A Construction of Maximal Commutative Subalgebra of Matrix Algebras

Abstract

Let (B,m_{B},k) be a maximal commutative k-subalgebra ofM_{m}(k). Then, for some element z in Soc(B), a k-algebraR=B[X,Y]/I, where I=(m_{B}X,m_{B}Y,X^{2}-z,Y^{2}-z,XY) willcreate an interesting maximal commutative k-subalgebra of amatrix algebra which is neither a C_{1}-construction nor aC_{2}-construction. This construction will also be useful toembed a maximal commutative k-subalgebra of matrix algebra to amaximal commutative k-subalgebra of a larger size matrixalgebra.