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The algebra tex2html_wrap_inline1137

Let tex2html_wrap_inline1139 be the Grassmann algebra over tex2html_wrap_inline1195 with two generators, i.e. the linear span of tex2html_wrap_inline1197 with arbitrary complex coefficients, where the generators satisfy the relations tex2html_wrap_inline1199 and tex2html_wrap_inline1201 . This algebra has an even part, generated by 1 and tex2html_wrap_inline1205 and an odd part generated by tex2html_wrap_inline1207 and tex2html_wrap_inline1209 . We call tex2html_wrap_inline1169 the algebra of tex2html_wrap_inline1171 matrices over the complex numbers and tex2html_wrap_inline1215 another copy of this algebra that we grade as follows: A matrix tex2html_wrap_inline1217 is called even if it is of the type


and odd if it is of the type


We call tex2html_wrap_inline1173 the Grassmann envelope of tex2html_wrap_inline1215 which is defined as the even part of the tensor product of tex2html_wrap_inline1215 and tex2html_wrap_inline1139 , i.e. the space of matrices tex2html_wrap_inline1171 matrices V with entries tex2html_wrap_inline1235 , tex2html_wrap_inline1237 , tex2html_wrap_inline1239 , tex2html_wrap_inline1241 , tex2html_wrap_inline1243 , that are even Grassmann elements (of the kind tex2html_wrap_inline1245 ) and entries tex2html_wrap_inline1247 , tex2html_wrap_inline1249 , tex2html_wrap_inline1251 , tex2html_wrap_inline1253 that are odd Grassmann elements (i.e. of the kind tex2html_wrap_inline1255 ). We define tex2html_wrap_inline1123 as




All entries besides the tex2html_wrap_inline1263 's are complex numbers (the above tex2html_wrap_inline1265 sign is a direct sum sign: these matrices are tex2html_wrap_inline1267 matrices written as a direct sum of two blocks of size tex2html_wrap_inline1171 ).

It is obvious that this is an associative algebra, with usual matrix multiplication, of dimension 27 (just count the number of arbitrary parameters). tex2html_wrap_inline1123 is not semi-simple (because of the appearance of Grassmann numbers in the entries of the matrices) and its semi-simple part tex2html_wrap_inline1275 , given by the direct sum of its block-diagonal tex2html_wrap_inline1263 -independent parts is equal to the 9+4+1= 14-dimensional algebra tex2html_wrap_inline1281 . The radical (more precisely the Jacobson radical) J of tex2html_wrap_inline1123 is the left-over piece that contains all the Grassmann entries, and only the Grassmann entries, so that tex2html_wrap_inline1287 . J has therefore dimension 13.

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