Chevalley groups

Arun Ram
Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 27 June 2007. This page is the result of joint work with James Parkinson and Christoph Schwer.

Chevalley groups

A Chevalley group is a group in which row reduction works. This means that it is a group with a special set of generators (the "elementary matrices") and relations which are generalizations of the usual row reduction operations. One way to efficiently encode these generators and relations is with a Kac-Moody Lie algebra $𝔤$. A $𝔤$-module is integrable if elements of ${𝔤}_{\alpha }$, $\alpha \in R_{\mathrm{re}}$, act locally nilpotently. The Kac-Moody group is the group $G$ generated by symbols

A prenilpotent pair is a pair of real roots $\alpha ,\beta \in R_{\mathrm{re}}$ such that the Lie subalgebra of $𝔤$ generated by ${𝔤}_{\alpha }$ and ${𝔤}_{\beta }$ is finite dimensional?? nilpotent???. The Steinberg group is the group given by generators $x_{\alpha }\left(f\right),\alpha \in R_{\mathrm{re}},f\in 𝔽,$ and relations

$\begin{array}{rl}\text{(R1)}& x_{\alpha }\left(f_1\right)x_{\alpha }\left(f_2\right)=x_{\alpha }(f_1+f_2),\text{ and}\\ \text{(R3)}& x_{\alpha }\left(f_1\right)x_{\beta }\left(f_2\right)=x_{\beta }\left(f_2\right)x_{\alpha }\left(f_1\right)x_{\alpha +\beta }\left(c_{11}{f}_1{f}_2\right)x_{2\alpha +\beta }\left(c_{21}{f}_1^2{f}_2\right)x_{\alpha +2\beta }\left(c_{12}{f}_1{f}_2^2\right)\cdots ,\end{array}$

for prenilpotent pairs $\alpha ,\beta$. The proof of this relation is found in [St Lemma 15] (is it also in [Bou]??, perhaps near [Bou, VIII 12.5 Lemma 3]? or maybe [Se I 1.1]).

For $1\le i\le n$ and $g\in {𝔽}^{\times }$ define

Let $P^{\vee }$ be a $\mathbb{Z}$-sublattice of $𝔥$ which is stable under the $W$ action and which contains $h_1,\dots ,h_n$. Let $T$ be the group given by generators $h_{{\lambda }^{\vee }}\left(g\right),\text{ for }{\lambda }^{\vee }\in P^{\vee },g\in {𝔽}^{\times }$ and relations

$\begin{array}{rl}\text{(R2)}& h_{{\lambda }^{\vee }}\left(g_1\right)h_{{\lambda }^{\vee }}\left(g_2\right)=h_{{\lambda }^{\vee }}\left(g_1{g}_2\right),\\ \text{(R7)}& h_{{\lambda }^{\vee }}\left(g\right)h_{{\mu }^{\vee }}\left(g\right)=h_{{\lambda }^{\vee }+{\mu }^{\vee }}\left(g\right),\end{array}$

The semidirect product $\mathrm{St}⋉T$ is the group generated by and $T$ with the additional relations

$\begin{array}{rl}\text{(4)}& h_{{\lambda }^{\vee }}\left(g\right)x_{{\alpha }_i}\left(f\right)h_{{\lambda }^{\vee }}(g{)}^{–1}=x_{{\alpha }_i}(g^{⟨{\lambda }^{\vee },\alpha ⟩}f),\end{array}$

Let $G$ be the group $\mathrm{St}⋉T$ with the additional relations

$\begin{array}{rl}\text{(6)}& n_i\left(g\right)=n_i{h}_{{\alpha }_i^{\vee }}\left(g\right),\\ \text{(7)}& n_i{x}_{\alpha }\left(f\right)n_i^{–1}=x_{s_i\alpha }\left(\pm f\right),\\ \text{(5)}& n_i{h}_{{\lambda }^{\vee }}\left(g\right)n_i^{–1}=h_{s_i{\lambda }^{\vee }}\left(\pm g\right),\end{array}$

Strangely?, Tits [Ti, 3.6] does not have the sign in relation (7). Relation (7) is dependent on some rigidity and finiteness of underlying Lie algebra, see [St Lemma 37(a)] and [Ti 3.7(a)]. According to [CC, ???] the sign is given by the choice of basis in $𝔤$ with $\left(\exp \hspace{0.17em}\mathrm{ad}{e}_i\right)\left(\exp \hspace{0.17em}\mathrm{ad}{f}_i\right)\left(\exp \hspace{0.17em}\mathrm{ad}{e}_i\right)\left(e_{\alpha }\right)=\pm e_{s_i\alpha }.$ It follows from (5) and (7) that $G$ has a symmetry under the group

