- Source: Steinberg group (K-theory)
In algebraic K-theory, a field of mathematics, the Steinberg group
St
(
A
)
{\displaystyle \operatorname {St} (A)}
of a ring
A
{\displaystyle A}
is the universal central extension of the commutator subgroup of the stable general linear group of
A
{\displaystyle A}
.
It is named after Robert Steinberg, and it is connected with lower
K
{\displaystyle K}
-groups, notably
K
2
{\displaystyle K_{2}}
and
K
3
{\displaystyle K_{3}}
.
Definition
Abstractly, given a ring
A
{\displaystyle A}
, the Steinberg group
St
(
A
)
{\displaystyle \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
p
q
(
λ
)
:=
1
+
a
p
q
(
λ
)
{\displaystyle {e_{pq}}(\lambda ):=\mathbf {1} +{a_{pq}}(\lambda )}
, where
1
{\displaystyle \mathbf {1} }
is the identity matrix,
a
p
q
(
λ
)
{\displaystyle {a_{pq}}(\lambda )}
is the matrix with
λ
{\displaystyle \lambda }
in the
(
p
,
q
)
{\displaystyle (p,q)}
-entry and zeros elsewhere, and
p
≠
q
{\displaystyle p\neq q}
— satisfy the following relations, called the Steinberg relations:
e
i
j
(
λ
)
e
i
j
(
μ
)
=
e
i
j
(
λ
+
μ
)
;
[
e
i
j
(
λ
)
,
e
j
k
(
μ
)
]
=
e
i
k
(
λ
μ
)
,
for
i
≠
k
;
[
e
i
j
(
λ
)
,
e
k
l
(
μ
)
]
=
1
,
for
i
≠
l
and
j
≠
k
.
{\displaystyle {\begin{aligned}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{aligned}}}
The unstable Steinberg group of order
r
{\displaystyle r}
over
A
{\displaystyle A}
, denoted by
St
r
(
A
)
{\displaystyle {\operatorname {St} _{r}}(A)}
, is defined by the generators
x
i
j
(
λ
)
{\displaystyle {x_{ij}}(\lambda )}
, where
1
≤
i
≠
j
≤
r
{\displaystyle 1\leq i\neq j\leq r}
and
λ
∈
A
{\displaystyle \lambda \in A}
, these generators being subject to the Steinberg relations. The stable Steinberg group, denoted by
St
(
A
)
{\displaystyle \operatorname {St} (A)}
, is the direct limit of the system
St
r
(
A
)
→
St
r
+
1
(
A
)
{\displaystyle {\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
i
j
(
λ
)
↦
e
i
j
(
λ
)
{\displaystyle {x_{ij}}(\lambda )\mapsto {e_{ij}}(\lambda )}
yields a group homomorphism
φ
:
St
(
A
)
→
GL
∞
(
A
)
{\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.
= 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
)
{\displaystyle \operatorname {GL} (A)}
.
Relation to K-theory
= K1
=K
1
(
A
)
{\displaystyle {K_{1}}(A)}
is the cokernel of the map
φ
:
St
(
A
)
→
GL
∞
(
A
)
{\displaystyle \varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)}
, as
K
1
{\displaystyle K_{1}}
is the abelianization of
GL
∞
(
A
)
{\displaystyle {\operatorname {GL} _{\infty }}(A)}
and the mapping
φ
{\displaystyle \varphi }
is surjective onto the commutator subgroup.
= K2
=K
2
(
A
)
{\displaystyle {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
{\displaystyle K}
-groups.
It is also the kernel of the mapping
φ
:
St
(
A
)
→
GL
∞
(
A
)
{\displaystyle \varphi :\operatorname {St} (A)\to {\operatorname {GL} _{\infty }}(A)}
. Indeed, there is an exact sequence
1
→
K
2
(
A
)
→
St
(
A
)
→
GL
∞
(
A
)
→
K
1
(
A
)
→
1.
{\displaystyle 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
)
;
Z
)
{\displaystyle {K_{2}}(A)={H_{2}}(E(A);\mathbb {Z} )}
.
= K3
=Gersten (1973) showed that
K
3
(
A
)
=
H
3
(
St
(
A
)
;
Z
)
{\displaystyle {K_{3}}(A)={H_{3}}(\operatorname {St} (A);\mathbb {Z} )}
.
References
Gersten, S. M. (1973), "
K
3
{\displaystyle K_{3}}
of a Ring is
H
3
{\displaystyle H_{3}}
of the Steinberg Group", Proceedings of the American Mathematical Society, 37 (2), American Mathematical Society: 366–368, doi:10.2307/2039440, JSTOR 2039440
Milnor, John Willard (1971), Introduction to Algebraic
K
{\displaystyle K}
-theory, Annals of Mathematics Studies, vol. 72, Princeton University Press, MR 0349811
Steinberg, Robert (1968), Lectures on Chevalley Groups, Yale University, New Haven, Conn., MR 0466335, archived from the original on 2012-09-10
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