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Polynomial detection of matrix subalgebras

Authors:	Daniel Birmajer

Institution:	Department of Mathematics and Computer Science, Nazareth College, 4245 East Avenue, Rochester, New York 14618

Abstract:	The double Capelli polynomial of total degree is $\begin{displaymath}\sum \left\{ (\mathrm{sg}\, \sigma\tau) x_{\sigma(1)}y_{\tau(... ...\sigma(t)}y_{\tau(t)} \vert \sigma,\, \tau \in S_t\right\}. \end{displaymath}$ It was proved by Giambruno-Sehgal and Chang that the double Capelli polynomial of total degree is a polynomial identity for . (Here, is a field and is the algebra of $n \times n$ matrices over .) Using a strengthened version of this result obtained by Domokos, we show that the double Capelli polynomial of total degree is a polynomial identity for any proper -subalgebra of . Subsequently, we present a similar result for nonsplit inequivalent extensions of full matrix algebras.

Keywords:	Polynomial identity polynomial test matrix subalgebra double Capelli polynomial

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