Model algebras, multiplicities, and representability indices of varieties of associative algebras.

*(English. Russian original)*Zbl 1077.16027
Sb. Math. 195, No. 1, 1-18 (2004); translation from Mat. Sb. 195, No. 1, 3-20 (2004).

The algebras considered in the paper under review are associative, with unity, and over an infinite field \(k\). Let \(M\) be a variety of algebras, then the positive integer \(r\) such that \(k_r\in M\) and \(k_{r+1}\notin M\) is the complexity of \(M\). (Here \(k_r\) is the full matrix algebra of order \(r\) over \(k\).) The nilpotence index of the radical of the relatively free algebras of countable rank in \(M\) is the multiplicity of \(M\). The author introduces the representability index of \(M\) and the notion of a model algebra for the variety \(M\), see for more details the papers by the same author [Mat. Sb. 193, No. 3, 25–36 (2002; Zbl 1042.16010) and Usp. Mat. Nauk 57, No. 4, 169–170 (2002; Zbl 1069.16506)].

The author constructs model algebras generating a series of nonmatrix varieties. These include the varieties generated by finite-dimensional Grassmann algebras, by upper triangular matrices, and some other. Furthermore bounds for the multiplicities of certain varieties are given.

The author constructs model algebras generating a series of nonmatrix varieties. These include the varieties generated by finite-dimensional Grassmann algebras, by upper triangular matrices, and some other. Furthermore bounds for the multiplicities of certain varieties are given.

Reviewer: Plamen Koshlukov (Campinas)

##### MSC:

16R10 | \(T\)-ideals, identities, varieties of associative rings and algebras |

16N40 | Nil and nilpotent radicals, sets, ideals, associative rings |

16R40 | Identities other than those of matrices over commutative rings |

16R50 | Other kinds of identities (generalized polynomial, rational, involution) |