Groups with 𝖠_{ℓ}-commutator relations

Abstract

If A A is a unital associative ring and ℓ ≥ 2 \ell \geq 2 , then the general linear group GL ⁡ ( ℓ , A ) \operatorname {GL}(\ell , A) has root subgroups U α U_\alpha and Weyl elements n α n_\alpha for α \alpha from the root system of type A ℓ − 1 \mathsf A_{\ell - 1} . Conversely, if an arbitrary group has such root subgroups and Weyl elements for ℓ ≥ 4 \ell \geq 4 satisfying natural conditions, then there is a way to recover the ring A A . A generalization of this result not involving the Weyl elements is proved, so instead of the matrix ring M ⁡ ( ℓ , A ) , \operatorname {M}(\ell , A), a nonunital associative ring with a well-behaved Peirce decomposition is provided.