Semimagic Squares and Non-Semisimple Algebras

Abstract

Now, define A 1 to be the matrix with 1 in the first row and column, -1 in the first row and jth column, -1 in the ith row and first column, 1 in the ith row and jth column, and 0 elsewhere for i, j = 2, , n. Then, as he remarked, the algebra (n has the dimension (n 1)2, and the matrices A q form a basis for .n These results are all he mentioned in his paper. But, in the following, it will be shown that (n iS isomorphic to the total matric algebra of degree n -1 when the characteristic p of the field Q of elements of matrices does not divide the order n, and that when n is divisible by p, the decomposition (1) does not hold, 2In is not semisimple, and there is the radical 91 such that 9 = 0. Moreover, by taking half of the condition, either E1 aJ==o(A) or Z=i aX1=o(A), for allj, we may obtain larger subalgebras of 9)1l, because the conditions (a), (b), (c) also hold. These algebras will be shown to be non-semisimple, having the radical 9 such that 92 = 0. The investigation of these algebras offers us a suitable illustration of the general theory of non-semisimple algebras [2]. 2. We start from the half of the condition for semimagic squares.