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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s1305401.png" /> be the [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s1305402.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s1305403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s1305404.png" /> any [[Field|field]]). (Much of what follows holds for arbitrary simple algebraic groups, not just for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s1305405.png" />.) For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s1305406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s1305407.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s1305408.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s1305409.png" /> denote the element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054010.png" /> which differs from the identity matrix only in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054011.png" />-entry, which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054012.png" /> rather than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054013.png" />. The following relations hold for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054014.png" /> as above and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054015.png" />:
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a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054016.png" />;
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b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054017.png" /> Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054018.png" /> denotes the commutator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054019.png" />.
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Let $G$ be the [[Group|group]] $\operatorname {SL} _ { n } ( F )$ ($n \geq 3$, $F$ any [[Field|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 $a , b \in F$:
  
R. Steinberg [[#References|[a4]]] proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054020.png" /> denotes the abstract group defined by these generators and relations and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054021.png" /> is the resulting [[Homomorphism|homomorphism]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054022.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054024.png" /> is a universal central extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054025.png" />: its kernel is central and it covers all central extensions uniquely (cf. also [[Extension of a group|Extension of a group]]). It follows that every [[Projective representation|projective representation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054026.png" /> lifts uniquely to a [[Linear representation|linear representation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054027.png" />, and, at least when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054028.png" /> is finite, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054029.png" /> is just the [[Schur multiplicator|Schur multiplicator]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054030.png" />, which was the motivation for Steinberg's study.
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a) $x _ { i j } ( a ) x _ {i j } ( b ) = x _ { i j } ( a + b )$;
  
Now, in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054031.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054035.png" /> and finally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054036.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054037.png" />, the group of units of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054038.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054039.png" /> works out to the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054040.png" />, it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054041.png" /> is always in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054042.png" />. As is mostly shown in [[#References|[a4]]], these elements generate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054043.png" /> and they satisfy:
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b) $( x _ { i j } ( a ) , x _ { k \text{l} } ( b ) ) = \left\{ \begin{array} { l l } { 1 } &amp; { \text { if } i \neq \text{l} , j \neq k }, \\ { x _ { i \text{l} } ( a b ) } &amp; { \text { if } i \neq \text{l} , j = k }. \end{array} \right.$ Here, $( x , y )$ denotes the commutator $x y x ^ { - 1 } y ^ { - 1 }$.
  
c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054044.png" /> is multiplicative as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054045.png" /> or of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054046.png" />;
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R. Steinberg [[#References|[a4]]] proved that if $H$ denotes the abstract group defined by these generators and relations and $\pi$ is the resulting [[Homomorphism|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|Extension of a group]]). It follows that every [[Projective representation|projective representation]] of $G$ lifts uniquely to a [[Linear representation|linear representation]] of $H$, and, at least when $F$ is finite, that $\operatorname{Ker} \pi$ is just the [[Schur multiplicator|Schur multiplicator]] of $G$, which was the motivation for Steinberg's study.
  
d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054047.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054048.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054049.png" />). Matsumoto's theorem [[#References|[a2]]] states that c) and d) form a presentation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054050.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054051.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054052.png" /> and hence may be (and will be) written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054053.png" />. The symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054054.png" /> is called the Steinberg symbol, as is also any symbol in any [[Abelian group|Abelian group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054055.png" /> for which c) and d) hold (which corresponds to a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054056.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054057.png" />).
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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 [[#References|[a4]]], these elements generate $\operatorname{Ker} \pi$ and they satisfy:
  
As a first example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054058.png" /> is finite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054059.png" /> is trivial, with a few exceptions (see [[#References|[a4]]]). Hence a) and b) form a presentation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054060.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054061.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054062.png" />, as above, is an isomorphism.
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c) $\{ a , b \}$ is multiplicative as a function of $a$ or of $b$;
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054063.png" /> is a [[Differential field|differential field]], then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054064.png" /> defines a symbol into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054065.png" />.
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d) $\{ a , b \} = 1$ if $a + b = 1$ (and $a , b \in F ^ { * }$). Matsumoto's theorem [[#References|[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|Abelian group]] $A$ for which c) and d) hold (which corresponds to a homomorphism of $K _ { 2 } F$ into $A$).
  
