Polynomial detection of matrix subalgebras
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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.References
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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
- 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
- © Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2117201