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Group Gradings on Matrix Algebras

Published online by Cambridge University Press:  20 November 2018

Yu. A. Bahturin  and
M. V. Zaicev
Yu. A. Bahturin
Affiliation:
Department of Mathematics and Statistics Memorial University of Newfoundland St. John's, NF A1A 5K9 and Department of Algebra Faculty of Mathematics and Mechanics Moscow State University Moscow 119899 Russia, email: yuri@math.mun.ca
M. V. Zaicev
Affiliation:
Department of Algebra Faculty of Mathematics and Mechanics Moscow State University Moscow 119899 Russia, email: zaicev@mech.math.msu.su
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Abstract

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Let $\Phi $ be an algebraically closed field of characteristic zero, $G$ a finite, not necessarily abelian, group. Given a $G$-grading on the full matrix algebra $A\,=\,{{M}_{n}}\left( \Phi \right)$, we decompose $A$ as the tensor product of graded subalgebras $A\,=\,B\,\otimes \,C,\,B\,\cong \,{{M}_{p}}\left( \Phi \right)$ being a graded division algebra, while the grading of $C\,\cong \,{{M}_{q}}\left( \Phi \right)$ is determined by that of the vector space ${{\Phi }^{n}}$. Now the grading of $A$ is recovered from those of $A$ and $B$ using a canonical “induction” procedure.

Keywords

16W50
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 45 , Issue 4 , 01 December 2002 , pp. 499 - 508
Copyright
Copyright © Canadian Mathematical Society 2002

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