In algebraic K-theory, a field of mathematics, the Steinberg group
is the universal central extension of the commutator subgroup of the stable general linear group of
It is named after Robert Steinberg, and it is connected with lower
Abstractly, given a ring
, the Steinberg group
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).
A concrete presentation using generators and relations is as follows.
is the identity matrix,
— satisfy the following relations, called the Steinberg relations: The unstable Steinberg group of order
, is defined by the generators
, these generators being subject to the Steinberg relations.
The stable Steinberg group, denoted by
, is the direct limit of the system
It can also be thought of as the Steinberg group of infinite order.
yields a group homomorphism
φ : St (
{\displaystyle \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.
The Steinberg group is the fundamental group of the Volodin space, which is the union of classifying spaces of the unipotent subgroups of
is the cokernel of the map
φ : St (
{\displaystyle \varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)}
is surjective onto the commutator subgroup.
is the center of the Steinberg group.
This was Milnor's definition, and it also follows from more general definitions of higher
It is also the kernel of the mapping
φ : St (
{\displaystyle \varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)}
Indeed, there is an exact sequence Equivalently, it is the Schur multiplier of the group of elementary matrices, so it is also a homology group:
Gersten (1973) showed that