# Moscow Mathematical Journal

Volume 21, Issue 1, January–March 2021 pp. 99–127.

Borel–de Siebenthal Theory for Affine Reflection Systems

**Authors**:
Deniz Kus (1) and R. Venkatesh (2)

**Author institution:**(1) University of Bochum, Faculty of Mathematics, Universitätsstr. 150, 44801 Bochum, Germany

(2) Department of Mathematics, Indian Institute of Science, Bangalore 560012

**Summary: **

We develop a Borel–de Siebenthal theory for affine reflection systems by describing their maximal closed subroot systems. Affine reflection systems (introduced by Loos and Neher) provide a unifying framework for root systems of finite-dimensional semi-simple Lie algebras, affine and toroidal Lie algebras, and extended affine Lie algebras. In the special case of nullity $k$ toroidal Lie algebras, we obtain a one-to-one correspondence between maximal closed subroot systems with full gradient and triples $(q,(b_i),H)$, where $q$ is a prime number, $(b_i)$ is a $n$-tuple of integers in the interval $[0,q-1]$ and $H$ is a $(k\times k)$ Hermite normal form matrix with determinant $q$. This generalizes the $k=1$ result of Dyer and Lehrer in the setting of affine Lie algebras.

2010 Math. Subj. Class. 17B67, 17B22.

**Keywords:**Extended affine Lie algebras, affine reflection systems, regular subalgebras.

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