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Graded central polynomials for the matrix algebra of order two. (English) Zbl 1184.16022

In the paper under review the authors study \(\mathbb{Z}_2\)-graded polynomial identities of the \(2\times 2\) matrix algebra \(M_2(K)\) over an infinite integral domain \(K\). The algebra \(M_2(K)\) is given the natural \(\mathbb{Z}_2\)-grading by assuming that the diagonal matrices are the 0-component while the off-diagonal ones form the 1-component.
Before, the \(\mathbb{Z}_2\)-graded polynomial identities for \(M_2(K)\) were described by O. M. Di Vincenzo [Isr. J. Math. 80, No. 3, 323-335 (1992; Zbl 0784.16015)] when \(K\) is a field of characteristic 0 and by P. Koshlukov, S. S. de Azevedo [Isr. J. Math. 128, 157-176 (2002; Zbl 1015.16023)] when \(K\) is an infinite field of odd characteristic. The description of the graded central polynomials was given by A. P. Brandaõ jun., P. Koshlukov [J. Pure Appl. Algebra 208, No. 3, 877-886 (2007; Zbl 1110.16018)] for infinite fields of characteristic different from 2.
The main results of the paper under review show that the behaviour of the graded identities and central polynomials when \(K\) is an infinite integral domain is similar to that when \(K\) is an infinite field of characteristic \(\neq 2\). The proofs use various combinatorial techniques and are characteristic-free. In particular, they work also when \(K\) is an infinite field of characteristic 2 and show that the graded identities and central polynomials follow from a finite number of them. On the contrary, it is still unknown whether the ordinary polynomial identities of \(M_2(K)\) over an infinite field of characteristic 2 have a finite basis.
Among the corollaries of the main results is a fact in the spirit of the problem of Procesi whether the algebra of generic \(n\times n\) matrices over \(\mathbb{Z}_p\) is isomorphic to the algebra of generic \(n\times n\) matrices over \(\mathbb{Z}\) modulo the ideal generated by \(p\). It is known that for any \(n>1\) there is a counterexample for a suitable prime \(p\). The authors show that in the graded case the analogue of the problem of Procesi is solved in the affirmative for generic \(2\times 2\) matrices over any commutative ring \(K\) and an arbitrary ideal of \(K\).

MSC:

16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
16R50 Other kinds of identities (generalized polynomial, rational, involution)
16W50 Graded rings and modules (associative rings and algebras)
16W55 “Super” (or “skew”) structure
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References:

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