Steinberg symbol

Let $G$ be the group $\operatorname {SL} _ { n } ( F )$ ($n \geq 3$, $F$ any field). (Much of what follows holds for arbitrary simple algebraic groups, not just for $\operatorname {SL} _ { n }$.) For $i,j = 1 , \ldots , n$, $i \neq j$, $a \in F$, let $x _ { i j } ( a )$ denote the element of $G$ which differs from the identity matrix only in the $( i , j )$-entry, which is $a$ rather than $0$. The following relations hold for all $( i , j )$ as above and $a , b \in F$:

a) $x _ { i j } ( a ) x _ {i j } ( b ) = x _ { i j } ( a + b )$;

b) $( x _ { i j } ( a ) , x _ { k \text{l} } ( b ) ) = \left\{ \begin{array} { l l } { 1 } & { \text { if } i \neq \text{l} , j \neq k }, \\ { x _ { i \text{l} } ( a b ) } & { \text { if } i \neq \text{l} , j = k }. \end{array} \right.$ Here, $( x , y )$ denotes the commutator $x y x ^ { - 1 } y ^ { - 1 }$.

R. Steinberg [a4] proved that if $H$ denotes the abstract group defined by these generators and relations and $\pi$ is the resulting homomorphism of $H$ onto $G$, then $\pi : H \rightarrow G$ is a universal central extension of $G$: its kernel is central and it covers all central extensions uniquely (cf. also Extension of a group). It follows that every projective representation of $G$ lifts uniquely to a linear representation of $H$, and, at least when $F$ is finite, that $\operatorname{Ker} \pi$ is just the Schur multiplicator of $G$, which was the motivation for Steinberg's study.

Now, in the group $H$, let $x ( \alpha ) = x _ { 12 } ( \alpha )$, $y ( a ) = x _ { 21 } ( a )$, $w ( a ) = x ( a ) y ( - a ^ { - 1 } ) x ( a )$, $h ( a ) = w ( a ) w ( 1 ) ^ { - 1 }$ and finally $\{ a , b \} = h ( a b ) h ( a ) ^ { - 1 } h ( b ) ^ { - 1 }$ for all $a , b \in F ^ { * }$, the group of units of $F$. Since $\pi h ( a )$ works out to the matrix $\operatorname { diag } ( a , a ^ { - 1 } , 1,1 , \ldots )$, it follows that $\{ a , b \}$ is always in $\operatorname{Ker} \pi$. As is mostly shown in [a4], these elements generate $\operatorname{Ker} \pi$ and they satisfy:

c) $\{ a , b \}$ is multiplicative as a function of $a$ or of $b$;

d) $\{ a , b \} = 1$ if $a + b = 1$ (and $a , b \in F ^ { * }$). Matsumoto's theorem [a2] states that c) and d) form a presentation of $\operatorname{Ker} \pi$. Thus, $\operatorname{Ker} \pi$ is independent of $n \geq 3$ and hence may be (and will be) written $K _ { 2 } F$. The symbol $\{ ., . \}$ is called the Steinberg symbol, as is also any symbol in any Abelian group $A$ for which c) and d) hold (which corresponds to a homomorphism of $K _ { 2 } F$ into $A$).

As a first example, if $F$ is finite, then $K _ { 2 } F$ is trivial, with a few exceptions (see [a4]). Hence a) and b) form a presentation of $\operatorname {SL} _ { n } ( F )$ ($n \geq 3$) and $\pi$, as above, is an isomorphism.

If $F$ is a differential field, then $\{ a , b \} = d a / a \wedge d b / b$ defines a symbol into $\Delta ^ { 2 } F$.

Consider next the field $\mathbf{Q}$ and its completions $\mathbf{R}$ and $\mathbf{Q} _ { p }$ (one for each prime number $p$), which are topological fields (cf. also Topological field). According to J. Tate (see [a3]),

\begin{equation*} K _ { 2 } {\bf Q} = \coprod _ { p } \mu _ { p }, \end{equation*}

where $\mu _ { p }$ is the group of roots of unity in $\mathbf{Q} _ { p }$, which is cyclic, of order $2$ if $p = 2$ and of order $p - 1$ if $p$ is odd. The factor for $p$ odd arises from the symbol $\{ a , b \} _ { p } = ( - 1 ) ^ { \alpha \beta } r ^ { \beta } s ^ { \alpha }$ on $\mathbf{Q} _ { p }$, and hence also on $\mathbf{Q}$, in which $a, b = p ^ { \alpha } r , p ^ { \beta } s$, with $r$, $s$ units in $\mathbf{Z} _ { p }$. Since $\{ \cdot , \cdot \}_p$ generates the group of continuous symbols on $\mathbf{Q} _ { p }$ into $\mathbf{C} ^ { * }$ [a3], one of the interpretations of this result is that the fundamental group of $\operatorname {SL} _ { n} ( \mathbf{Q} _ { p } )$ is cyclic of order $p - 1$. And similarly for $p = 2$. For $K _ { 2 } \mathbf{R}$ one again gets the group of roots of unity, generated by $\{ a , b \} _ { \infty }$, which is $- 1$ if $a$ and $b$ are both negative and is $1$ otherwise. Fitting $\{ a , b \} _ { \infty }$ 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 $\operatorname {SL} _ { n }$ (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 $K _ { 2 }$ has been extended by J. Milnor [a3] to arbitrary commutative rings $R$ as follows. Let $G = E ( R )$ denote the group of $( \infty \times \infty )$-matrices over $R$ generated by the matrices $x _ { i j }( \cdot )$ defined earlier, but with no upper bound on $i$ or $j$. The relations a) and b) continue to hold and they again define a universal central extension, whose kernel is called $K _ { 2 } R$. The motivation comes from algebraic $K$-theory, where this definition fits in well with earlier definitions of $K _ { 0 } R$ and $K _ { 1 } R$ (see [a3]) via natural exact sequences, product formulas and so on. The Steinberg symbol $\{ a , b \}$ still exists, but only if $a$ and $b$ commute and are in $R ^ { * }$. For some rings there are enough values of $\{ ., . \}$ to generate $K _ { 2 } R$, e.g., for $R = \mathbf{Z}$ (in which case $K _ { 2 } R$ is of order $2$ generated by $\{ - 1 , - 1 \}$), 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

\begin{equation*} \langle a , b \rangle = \end{equation*}

\begin{equation*} = y ( - b ( 1 + a b ) ^ { - 1 } ) x ( a ) y ( b ) x ( - ( 1 + a b ) ^ { - 1 } a ) h ( 1 + a b ) ^ { - 1 } \end{equation*}

for all commuting pairs $a , b \in R$ such that $1 + a b \in R ^ { * }$. There are various identities pertaining to $\langle \, .\, ,\, . \, \rangle$ and connecting it to $\{ ., . \}$.

These symbols, and yet others not defined here, have been used to calculate $K _ { 2 } R$, or at least to prove that it is non-trivial, for many rings arising in $K$-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 $K _ { 2 }$: A survey of computations and problems" , Algebraic $K$-Theory II , Lecture Notes in Mathematics , 342 , Springer (1973) pp. 243–280 MR354815
[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 $K$-theory" , Ann. of Math. Stud. , 72 , Princeton Univ. Press (1971) MR349811
[a4]	R. Steinberg, "Générateurs, relations et revêtements de groupes algébriques" , Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962) , Gauthier-Villars (1962) pp. 113–127 MR0153677 Zbl 0272.20036