Representations of the Complex Classical Cayley-Klein Categories

Abstract

Complex classical Cayley-Klein categories ${\bf A({\bj})}$, ${\bf B({\bj})}$, ${\bf C({\bj})}$ and ${\bf D({\bj})}$ are constructed by the method of categorical extension of the complex classical Cayley-Klein groups $SL(2n;{\bj};{\Bbb C}), SO(2n+1;{\bj};{\Bbb C}), Sp(2n;{\bj};{\Bbb C})$ and $SO(2n;{\bj};{\Bbb C})$, respectively. The explicit construction of the irreducible representations of the complex classical Cayley-Klein categories is received. Completeness of lists of the irreducible representations of the complex classical Cayley-Klein categories and the classification theorems are proved.