In algebraic K-theory, a field of mathematics, the Steinberg group St ( A ) {\displaystyle \operatorname {St} (A)} of a ring A {\displaystyle A} is the universal central extension of the commutator subgroup of the stable general linear group of A {\displaystyle A} . It is named after Robert Steinberg, and it is connected with lower K {\displaystyle K} -groups, notably K 2 {\displaystyle K_{2}} and K 3 {\displaystyle K_{3}} .
Definition Abstractly, given a ring A {\displaystyle A} , the Steinberg group St ( A ) {\displaystyle \operatorname {St} (A)} is the universal central extension of the commutator subgroup of the stable general linear group (the commutator subgroup is perfect and so has a universal central extension).
Presentation using generators and relations A concrete presentation using generators and relations is as follows. Elementary matrices — i.e. matrices of the form e p q ( λ ) := 1 + a p q ( λ ) {\displaystyle {e_{pq}}(\lambda ):=\mathbf {1} +{a_{pq}}(\lambda )} , where 1 {\displaystyle \mathbf {1} } is the identity matrix, a p q ( λ ) {\displaystyle {a_{pq}}(\lambda )} is the matrix with λ {\displaystyle \lambda } in the ( p , q ) {\displaystyle (p,q)} -entry and zeros elsewhere, and p ≠ q {\displaystyle p\neq q} — satisfy the following relations, called the Steinberg relations:
e i j ( λ ) e i j ( μ ) = e i j ( λ + μ ) ; [ e i j ( λ ) , e j k ( μ ) ] = e i k ( λ μ ) , for i ≠ k ; [ e i j ( λ ) , e k l ( μ ) ] = 1 , for i ≠ l and j ≠ k . {\displaystyle {\begin{aligned}e_{ij}(\lambda )e_{ij}(\mu )&=e_{ij}(\lambda +\mu );&&\\\left[e_{ij}(\lambda ),e_{jk}(\mu )\right]&=e_{ik}(\lambda \mu ),&&{\text{for }}i\neq k;\\\left[e_{ij}(\lambda ),e_{kl}(\mu )\right]&=\mathbf {1} ,&&{\text{for }}i\neq l{\text{ and }}j\neq k.\end{aligned}}}
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