Consider next the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054066.png" /> and its completions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054068.png" /> (one for each prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054069.png" />), which are topological fields (cf. also [[Topological field|Topological field]]). According to J. Tate (see [[#References|[a3]]]),
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As a first example, if $F$ is finite, then $K _ { 2 } F$ is trivial, with a few exceptions (see [[#References|[a4]]]). Hence a) and b) form a presentation of $\operatorname {SL} _ { n } ( F )$ ($n \geq 3$) and $\pi$, as above, is an isomorphism.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054070.png" /></td> </tr></table>
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If $F$ is a [[Differential field|differential field]], then $\{ a , b \} = d a / a \wedge d b / b$ defines a symbol into $\Delta ^ { 2 } F$.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054071.png" /> is the group of roots of unity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054072.png" />, which is cyclic, of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054073.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054074.png" /> and of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054075.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054076.png" /> is odd. The factor for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054077.png" /> odd arises from the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054078.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054079.png" />, and hence also on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054080.png" />, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054081.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054083.png" /> units in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054084.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054085.png" /> generates the group of continuous symbols on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054086.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054087.png" /> [[#References|[a3]]], one of the interpretations of this result is that the [[Fundamental group|fundamental group]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054088.png" /> is cyclic of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054089.png" />. And similarly for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054090.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054091.png" /> one again gets the group of roots of unity, generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054092.png" />, which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054093.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054095.png" /> are both negative and is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054096.png" /> otherwise. Fitting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054097.png" /> into Tate's formula above is the last step in a beautiful proof by him (see [[#References|[a3]]]) of Gauss' quadratic reciprocity law (cf. also [[Quadratic reciprocity law|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054098.png" /> (and other groups) over arbitrary algebraic number fields and their completions initiated by C. Moore and completed by H. Matsumoto in [[#References|[a2]]].
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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|Topological field]]). According to J. Tate (see [[#References|[a3]]]),
  
The definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s13054099.png" /> has been extended by J. Milnor [[#References|[a3]]] to arbitrary commutative rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540100.png" /> as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540101.png" /> denote the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540102.png" />-matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540103.png" /> generated by the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540104.png" /> defined earlier, but with no upper bound on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540105.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540106.png" />. The relations a) and b) continue to hold and they again define a universal central extension, whose kernel is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540107.png" />. The motivation comes from [[Algebraic K-theory|algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540108.png" />-theory]], where this definition fits in well with earlier definitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540110.png" /> (see [[#References|[a3]]]) via natural exact sequences, product formulas and so on. The Steinberg symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540111.png" /> still exists, but only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540112.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540113.png" /> commute and are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540114.png" />. For some rings there are enough values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540115.png" /> to generate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540116.png" />, e.g., for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540117.png" /> (in which case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540118.png" /> is of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540119.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540120.png" />), or for any semi-local ring or for any discrete valuation ring (in which case R.K. Dennis and M.R. Stein [[#References|[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*} K _ { 2 } {\bf Q} = \coprod _ { p } \mu _ { p }, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540121.png" /></td> </tr></table>
+
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 $\mathbf{C} ^ { * }$ [[#References|[a3]]], one of the interpretations of this result is that the [[Fundamental group|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 [[#References|[a3]]]) of Gauss' quadratic reciprocity law (cf. also [[Quadratic reciprocity law|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 [[#References|[a2]]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540122.png" /></td> </tr></table>
+
The definition of $K _ { 2 }$ has been extended by J. Milnor [[#References|[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|algebraic $K$-theory]], where this definition fits in well with earlier definitions of $K _ { 0 } R$ and $K _ { 1 } R$ (see [[#References|[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 [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540123.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540124.png" />. There are various identities pertaining to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540125.png" /> and connecting it to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540126.png" />.
+
\begin{equation*} \langle a , b \rangle = \end{equation*}
  
These symbols, and yet others not defined here, have been used to calculate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540127.png" />, or at least to prove that it is non-trivial, for many rings arising in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540128.png" />-theory, number theory, algebraic geometry, topology, and other parts of mathematics.
+
\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 [[#References|[a1]]] and [[#References|[a3]]] give good overall views of the subjects discussed.
 
References [[#References|[a1]]] and [[#References|[a3]]] give good overall views of the subjects discussed.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.K. Dennis, M.R. Stein, "The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540129.png" />: A survey of computations and problems" , ''Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540130.png" />-Theory II'' , ''Lecture Notes in Mathematics'' , '''342''' , Springer (1973) pp. 243–280 {{MR|354815}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Matsumoto, "Sur les sous-groupes arithmétiques des groupes semisimples déployés" ''Ann. Sci. École Norm. Sup. (4)'' , '''2''' (1969) pp. 1–62 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Milnor, "Introduction to algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130540/s130540131.png" />-theory" , ''Ann. of Math. Stud.'' , '''72''' , Princeton Univ. Press (1971) {{MR|349811}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> R. Steinberg, "Générateurs, relations et revêtements de groupes algébriques" , ''Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962)'' , Gauthier-Villars (1962) pp. 113–127 {{MR|0153677}} {{ZBL|0272.20036}} </TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top"> 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 {{MR|354815}} {{ZBL|}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> H. Matsumoto, "Sur les sous-groupes arithmétiques des groupes semisimples déployés" ''Ann. Sci. École Norm. Sup. (4)'' , '''2''' (1969) pp. 1–62 {{MR|}} {{ZBL|}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> J. Milnor, "Introduction to algebraic $K$-theory" , ''Ann. of Math. Stud.'' , '''72''' , Princeton Univ. Press (1971) {{MR|349811}} {{ZBL|}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> R. Steinberg, "Générateurs, relations et revêtements de groupes algébriques" , ''Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962)'' , Gauthier-Villars (1962) pp. 113–127 {{MR|0153677}} {{ZBL|0272.20036}} </td></tr></table>

Latest revision as of 16:56, 1 July 2020

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
How to Cite This Entry:
Steinberg symbol. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steinberg_symbol&oldid=50120
This article was adapted from an original article by Robert Steinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article