Steinberg group (K-theory)

In algebraic K-theory, a field of mathematics, the Steinberg group $\operatorname(A)$ 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 $K_$ and $K_$.

Definition

Abstractly, given a ring $A$, the Steinberg group $\operatorname(A)$ 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 $(\lambda) := \mathbf + (\lambda)$, where $\mathbf$ is the identity matrix, $(\lambda)$ 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
e_(\lambda) e_(\mu) & = e_(\lambda+\mu); && \\
\left e_(\lambda),e_(\mu) \right& = e_(\lambda \mu), && \text i \neq k; \\
\left e_(\lambda),e_(\mu) \right& = \mathbf, && \text i \neq l \text j \neq k.
\end

The unstable Steinberg group of order $r$ over $A$, denoted by $(A)$, is defined by the generators $(\lambda)$, 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 $\operatorname(A)$, is the direct limit of the system $(A) \to (A)$. It can also be thought of as the Steinberg group of infinite order.

Mapping $(\lambda) \mapsto (\lambda)$ yields a group homomorphism $\varphi: \operatorname(A) \to (A)$. 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 GL(''A'').

Relation to ''K''-theory

''K''₁

$(A)$ is the cokernel of the map $\varphi: \operatorname(A) \to (A)$, as $K_$ is the abelianization of $(A)$ and the mapping $\varphi$ is surjective onto the commutator subgroup.

''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 $\varphi: \operatorname(A) \to (A)$. Indeed, there is an exact sequence
:$1 \to (A) \to \operatorname(A) \to (A) \to (A) \to 1.$

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

''K''₃

showed that $(A) = (\operatorname(A);\mathbb)$.

References

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*{{Citation , last1 = Steinberg , first1 = Robert , title = Lectures on Chevalley Groups , url = https://www.math.ucla.edu/~rst/ , publisher = Yale University, New Haven, Conn. , mr = 0466335 , year = 1968 , url-status = dead , archiveurl = https://web.archive.org/web/20120910032654/http://www.math.ucla.edu/~rst/ , archivedate = 2012-09-10

K-theory