Steinberg group (K-theory)

In algebraic K-theory, a field of mathematics, the Steinberg group [math]\displaystyle{ \operatorname{St}(A) }[/math] of a ring [math]\displaystyle{ A }[/math] is the universal central extension of the commutator subgroup of the stable general linear group of [math]\displaystyle{ A }[/math]. It is named after Robert Steinberg, and it is connected with lower [math]\displaystyle{ K }[/math]-groups, notably [math]\displaystyle{ K_{2} }[/math] and [math]\displaystyle{ K_{3} }[/math].

Definition

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

[math]\displaystyle{ \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} }[/math]

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

Mapping [math]\displaystyle{ {x_{ij}}(\lambda) \mapsto {e_{ij}}(\lambda) }[/math] yields a group homomorphism [math]\displaystyle{ \varphi: \operatorname{St}(A) \to {\operatorname{GL}_{\infty}}(A) }[/math]. 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 [math]\displaystyle{ \operatorname{GL}(A) }[/math].

Relation to K-theory

K₁

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

K₂

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

It is also the kernel of the mapping [math]\displaystyle{ \varphi: \operatorname{St}(A) \to {\operatorname{GL}_{\infty}}(A) }[/math]. Indeed, there is an exact sequence

[math]\displaystyle{ 1 \to {K_{2}}(A) \to \operatorname{St}(A) \to {\operatorname{GL}_{\infty}}(A) \to {K_{1}}(A) \to 1. }[/math]

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

K₃

(Gersten 1973) showed that [math]\displaystyle{ {K_{3}}(A) = {H_{3}}(\operatorname{St}(A);\mathbb{Z}) }[/math].

References

• Gersten, S. M. (1973), "[math]\displaystyle{ K_{3} }[/math] of a Ring is [math]\displaystyle{ H_{3} }[/math] of the Steinberg Group", Proceedings of the American Mathematical Society (American Mathematical Society) 37 (2): 366–368, doi:10.2307/2039440 
• Introduction to Algebraic [math]\displaystyle{ K }[/math]-theory, Annals of Mathematics Studies, 72, Princeton University Press, 1971 
• Steinberg, Robert (1968), Lectures on Chevalley Groups, Yale University, New Haven, Conn., archived from the original on 2012-09-10, https://web.archive.org/web/20120910032654/http://www.math.ucla.edu/~rst/