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The algebraic and geometric classification of nilpotent weakly associative and symmetric Leibniz algebras. (English) Zbl 1483.17004

An algebra \(A\) is a vector space with a bilinear multiplication. In the paper under review all algebras are vector spaces over the complex numbers. Let \(L_a:A\to A\), \(x\mapsto ax\) denote the left multiplication operator on \(A\), and let \(R_a:A\to A\), \(x\mapsto xa\) denote the right multiplication operator on \(A\). A left Leibniz algebra is an algebra such that every left multiplication operator is a derivation, a right Leibniz algebra is an algebra such that every right multiplication operator is a derivation, and a left and right Leibniz algebra is called symmetric. Note that every Lie algebra is a symmetric Leibniz algebra. Slightly more general than symmetric Leibniz algebras are weakly associative algebras which are algebras such that \(L_a-R_a\) is a derivation for every element \(a\). Note that every commutative algebra is weakly associative. Moreover, in the class of anti-commutative algebras, Lie algebras, left and right Leibniz algebras, symmetric Leibniz algebras, and weakly associative algebras all are the same.
The classification of complex algebras of a fixed dimension satisfying certain identities can be divided into a purely algebraic part (i.e., the classification of the isomorphism classes of the algebras) and an algebro-geometric part (i.e., finding the dimension of the affine variety \(\mathbb{L}(T)\) of the algebra structures on \(\mathbb{C}^n\) satisfying the identities \(T\) as well as a description of the irreducible components of \(\mathbb{L}(T)\), the \(\mathrm{GL}_n(\mathbb{C} )\)-orbits of the points in \(\mathbb{L}(T)\), and the closures of these orbits). In general, a classification of the isomorphism classes or an explicit description of the irreducible components and the orbit (closures) is only possible in small dimensions. In the first section of the paper the authors classify the isomorphism classes of the four-dimensional symmetric Leibniz algebras and the five-dimensional nilpotent symmetric Leibniz algebras. In the second much shorter section of the paper the dimension and the irreducible components of the affine varieties of these classes of algebras are determined.
In the four-dimensional case the classification is divided into nilpotent and non-nilpotent algebras. In a previous paper I. Kaygorodov et al. [Linear Multilinear Algebra 66, No. 4, 704–716 (2018; Zbl 1472.17100)] has already described the four-dimensional 2-step nilpotent symmetric Leibniz algebras. There are five isomorphism classes of non-2-step four-dimensional nilpotent symmetric Leibniz algebras that are explicitly described in the present paper. More generally, the authors classify the isomorphism classes of four-dimensional nilpotent weakly associative algebras. The four-dimensional nilpotent commutative algebras have already been classified by A. Fernández Ouaridi et al. [J. Pure Appl. Algebra 226, No. 3, Article ID 106850, 21 p. (2022; Zbl 1486.17001)]. Any other four-dimensional nilpotent weakly associative algebra is either a symmetric Leibniz algebra or it belongs to two 1-parameter families or nine additional isomorphism classes. Furthermore, a five-dimensional nilpotent symmetric Leibniz algebra is either 2-step nilpotent or it belongs to one 2-parameter family, six 1-parameter families or forty-two additional isomorphism classes. Finally, the isomorphism classes of non-nilpotent four-dimensional symmetric Leibniz algebras are derived from the classification of the non-nilpotent four-dimensional right Leibniz algebras obtained previously by N. Ismailov et al. [Int. J. Math. 29, No. 5, Article ID 1850035, 12 p. (2018; Zbl 1423.17005)].
The set of algebra structures on a given \(n\)-dimensional complex vector space satisfying the identities \(T\) identified with the structure constants with respect to a fixed basis is a (Zariski-)closed subset \(\mathbb{L}(T)\) of the affine space \(\mathbb{C}^{n^3}\). The general linear group \(\mathrm{GL}_n(\mathbb{C})\) acts on the affine variety \(\mathbb{L}(T)\) via change of basis, and the \(\mathrm{GL}_n(\mathbb{C})\)-orbits correspond to the isomorphism classes of the \(n\)-dimensional algebras satisfying the identities \(T\). An algebra is called rigid in \(\mathbb{L}(T)\) when the \(\mathrm{GL}_n (\mathbb{C})\)-orbit of the corresponding point in \(\mathbb{L} (T)\) is open. For the affine varieties of the classes of algebras considered in the first part of their paper the authors obtain the following results:
1) The variety of four-dimensional nilpotent symmetric Leibniz algebras has dimension 11, and it has three irreducible components of which one contains rigid algebras.
2) The variety of four-dimensional nilpotent weakly associative algebras has dimension 16, and it also has three irreducible components of which one consists of the rigid four-dimensional nilpotent symmetric Leibniz algebras. (Remark of the reviewer: In the last paragraph on page 307 is a typo when the authors state that the variety of four-dimensional nilpotent weakly associative algebras has 4 irreducible components.)
3) The variety of all four-dimensional symmetric Leibniz algebras has dimension 13, and it has five irreducible components of which one contains rigid algebras. (Remark of the reviewer: In the proof of Theorem E on page 309 there is another typo when the authors write \(\mathfrak{L}_2\) instead of \(\mathfrak{L}_{02}\).)
4) The variety of five-dimensional nilpotent symmetric Leibniz algebras has dimension 24, six irreducible components, and no rigid algebras.
Note that in the results 1)–4) the authors describe the irreducible components explicitly as closures of orbits of some specific algebra structures.
A degeneration of an algebra \(A\) is an algebra that belongs to the (Zariski-)closure of the orbit of the point in \(\mathbb{L}(T)\) corresponding to \(A\). F. Grunewald and J. O’Halloran [J. Algebra 162, No. 1, 210–224 (1993; Zbl 0799.17007)] conjectured that every nilpotent Lie algebra is the degeneration of a non-nilpotent Lie algebra of the same dimension. It follows from the classification of orbit closures of three- and four-dimensional Lie algebras due to D. Burde and C. Steinhoff [J. Algebra 214, No. 2, 729–739 (1999; Zbl 0932.17005)] that the Grunewald-O’Halloran conjecture is true in dimensions 3 and 4. Furthermore, J. M. Casas et al. [Linear Algebra Appl. 439, No. 2, 472–487 (2013; Zbl 1305.17001)] showed that the analogue of the Grunewald-O’Halloran conjecture is true for (left or right) Leibniz algebras of dimension at most 3. On the other hand, M. Vergne [Bull. Soc. Math. Fr. 98, 81–116 (1970; Zbl 0244.17011)]conjectured that there are no nilpotent algebras that are rigid in the variety of arbitrary \(n\)-dimensional Lie algebras. The authors obtain from their explicit description of the irreducible components of the affine variety of the four-dimensional symmetric Leibniz algebras that the analogue of Vergne’s conjecture is true for four-dimensional symmetric Leibniz algebras. Moreover, since there is an irreducible component that only contains nilpotent algebras, the analogue of the Grunewald-O’Halloran conjecture fails for four-dimensional symmetric Leibniz algebras.

MSC:

17A30 Nonassociative algebras satisfying other identities
17A32 Leibniz algebras
14D06 Fibrations, degenerations in algebraic geometry
14L30 Group actions on varieties or schemes (quotients)
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References:

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