Next: A system of generators Up: No Title Previous: Introduction

The algebra

Let be the Grassmann algebra over with two generators, i.e. the linear span of with arbitrary complex coefficients, where the generators satisfy the relations and . This algebra has an even part, generated by 1 and and an odd part generated by and . We call the algebra of matrices over the complex numbers and another copy of this algebra that we grade as follows: A matrix is called even if it is of the type

and odd if it is of the type

We call the Grassmann envelope of which is defined as the even part of the tensor product of and , i.e. the space of matrices matrices V with entries , , , , , that are even Grassmann elements (of the kind ) and entries , , , that are odd Grassmann elements (i.e. of the kind ). We define as

Explicitly,

All entries besides the 's are complex numbers (the above sign is a direct sum sign: these matrices are matrices written as a direct sum of two blocks of size ).

It is obvious that this is an associative algebra, with usual matrix multiplication, of dimension 27 (just count the number of arbitrary parameters). is not semi-simple (because of the appearance of Grassmann numbers in the entries of the matrices) and its semi-simple part , given by the direct sum of its block-diagonal -independent parts is equal to the 9+4+1= 14-dimensional algebra . The radical (more precisely the Jacobson radical) J of is the left-over piece that contains all the Grassmann entries, and only the Grassmann entries, so that . J has therefore dimension 13.

Robert Coquereaux
Tue Nov 5 15:18:21 MET 1996