From

$\begin{array}{ccc}{h}_{{\lambda }^{\vee }}\left(g\right)h_{{\alpha }^{\vee }}\left(f\right)n_{\alpha }{h}_{{\lambda }^{\vee }}(g{)}^{–1}& =& x_{\alpha }\left(g^{⟨{\lambda }^{\vee },\alpha ⟩}f\right)x_{–\alpha }(–g^{⟨{\lambda }^{\vee },–\alpha ⟩}{f}^{–1})x_{\alpha }\left(g^{⟨{\lambda }^{\vee },\alpha ⟩}f\right)\\ & =& h_{{\alpha }^{\vee }}\left(g^{⟨{\lambda }^{\vee },\alpha ⟩}\right)h_{{\alpha }^{\vee }}\left(f\right)n_{\alpha }=h_{⟨{\lambda }^{\vee },\alpha ⟩{\alpha }^{\vee }}\left(g\right)h_{{\alpha }^{\vee }}\left(f\right)n_{\alpha }\text{,}\end{array}$

it follows that $h_{{\lambda }^{\vee }–⟨{\lambda }^{\vee },\alpha ⟩{\alpha }^{\vee }}\left(g\right)n_{\alpha }=n_{\alpha }{h}_{{\lambda }^{\vee }}(g{)}^{–1}$, so that

$\begin{array}{rcl}{h}_{{\lambda }^{\vee }}\left(f\right)h_{{\mu }^{\vee }}\left(g\right)& =& h_{{\mu }^{\vee }}\left(f^{–1}\right)h_{{\lambda }^{\vee }+{\mu }^{\vee }}\left(f\right)h_{{\lambda }^{\vee }+{\mu }^{\vee }}\left(g\right)h_{{\lambda }^{\vee }}\left(g^{–1}\right)\\ & =& h_{{\mu }^{\vee }}\left(f^{–1}\right)h_{{\lambda }^{\vee }+{\mu }^{\vee }}\left(fg\right)h_{{\lambda }^{\vee }}{g}^{–1})\\ & =& h_{{\mu }^{\vee }}\left(f^{–1}\right)h_{{\mu }^{\vee }}\left(fg\right)h_{{\lambda }^{\vee }}\left(fg\right)h_{{\lambda }^{\vee }}\left(g^{–1}\right)\\ & =& h_{{\mu }^{\vee }}\left(g\right)h_{{\lambda }^{\vee }}\left(f\right),\end{array}$

if ${\omega }_1^{\vee },\dots ,{\omega }_n^{\vee }$ is a $\mathbb{Z}$-basis of $P^{\vee }$ then

If

$\begin{array}{rl}\text{(R2)}& h_{{\lambda }^{\vee }}\left(g_1\right)h_{{\lambda }^{\vee }}\left(g_2\right)=h_{{\lambda }^{\vee }}\left(g_1{g}_2\right),\\ \text{(R7)}& h_{{\lambda }^{\vee }}\left(g\right)h_{{\mu }^{\vee }}\left(g\right)=h_{{\lambda }^{\vee }+{\mu }^{\vee }}\left(g\right),\\ \text{(c)}& h_{{\alpha }_1^{\vee }}\left(g_1\right)\cdots h_{{\alpha }_n^{\vee }}\left(g_n\right)=1\iff g_1^{⟨\mu ,{\alpha }_1^{\vee }⟩}\cdots g_n^{⟨\mu ,{\alpha }_n^{\vee }⟩}=1,\text{for all weights}\hspace{0.17em}\mu \hspace{0.17em}\text{of}\hspace{0.17em}V\\ \text{(d)}& Z\left(G\right)=\left\{h_{{\alpha }_1^{\vee }}\right(g_1\left)\cdots h_{{\alpha }_n^{\vee }}\right(g_n)\mid g_1^{⟨\beta ,{\alpha }_1^{\vee }⟩}\cdots g_n^{⟨\beta ,{\alpha }_n^{\vee }⟩}=1, \text{for all}\hspace{0.17em}\beta \in R\},\\ \text{(e)}& H=\left\{h_{{\omega }_1^{\vee }}\right(g_1\left)\cdots h_{{\omega }_n^{\vee }}\right(g_n)\mid g_1,\dots ,g_n\in {𝔽}^{\times }\},\end{array}$

if ${\omega }_1^{\vee },\dots ,{\omega }_n^{\vee }$ is a $\mathbb{Z}$-basis of the $\mathbb{Z}$-span of the weights of $V$ and $𝔽$ is big enough.

Examples

Example 1: Let $𝔤=\mathrm{span}\{X_{\alpha },X_{–\alpha },H_{{\alpha }^{\vee }}\}$ with

Example 2: Let $𝔤=𝔰{𝔩}_2=\mathrm{span}\{X_{\alpha },H_{{\alpha }^{\vee }},X_{–\alpha }\}$ and let $V=𝔤$ be the adjoint representation so that

References

[Bou] N. Bourbaki, Groupes et Algebres de Lie Hermann, Paris, 1975.

[CC] R. Carter and Y. Chen, Automorphisms of affine Kac-Moody groups and related Chevalley groups over rings, J. Algebra 155 (1993), no. 1, 44-94. MR1206622 (94e:17032)

[Kac] V. Kac, Infinite dimensional Lie algebras,Second edition, Cambridge University Press, 1985.

[Se] J.-P. Serre, Arbres, amalgames, SL_2 , Asterisque 46 (1977).

[St] R. Steinberg, Lecture Notes on Chevalley groups, Yale University, 1967.

[T] J. Tits, Uniqueness and Presentation of Kac-Moody groups over Fields, J. Algebra 105 No. 2 (1987) 542-573; MR0873684 (89b:17020)