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

TL;DR

This paper characterizes intersection matrix algebras derived from affine extensions of type B_r Cartan matrices, showing they are isomorphic to universal coverings of certain orthogonal Lie algebras with detailed algebraic structures.

Contribution

It provides a complete description of the algebraic components underlying BC_r graded intersection matrix algebras and establishes their isomorphism to universal covering algebras of orthogonal Lie algebras.

Findings

im(A^d) is isomorphic to the universal covering algebra of so_{2r+1}(a,η,C,χ)

Explicit descriptions of algebra components a, η, C, and χ

Generalization to arbitrary long roots in the root system Δ_{B_r}

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 $\Delta_{B_r}$, where $r \geq 3$. We show that im(A^d) is isomorphic to the universal covering algebra of $so_{2r+1}(a,\eta,C,\chi)$, where $a$ is an associative algebra with involution $\eta$, and $C$ is an $a$-module with hermitian form $\chi$. We provide a description of all four of the components $a$, $\eta$, $C$, and $\chi$.