Steinberg group (K-theory)

In algebraic K-theory, a field of mathematics, the Steinberg group $$ \operatorname{St}(A) $$ of a ring $$ A $$ is the universal central extension of the commutator subgroup of the stable general linear group of $$ A $$.

It is named after Robert Steinberg, and it is connected with lower $ K $-groups, notably $$ K_{2} $$ and $$ K_{3} $$.

Definition
Abstractly, given a ring $$ A $$, the Steinberg group $$ \operatorname{St}(A) $$ 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 $$ {e_{pq}}(\lambda) := \mathbf{1} + {a_{pq}}(\lambda) $$, where $$ \mathbf{1} $$ is the identity matrix, $$ {a_{pq}}(\lambda) $$ is the matrix with $$ \lambda $$ in the $$ (p,q) $$-entry and zeros elsewhere, and $$ p \neq q $$ — satisfy the following relations, called the Steinberg relations:

\begin{align} 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{align} $$

The unstable Steinberg group of order $$ r $$ over $$ A $$, denoted by $$ {\operatorname{St}_{r}}(A) $$, is defined by the generators $$ {x_{ij}}(\lambda) $$, where $$ 1 \leq i \neq j \leq r $$ and $$ \lambda \in A $$, these generators being subject to the Steinberg relations. The stable Steinberg group, denoted by $$ \operatorname{St}(A) $$, is the direct limit of the system $$ {\operatorname{St}_{r}}(A) \to {\operatorname{St}_{r + 1}}(A) $$. It can also be thought of as the Steinberg group of infinite order.

Mapping $$ {x_{ij}}(\lambda) \mapsto {e_{ij}}(\lambda) $$ yields a group homomorphism $$ \varphi: \operatorname{St}(A) \to {\operatorname{GL}_{\infty}}(A) $$. As the elementary matrices generate the commutator subgroup, this mapping is surjective onto the commutator subgroup.

Interpretation as a fundamental group
The Steinberg group is the fundamental group of the Volodin space, which is the union of classifying spaces of the unipotent subgroups of $$\operatorname{GL}(A)$$.

K1
$$ {K_{1}}(A) $$ is the cokernel of the map $$ \varphi: \operatorname{St}(A) \to {\operatorname{GL}_{\infty}}(A) $$, as $$ K_{1} $$ is the abelianization of $$ {\operatorname{GL}_{\infty}}(A) $$ and the mapping $$ \varphi $$ is surjective onto the commutator subgroup.

K2
$$ {K_{2}}(A) $$ is the center of the Steinberg group. This was Milnor's definition, and it also follows from more general definitions of higher $$ K $$-groups.

It is also the kernel of the mapping $$ \varphi: \operatorname{St}(A) \to {\operatorname{GL}_{\infty}}(A) $$. Indeed, there is an exact sequence
 * $$ 1 \to {K_{2}}(A) \to \operatorname{St}(A) \to {\operatorname{GL}_{\infty}}(A) \to {K_{1}}(A) \to 1. $$

Equivalently, it is the Schur multiplier of the group of elementary matrices, so it is also a homology group: $$ {K_{2}}(A) = {H_{2}}(E(A);\mathbb{Z}) $$.

K3
showed that $$ {K_{3}}(A) = {H_{3}}(\operatorname{St}(A);\mathbb{Z}) $$.