View source for Special linear group

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{{MSC|20Gxx|19-XX,11E57,20G15}}

The special linear group 
''of degree (order) $\def\SL{\textrm{SL}}\def\GL{\textrm{GL}} n$ over a ring $R$''
is the subgroup $\SL(n,R)$ of the
[[General linear group|general linear group]] $\GL(n,R)$ which is the kernel of a determinant homomorphism $\det_n$. The structure of $\SL(n,R)$ depends on $R$, $n$ and the type of determinant defined on $\GL(n,R)$. There are three main types of determinants of importance here: the ordinary determinant in the case when $R$ is a commutative ring, the non-commutative Dieudonné determinant (cf.
[[Determinant|Determinant]]) when $R$ is a division ring (see
{{Cite|Ar}}), and the [[Reduced norm|reduced norm homomorphism]] for a division ring $R$ that is finite dimensional over its centre (see
{{Cite|Ba}}).

$\SL(n,R)$ has the following noteworthy subgroups: the group $\def\E{\textrm{E}}\E(n,R$ generated by the elementary matrices $\def\l{\lambda} e_{ij}^\l$ (see
[[Algebraic K-theory|Algebraic $K$-theory]]) and, for every two-sided ideal $q$ of $R$, the congruence subgroup $\SL(n,R,q)$ and the group $\def\E{\textrm{E}}\E(n,R,q)$ which is the normal subgroup of $\E(n,R)$ generated by the matrices $e_{ij}^\l$ for $\l\in q$. Let $A\in \E(n,R,q)$ and let

$$A \mapsto \begin{pmatrix}A & 0\\ 0 & 1\end{pmatrix}$$
be the imbedding of $\E(n,R,q)$ into $\E(n+1,R,q)$. Then passage to the direct limit gives the group $\E(R,q)$. The group $\SL(R,q)$ is defined in a similar way. When $q=R$ one writes $\E(R)$ and $\SL(R)$ instead of $\E(R,R)$ and $\SL(R,R)$, respectively. The latter is called the stable special linear group of the ring $R$. The normal subgroup structure of $\SL(R)$ is closely connected with the structure of the groups $\SL(n,R)$: A subgroup $H$ is normal in $\SL(R)$ if and only if, for some (unique) two-sided ideal $q$ of $R$, the following inclusions hold:

$$\E(R,q)\subset H\subset \SL(R,q)$$
Thus, the Abelian groups $\def\SK{\textrm{SK}}\SK_1(R,q) = \SL(R,q)/\E(R,q)$ classify the normal subgroups of $\SL(R)$. The group $\SK_1(R) = \SK(R,R)$ is called the reduced Whitehead group of $R$. A satisfactory description of the normal subgroup structure of $\SL(n,R)$ for an arbitrary ring $R$ uses a condition involving the
[[Stable rank|stable rank]] of the ideal $q$ ($\def\str{\textrm{str}\;}\str q$). Namely, if $n\ge \str q +1$, then there is an isomorphism

$$\SL(n,R,q)/\E(nR,q) \simeq \SK_1(R,q)$$
In addition, if the conditions $n\ge \str R + 1 $, $n\ge 3$ hold, then for every normal subgroup $H$ of $\SL(n,R)$ the inclusions

$$\E(n,R,q)\subset H\subset \SL'(n,R,q)$$
hold for an appropriate $q$, where $\SL'(nR,q) = \GL'(n,R,q)\cap \SL(n,R)$, and $\GL'(n,R,q)$ is the pre-image of the centre of $\GL(n,R/q)$ in $\GL(n,R)$. For certain special rings definitive results are known (see
{{Cite|Ba}},
{{Cite|Su}}, for example).

