Special linear group

2020 Mathematics Subject Classification: Primary: 20Gxx Secondary: 19-XX11E5720G15 [MSN][ZBL]

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 $\GL(n,R)$ 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) when $R$ is a division ring (see [Ar]), and the reduced norm homomorphism for a division ring $R$ that is finite dimensional over its centre (see [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) 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 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 [Ba], [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 $S_3$ of degree 3, while $\SL(2,\F_3)/Z_2$ is isomorphic to the 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 [Pl]). The groups $\SK_1(R)$ play an important role in algebraic geometry (see [Pl2], [Pl3]). There are also generalizations of the reduced norm homomorphism, which have stimulated a series of new investigations into special linear groups.

References

[Ar]	E. Artin, "Geometric algebra", Interscience (1957) MR0082463 Zbl 0077.02101
[Ba]	H. Bass, "Algebraic K-theory", Benjamin (1968) MR0249491 Zbl 0174.30302
[HaOM]	A.J. Hahn, O.T. O'Meara, "The classical groups and K-theory", Springer (1989) MR1007302 Zbl 0683.20033
[Mi]	J.W. Milnor, "Introduction to algebraic K-theory", Princeton Univ. Press (1971) MR0349811 Zbl 0237.18005
[Pl]	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
[Pl2]	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 MR0424872 MR0407082
[Su]	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