Steinberg symbol
Let be the group
(
,
any field). (Much of what follows holds for arbitrary simple algebraic groups, not just for
.) For
,
,
, let
denote the element of
which differs from the identity matrix only in the
-entry, which is
rather than
. The following relations hold for all
as above and
:
a) ;
b) Here,
denotes the commutator
.
R. Steinberg [a4] proved that if denotes the abstract group defined by these generators and relations and
is the resulting homomorphism of
onto
, then
is a universal central extension of
: its kernel is central and it covers all central extensions uniquely (cf. also Extension of a group). It follows that every projective representation of
lifts uniquely to a linear representation of
, and, at least when
is finite, that
is just the Schur multiplicator of
, which was the motivation for Steinberg's study.
Now, in the group , let
,
,
,
and finally
for all
, the group of units of
. Since
works out to the matrix
, it follows that
is always in
. As is mostly shown in [a4], these elements generate
and they satisfy:
c) is multiplicative as a function of
or of
;
d) if
(and
). Matsumoto's theorem [a2] states that c) and d) form a presentation of
. Thus,
is independent of
and hence may be (and will be) written
. The symbol
is called the Steinberg symbol, as is also any symbol in any Abelian group
for which c) and d) hold (which corresponds to a homomorphism of
into
).
As a first example, if is finite, then
is trivial, with a few exceptions (see [a4]). Hence a) and b) form a presentation of
(
) and
, as above, is an isomorphism.
If is a differential field, then
defines a symbol into
.
Consider next the field and its completions
and
(one for each prime number
), which are topological fields (cf. also Topological field). According to J. Tate (see [a3]),
![]() |
where is the group of roots of unity in
, which is cyclic, of order
if
and of order
if
is odd. The factor for
odd arises from the symbol
on
, and hence also on
, in which
, with
,
units in
. Since
generates the group of continuous symbols on
into
[a3], one of the interpretations of this result is that the fundamental group of
is cyclic of order
. And similarly for
. For
one again gets the group of roots of unity, generated by
, which is
if
and
are both negative and is
otherwise. Fitting
into Tate's formula above is the last step in a beautiful proof by him (see [a3]) of Gauss' quadratic reciprocity law (cf. also Quadratic reciprocity law). All of these ideas (as well as the norm residue symbol, for which c) and d) also hold) figure in a deep study of the group
(and other groups) over arbitrary algebraic number fields and their completions initiated by C. Moore and completed by H. Matsumoto in [a2].
The definition of has been extended by J. Milnor [a3] to arbitrary commutative rings
as follows. Let
denote the group of
-matrices over
generated by the matrices
defined earlier, but with no upper bound on
or
. The relations a) and b) continue to hold and they again define a universal central extension, whose kernel is called
. The motivation comes from algebraic
-theory, where this definition fits in well with earlier definitions of
and
(see [a3]) via natural exact sequences, product formulas and so on. The Steinberg symbol
still exists, but only if
and
commute and are in
. For some rings there are enough values of
to generate
, e.g., for
(in which case
is of order
generated by
), or for any semi-local ring or for any discrete valuation ring (in which case R.K. Dennis and M.R. Stein [a1] have given a complete set of relations, which include c) and d) above). For other rings, new symbols are needed. The Dennis–Stein symbol is defined by
![]() |
![]() |
for all commuting pairs such that
. There are various identities pertaining to
and connecting it to
.
These symbols, and yet others not defined here, have been used to calculate , or at least to prove that it is non-trivial, for many rings arising in
-theory, number theory, algebraic geometry, topology, and other parts of mathematics.
References [a1] and [a3] give good overall views of the subjects discussed.
References
[a1] | R.K. Dennis, M.R. Stein, "The functor ![]() ![]() |
[a2] | H. Matsumoto, "Sur les sous-groupes arithmétiques des groupes semisimples déployés" Ann. Sci. École Norm. Sup. (4) , 2 (1969) pp. 1–62 |
[a3] | J. Milnor, "Introduction to algebraic ![]() |
[a4] | R. Steinberg, "Générateurs, relations et revêtements de groupes algébriques" , Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962) , Gauthier-Villars (1962) pp. 113–127 |
Steinberg symbol. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steinberg_symbol&oldid=16963