In the case of the non-commutative Dieudonné determinant (so that $R$ is a division ring), the results are exhaustive. The groups $\SL(n,R)$ and $E(n,R)$ coincide. $\SL(n,R)$ is the commutator subgroup of $\GL(n,R)$, except in the case of $\SL(2,\F_2)$ (where $\F_q$ denotes the field of $q$ elements). The centre $Z_n$ of $\SL(n,R)$ consists of the scalar matrices $\def\diag{\textrm{diag}}\def\a{\alpha} \diag(\a,\dots,\a)$, where $\a$ is an element of the centre of $R$ and $\a^n\in[R^*,R^*]$, $[R^*,R^*]$ being the commutator subgroup of the multiplicative group $R^*$ of the division ring $R$. The quotient group $SL(n,R)/Z_n$ is simple except when $n=2$ and $R=\F_2,\F_3$. When $n=2$, $\SL(2,\F_2) = \SL(2,\F_2)/Z_2$ and $\SL(2,\F_2)$ is isomorphic to the
[[Symmetric group|symmetric group]] $S_3$ of degree 3, while $\SL(2,\F_3)/Z_2$ is isomorphic to the
[[Alternating group|alternating group]] $A_4$ of degree 4.

If $\det_n$ is a reduced norm homomorphism, then

$$\SL(n,R)/\E(n,R) \simeq \SK_1(R)$$
and

$$\SK_1(R)\simeq \SL(1,R)/[R^*,R^*],$$
so that the group $\SK_1(R)$ is trivial when $R$ is a field. The conjecture that $\SK_1(R) = \{0\}$ for any division ring $R$ stood for a long time. However, in 1975 it was shown that this is not true (see
{{Cite|Pl}}). The groups $\SK_1(R)$ play an important role in algebraic geometry (see
{{Cite|Pl2}},
{{Cite|Pl3}}). There are also generalizations of the reduced norm homomorphism, which have stimulated a series of new investigations into special linear groups.

====References====
{|
|-
|valign="top"|{{Ref|Ar}}||valign="top"|  E. Artin,   "Geometric algebra", Interscience  (1957)    {{MR|0082463}}  {{ZBL|0077.02101}} 
|-
|valign="top"|{{Ref|Ba}}||valign="top"|  H. Bass,   "Algebraic K-theory", Benjamin  (1968)     {{MR|0249491}}  {{ZBL|0174.30302}} 
|-
|valign="top"|{{Ref|HaOM}}||valign="top"|  A.J. Hahn, O.T. O'Meara, "The classical groups and K-theory", Springer  (1989)    {{MR|1007302}}  {{ZBL|0683.20033}} 
|-
|valign="top"|{{Ref|Mi}}||valign="top"|  J.W. Milnor,   "Introduction to algebraic K-theory", Princeton Univ. Press  (1971)     {{MR|0349811}}  {{ZBL|0237.18005}} 
|-
|valign="top"|{{Ref|Pl}}||valign="top"|  V.P. Platonov,   "A problem of Tannaka–Artin and groups of projective conorms"  ''Soviet Math. Dokl.'', '''16'''  (1975)  pp. 781–786  ''Dokl. Akad. Nauk SSSR'', '''222''' :  2  (1975)  pp. 1299–1302   {{ZBL|0338.16004}} 
|-
|valign="top"|{{Ref|Pl2}}||valign="top"|  V.P. Platonov,   "The Tannaka–Artin problem and reduced K-theory"  ''Math. USSR Izv.'', '''10'''  (1976)  pp. 211–243  ''Izv. Akad. Nauk SSSR Ser. Mat.'', '''40''' :  2  (1976)  pp. 227–261     {{MR|0424872}} {{MR|0407082}}  
|-
|valign="top"|{{Ref|Su}}||valign="top"|  A.A. Suslin,   "On the structure of the special linear group over polynomial rings"  ''Math. USSR Izv.'', '''11'''  (1977)  pp. 211–238  ''Izv. Akad. Nauk SSSR Ser. Mat.'', '''41''' :  2  (1977)  pp. 235–252    {{ZBL|0378.13002}} {{ZBL|0354.13009}} 
|-
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