-ALGEBRAS FROM GROUP REPRESENTATIONS

Abstract

AbstractWe introduce certain$C^*$-algebras andk-graphs associated tokfinite-dimensional unitary representations$\rho _1,\ldots ,\rho _k$of a compact groupG. We define a higher rank Doplicher-Roberts algebra$\mathcal {O}_{\rho _1,\ldots ,\rho _k}$, constructed from intertwiners of tensor powers of these representations. Under certain conditions, we show that this$C^*$-algebra is isomorphic to a corner in the$C^*$-algebra of a row-finite rankkgraph$\Lambda $with no sources. ForGfinite and$\rho _i$faithful of dimension at least two, this graph is irreducible, it has vertices$\hat {G}$and the edges are determined bykcommuting matrices obtained from the character table of the group. We illustrate this with some examples when$\mathcal {O}_{\rho _1,\ldots ,\rho _k}$is simple and purely infinite, and with someK-theory computations.