In
algebraic K-theory
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
, a field of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Steinberg group
of a ring
is the
universal central extension
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations.
Examples and properties
The Schur multiplier \oper ...
of the
commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal s ...
of the stable
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
of
.
It is named after
Robert Steinberg
Robert Steinberg (May 25, 1922, Soroca, Bessarabia, Romania (present-day Moldova) – May 25, 2014) was a mathematician at the University of California, Los Angeles.
He introduced the Steinberg representation, the Lang–Steinberg theorem, t ...
, and it is connected with
lower -groups, notably
and
.
Definition
Abstractly, given a ring
, the Steinberg group
is the
universal central extension
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations.
Examples and properties
The Schur multiplier \oper ...
of the
commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal s ...
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
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
is as follows.
Elementary matrices — i.e. matrices of the form
, where
is the identity matrix,
is the matrix with
in the
-entry and zeros elsewhere, and
— satisfy the following relations, called the Steinberg relations:
:
The unstable Steinberg group of order
over
, denoted by
, is defined by the generators
, where
and
, these generators being subject to the Steinberg relations. The stable Steinberg group, denoted by
, is the
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...
of the system
. It can also be thought of as the Steinberg group of infinite order.
Mapping
yields a
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
wh ...
. As the elementary matrices generate the
commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal s ...
, this mapping is surjective onto the commutator subgroup.
Interpretation as a fundamental group
The Steinberg group is the
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of the
Volodin space, which is the union of
classifying space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free acti ...
s of the
unipotent
In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' − 1 is a nilpotent element; in other words, (''r'' − 1)''n'' is zero for some ''n''.
In particular, a square matrix ''M'' is a unipoten ...
subgroups of GL(''A'').
Relation to ''K''-theory
''K''1
is the
cokernel
The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of .
Cokernels are dual to the kernels of category theory, hence the name: ...
of the map
, as
is the abelianization of
and the mapping
is surjective onto the commutator subgroup.
''K''2
is the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
of the Steinberg group. This was Milnor's definition, and it also follows from more general definitions of higher
-groups.
It is also the kernel of the mapping
. Indeed, there is an
exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context o ...
:
Equivalently, it is the
Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations.
Examples and properties
The Schur multiplier \oper ...
of the group of
elementary matrices, so it is also a
homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
:
.
''K''3
showed that
.
References
*
*
*{{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