About $\ast$-representations of polynomial semilinear relations

Abstract

In the present paper we study $\ast$-representations of semilinear relations with polynomial characteristic functions. For any finite simple non-oriented graph $\Gamma$ we construct a polynomial characteristic function such that $\Gamma$ is its graph. Full description of graphs which satisfy polynomial (degree one and two) semilinear relations is obtained. We introduce the $G$-orthoscalarity condition and prove that any semili ear relation with quadratic characteristic function and condition of $G$-orthoscalarity is $\ast$-tame. This class of relations contains, in particular, $\ast$-representations of $U_{q}(so(3)).$