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 : A survey of computations and problems" , Algebraic -Theory II , Lecture Notes in Mathematics , 342 , Springer (1973) pp. 243–280
[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 -theory" , Ann. of Math. Stud. , 72 , Princeton Univ. Press (1971)
[a4]	R. Steinberg, "Générateurs, relations et revêtements de groupes algébriques" , Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962) , Gauthier-Villars (1962) pp. 113–127