Realizations of BC_r graded intersection matrix algebras with grading subalgebras of type B_r, $r \geq 3$

Abstract

We study intersection matrix algebras im(A^d) that arise from affinizing a Cartan matrix A of type B_r with d arbitrary long roots in the root system $Δ_{B_r}$, where $r \geq 3$. We show that im(A^d) is isomorphic to the universal covering algebra of $so_{2r+1}(a,η,C,χ)$, where $a$ is an associative algebra with involution $η$, and $C$ is an $a$-module with hermitian form $χ$. We provide a description of all four of the components $a$, $η$, $C$, and $χ$.