On cohomology and deformations of Jacobi-Jordan algebras. (English) Zbl 1533.17002
Jacobi-Jordan algebras (also known as mock-Lie algebras, or Jordan algebras of nilindex \(3\) in the literature) constitute a variety of algebras defined by two identities: commutativity and the Jacobi identity \((xy)z + (zx)y + (yz)x = 0\). Basing on the classification of such low-dimensional algebras from [D. Burde and A. Fialowski, Linear Algebra Appl. 459, 586–594 (2014; Zbl 1385.17013)], the author singles out Jacobi-Jordan algebras of dimension \(\le 5\) having either nondegenerate skew-symmetric bilinear form \(\omega\) satisfying the cocycle equation \(\omega(xy,z) + \omega(zx,y) + \omega(yz,x) = 0\), or having a nondegenerate symmetric bilinear invariant form.
The author then considers, in the standard way, the second degree cohomology of Jacobi-Jordan algebras with coefficients in the adjoint module, responsible for infinitesimal deformations, computes it for algebras of dimension \(\le 5\), and computes the corresponding Massey brackets, getting in this way some examples of parametric families of Jacobi-Jordan algebras, or establishing that some algebras are rigid.
Reviewer’s remark: To construct an adequate cohomology theory of Jacobi-Jordan algebras is not an entirely trivial task, as was noted by the reviewer in [Linear Algebra Appl. 518, 79–96 (2017; Zbl 1400.17015)], due to an apparent contradiction between the standard interpretations of low-degree cohomology as outer derivations and as infinitesimal deformations. An apparently satisfactory solution of this problem was given in [A. Baklouti et al., “Cohomology and deformations of Jacobi-Jordan algebras”, Preprint, arXiv:2109.12364].
The author then considers, in the standard way, the second degree cohomology of Jacobi-Jordan algebras with coefficients in the adjoint module, responsible for infinitesimal deformations, computes it for algebras of dimension \(\le 5\), and computes the corresponding Massey brackets, getting in this way some examples of parametric families of Jacobi-Jordan algebras, or establishing that some algebras are rigid.
Reviewer’s remark: To construct an adequate cohomology theory of Jacobi-Jordan algebras is not an entirely trivial task, as was noted by the reviewer in [Linear Algebra Appl. 518, 79–96 (2017; Zbl 1400.17015)], due to an apparent contradiction between the standard interpretations of low-degree cohomology as outer derivations and as infinitesimal deformations. An apparently satisfactory solution of this problem was given in [A. Baklouti et al., “Cohomology and deformations of Jacobi-Jordan algebras”, Preprint, arXiv:2109.12364].
Reviewer: Pasha Zusmanovich (Ostrava)
MSC:
| 17A30 | Nonassociative algebras satisfying other identities |
| 17A15 | Noncommutative Jordan algebras |
| 17C99 | Jordan algebras (algebras, triples and pairs) |
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