Groups with ${\mathsf{A}}_{\ell }$-commutator relations

Abstract

If $A$ is a unital associative ring and $\ell \ge 2$, then the general linear group $\mathrm{GL}(\ell ,A)$ has root subgroups $U_{\alpha }$ and Weyl elements $n_{\alpha }$ for $\alpha$ from the root system of type ${\mathsf{A}}_{\ell -1}$. Conversely, if an arbitrary group has such root subgroups and Weyl elements for $\ell \ge 4$ satisfying natural conditions, then there is a way to recover the ring $A$. We prove a generalization of this result not using the Weyl elements, so instead of the matrix ring $\mathrm{M}(\ell ,A)$ we construct a non-unital associative ring with a well-behaved Peirce decomposition.

arXiv commentSome misprints are fixed

References in corpus (1)

• Centrality of ${\mathrm{K}}_2$-functor revisited

Cited by in corpus (1)

• Groups with ${\mathsf{BC}}_{\ell }$-commutator relations