Extending structures for associative conformal algebras

Abstract

In this paper, we give a study of the $\mathbb{C}[\partial]$-split extending structures problem for associative conformal algebras. Using the unified product as a tool, which includes interesting products such as bicrossed product, cocycle semi-direct product and so on, a cohomological type object is constructed to characterize the $\mathbb{C}[\partial]$-split extending structures for associative conformal algebras. Moreover, using this theory, the extending structures of an associative conformal algebra $A$ which is free as a $\mathbb{C}[\partial]$-module by the $\mathbb{C}[\partial]$-module $Q=\mathbb{C}[\partial]x$ are described using flag datums of $A$. Furthermore, we give a classification of the extending structures of $A$ by $Q=\mathbb{C}[\partial]x$ in detail up to equivalence when $A$ is a free associative conformal algebra of rank 1.Comment: 17 pages, Linear and Multillinear Algebra, 201

$\mathcal{O}$-operators and Nijenhius operators of associative conformal algebras

“…a matching pair of associative conformal algebras if A 1 and A 2 are subalgebras of A (cf. [29,31]). If an associative algebra decomposes into two subalgebras, it is also called an associative twilled algebra or simply twilled algebra in the literature (see, for example, [16]).…”

On antisymmetric infinitesimal conformal bialgebras

“…We recall the notions of an associative conformal algebra, its bimodule and a matched pair of associative conformal algebras. The interested readers may consult [16] and [11] for more details.…”

“…The conformal analogues of associative algebras, namely, associative conformal algebras naturally appeared in the representation theory of Lie conformal algebras ( [9]). They were studied widely on the structure theory ( [4,5,6,8,10,11,18,20,22,26,27,28,29,30,31,32]) as well as representation theory ( [7,21,23]). We would like to point that there are the "conformal analogues" for certain algebras besides Lie and associative algebras or the "conformal structures" of these algebras such as left-symmetric conformal algebras ( [13]) and Jordan conformal algebras ( [19]).…”

“…Explicitly, we first introduce the notion of double constructions of Frobenius conformal algebras as a conformal analogue of double constructions of Frobenius algebras, which are associative conformal algebras that are decomposed into the direct sum of another associative conformal algebra and its conformal dual as a C[∂]-module such that both of them are subalgebras and the natural conformal bilinear form is invariant. Such structures are interpreted equivalently in terms of matched pairs of associative conformal algebras which were introduced in [11]. Finally the notion of antisymmetric infinitesimal (ASI) conformal bialgebras is introduced as equivalent structures of the aforementioned matched pairs of associative conformal algebras as well as the double constructions of Frobenius conformal algebras.…”

Unified products of Leibniz conformal algebras

“…The C[∂]-split extending structures problem for Leibniz algebras, Lie conformal algebras and associative conformal algebras were solved in [1,9,10], respectively. The purpose of the present paper is to do the same for Leibniz conformal algebras.…”