Group Gradings on Matrix Algebras

http://dx.doi.org/10.4153/CMB-2002-051-x
Canad. Math. Bull. 45(2002), 499-508

  Published:2002-12-01
 Printed: Dec 2002

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 $A = M_n(\Phi)$, we decompose $A$ as the tensor product of graded subalgebras $A = B\otimes C$, $B\cong M_p (\Phi)$ being a graded division algebra, while the grading of $C\cong M_q (\Phi)$ 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.