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The variety of dual mock-Lie algebras. (English) Zbl 1490.17003

Jordan algebras which satisfy the Jacobi identity were first considered by K. A. Zhevlakov [Algebra Logika 5, No. 3, 37–58 (1966; Zbl 0253.17015)]. D. Burde and A. Fialowski gave the name Jacobi-Jordan algebras in [Linear Algebra Appl. 459, 586–594 (2014; Zbl 1385.17013)], and described some of their properties. P. Zusmanovich [Linear Algebra Appl. 518, 79–96 (2017; Zbl 1400.17015)] gave the name for them mock-Lie algebras. In this paper the authors consider dual mock-Lie algebras, which are both anticommutative and antiassociative algebras. Low dimensional such algebras are already classified. Using the results of M. Alejandra Alvarez [Algebra Colloq. 25, No. 2, 349–360 (2018; Zbl 1434.17015)], the authors give the algebraic classification in dimension 7. With 1-dimensional central extensions they get the 8-dimensional algebras, and 2-dimensional central extensions get the 9-dimensional algebras. In Section 2 the authors classify degenerations of dual mock-Lie algebras of dimension 7 and 8.

MSC:

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