E up to isomorphism such that A is a subalgebra of E. Similar problems for some classical algebra objects such as groups, associative algebras, Hopf algebras, Lie algebras, Leibniz algebras and left-symmetric algebras have been studied in [1,3,2,4,5,13] respectively. Note that this problem is very difficult.…”