Supervarieties of small graded colength.
Let \(F\) be a field of characteristic zero. For an associative PI-algebra \(A\) over \(F\) the most important quantitative information on the polynomial identities of \(A\) can be derived from the cocharacter sequence \(\chi_n(A)=\sum_{\lambda\vdash n}m_\lambda(A)\chi_\lambda\), \(n=1,2,\dots\), where \(\chi_\lambda\) is the irreducible \(S_n\)-character associated with the partition \(\lambda\) of \(n\). Let \(c_n(A)\) be the \(n\)-th codimension of \(A\) (i.e., the degree of the \(S_n\)-character \(\chi_n(A)\)) and let \(l_n(A)=\sum_{\lambda\vdash n}m_\lambda(A)\) be the \(n\)-th colength of \(A\) (which counts the number of the irreducible characters appearing in \(\chi_n(A)\)). Results of A. R. Kemer from the late 1970's give that the codimension sequence \(c_n(A)\) is polynomially bounded if and only if the colength \(l_n(A)\) is bounded by a constant. Equivalently, the infinite dimensional Grassmann algebra \(G\) and the algebra \(UT_2(F)\) of \(2\times 2\) upper triangular matrices do not belong to the variety of algebras generated by \(A\). It is well known that the study of superpolynomial identities of associative superalgebras is both of independent interest and an essential ingredient of the modern theory of algebras with ordinary polynomial identities. Let \(A\) be a superalgebra over \(F\) which satisfies a superpolynomial identity. Then the \(n\)-th graded cocharacter of \(A\) is \(\chi^{gr}_n(A)=\sum_{|\lambda|+|\mu|=n}m_{\lambda,\mu}(A)\chi_{\lambda,\mu}\), where \(\chi_{\lambda,\mu}\) is the irreducible \(\mathbb Z_2\wr S_n\)-character associated with the pair of partitions \(\lambda\) of \(r\) and \(\mu\) of \(n-r\). As in the ordinary case one defines the \(n\)-th graded codimension \(c^{gr}_n(A)\) and the \(n\)-th graded colength \(l^{gr}_n(A)\). \textit{A. Giambruno, S. Mishchenko} and \textit{M. Zaicev} [Commun. Algebra 29, No. 9, 3787-3800 (2001; Zbl 1006.16024)] established that the sequence \(c^{gr}_n(A)\), \(n=1,2,\dots\), is polynomially bounded if and only if five explicitly given superalgebras do not belong to the variety of superalgebras generated by \(A\). The first result of the paper under review adds a new equivalent condition in the finite dimensional case in the spirit of the results of Kemer for ordinary PI-algebras. If \(A\) is a finite dimensional superalgebra then \(c^{gr}_n(A)\) is polynomially bounded if and only if the graded colength \(l^{gr}_n(A)\) is bounded by a constant. In the next result the author classifies the finite dimensional superalgebras \(A\) with the property that \(l^{gr}_n(A)\leq 2\) for sufficiently large \(n\). All such algebras \(A\) are characterized by two equivalent conditions: (1) Up to a nilpotent direct summand the superalgebra \(A\) has the same superidentities as a commutative superalgebra with trivial grading or as one of five two-dimensional superalgebras which are subalgebras of \(UT_2(F)\); (2) Several explicitly given superalgebras do not belong to the variety of superalgebras generated by \(A\). The list of forbidden algebras contains subalgebras of the algebra \(UT_3(F)\) of \(3\times 3\) upper triangular matrices and the Grassmann algebra \(G_2\) on the two-dimensional vector space, equipped with suitable \(\mathbb Z_2\)-gradings. -- This is a superanalogue of a result of \textit{A. Giambruno} and \textit{D. La Mattina} [J. Algebra 284, No. 1, 371-391 (2005; Zbl 1071.16021)] which describes the ordinary PI-algebras \(A\) with \(l_n(A)\leq 2\). As a by-product of the proofs the paper contains results on the superidentities of several subalgebras of the algebra \(UT_3(F)\) which may serve as a motivation and an experimental material for future research.
scientific article; zbMATH DE number 6155421
scientific article