Polynomial detection of matrix subalgebras

Author:
Daniel Birmajer

Journal:
Proc. Amer. Math. Soc. **133** (2005), 1007-1012

MSC (2000):
Primary 15A24, 15A99, 16R99

DOI:
https://doi.org/10.1090/S0002-9939-04-07631-2

Published electronically:
October 18, 2004

MathSciNet review:
2117201

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The double Capelli polynomial of total degree $2t$ is \begin{equation*} \sum \left \{ (\mathrm {sg} \sigma \tau ) x_{\sigma (1)}y_{\tau (1)}x_{\sigma (2)}y_{\tau (2)}\cdots x_{\sigma (t)}y_{\tau (t)} |\; \sigma , \tau \in S_t\right \}. \end{equation*} It was proved by Giambruno-Sehgal and Chang that the double Capelli polynomial of total degree $4n$ is a polynomial identity for $M_n(F)$. (Here, $F$ is a field and $M_n(F)$ is the algebra of $n \times n$ matrices over $F$.) Using a strengthened version of this result obtained by Domokos, we show that the double Capelli polynomial of total degree $4n-2$ is a polynomial identity for any proper $F$-subalgebra of $M_n(F)$. Subsequently, we present a similar result for nonsplit inequivalent extensions of full matrix algebras.

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Additional Information

**Daniel Birmajer**

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

Email:
abirmaj6@naz.edu

Keywords:
Polynomial identity,
polynomial test,
matrix subalgebra,
double Capelli polynomial

Received by editor(s):
November 13, 2003

Received by editor(s) in revised form:
December 22, 2003

Published electronically:
October 18, 2004

Communicated by:
Martin Lorenz

Article copyright:
© Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.