The Algebraic and Geometric Classification of Nilpotent Bicommutative Algebras
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The Algebraic and Geometric Classification of Nilpotent Bicommutative Algebras ´ ´ 2 · Vasily Voronin3 Ivan Kaygorodov1 · Pilar Paez-Guill an Received: 7 March 2019 / Accepted: 11 December 2019 / © Springer Nature B.V. 2020
Abstract We classify the complex 4-dimensional nilpotent bicommutative algebras from both algebraic and geometric approaches. Keywords Bicommutative algebras · Nilpotent algebras · Algebraic classification · Central extension · Geometric classification · Degeneration Mathematics Subject Classification (2010) 17A30 · 14D06 · 14L30
1 Introduction One of the classical problems in the theory of non-associative algebras is to classify (up to isomorphism) the algebras of dimension n from a certain variety defined by some family of polynomial identities. It is typical to focus on small dimensions, and there are two main directions for the classification: algebraic and geometric. Varieties as Jordan, Lie, Leibniz or Zinbiel algebras have been studied from these two approaches ([1, 11, 15–18, 26, 27, 31, 34, 45] and [3, 5, 8, 11, 13, 14, 25, 26, 28–30, 35–38, 41–48], respectively). In the present paper, we give the algebraic and geometric classification of 4-dimensional nilpotent bicommutative algebras.
Presented by: Michel Brion Ivan Kaygorodov
[email protected] Pilar P´aez-Guill´an [email protected] Vasily Voronin [email protected] 1
CMCC, Universidade Federal do ABC, Santo Andr´e, Brazil
2
University of Santiago de Compostela, Santiago de Compostela, Spain
3
Novosibirsk State University, Novosibirsk, Russia
I. Kaygorodov et al.
The variety of bicommutative algebras is defined by the following identities of right- and left-commutativity: (xy)z = (xz)y, x(yz) = y(xz). It contains the commutative associative algebras as a subvariety. One-sided commutative algebras first appeared in the paper by Cayley [9] in 1857. The structure of the free bicommutative algebra of countable rank and its main numerical invariants were described by Dzhumadildaev, Ismailov and Tulenbaev [23], see also the announcement [22]. Bicommutative algebras were also studied in [20, 21, 24], and in [6, 7] under the name of LR-algebras. The key step in our method for algebraically classifying bicommutative nilpotent algebras is the calculation of central extensions of smaller algebras. In short, an algebra A is called an extension of another algebra B by K if there exists a short exact sequence 0 → K → A → B → 0. The simplest example is the direct sum B ⊕ K with the inclusion and the projection. Imposing additional conditions, we find important special types of extensions such as the split extensions, the HNN-extensions and many others. In this paper, we will deal with central extensions, i.e. extensions in which the annihilator of A contains K. Some important algebras can be constructed as central extensions; for example, the Virasoro algebra is the universal central extension of the Witt algebra, and the Heisenberg algebra is a central extension of a commutative Lie algebra. The theory of central extensions
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