Talk on Chevalley's integral forms

introduction

• linear algebra : all bases are equal (half true. diagonalization)
• actually 'All bases are equal, but some bases are more equal than others'
• usually good bases : rich source of mathematics
• what are good bases?

integral forms

• Chevalley 1955, integral forms for finite-dimensional simple Lie algebras => construction of Chevalley groups (Chevalley integral form)
• Kostant 1966, integral forms for the UEAs of simple Lie algebras (see The fake monster formal group by Borcherds for more)
• \(A\) : algebra (or vector space) over \(\mathbb{C}\) (for any field \(\mathbb{F}\) of characteristic 0)

def

An integral form (or a \(\mathbb{Z}\)-form) \(A_\mathbb{Z}\) of \(A\) to be a \(\mathbb{Z}\)-algebra (\(\mathbb{Z}\)-module) such that \(\mathbb{C}\otimes_\mathbb{Z}A_\mathbb{Z}=A\).

An integral basis for \(A\) is a \(\mathbb{Z}\)-basis for \(A_\mathbb{Z}\).

review of basics on \(\mathfrak{sl}_2\)

Lie algebra \(\mathfrak{sl}(2)\)

• \(\mathfrak{g}=\mathbb{C}\langle E,F,H \rangle\)
• commutator

\[ [E,F]=H \\ [H,E]=2E \\ [H,F]=-2F \]

• \(\mathfrak{g}_{\mathbb{Z}}=\mathbb{\mathbb{Z}}\langle E,F,H \rangle\) is an integral form (\(\mathfrak{g}_{\mathbb{Z}}\) is a Lie algebra over \(\mathbb{Z}\))

UEA

• universal enveloping algebra \(U(\mathfrak{g})\) PBW basis

\[\{F^kH^lE^m|k,l,m\geq 0\}\]

• Hopf algebra with coproduct \(\Delta : U(\mathfrak{g})\to U(\mathfrak{g})\) defined by \(\Delta(x)=x\otimes 1+1\otimes x\) for \(x\in \mathfrak{g}\)
• integral form and integral basis ? answer later

finite dimensional representations

• \(V\) : irreducible finite dimensional module
• \(V=\oplus_{\mu\in\mathbb{C}}V_{\mu}\), \(V_{\mu}=\{v\in V|Hv=\mu v\}\)
• there exists \(v_0\neq 0\) such that

\[Ev_0=0\] \[Hv_0=\lambda v_0\]

• let \(F^{(j)}:=\frac{F^j}{j!}\), \(E^{(j)}:=\frac{E^j}{j!}\)
• define \(v_j:=F^{(j)}v_0, j \in\mathbb{Z}_{\geq 0}\), we have

\[H v_j=(\lambda -2j)v_j\] \[F v_j=(j+1)v_{j+1}\] \[E v_j=(\lambda -j+1)v_{j-1}\]

• as \(V\) is finite dimensional, there exists \(l\in \in\mathbb{Z}_{\geq 0}\) such that \(v_m\neq 0\) and \(v_{m+1}=0\)
• then \(Ev_{m+1}=(\lambda-m)v_{m}=0\) and so \(\lambda-m=0\). So \(\lambda \in\mathbb{Z}_{\geq 0}\)
• Let \(V_{\mathbb{Z}}\) be the \(\mathbb{Z}\)-span of \(\{v_j|j\geq 0\}\)
• as \(V\) is irreducible, \(V=\mathbb{C}\otimes_{\mathbb{Z}}V_{\mathbb{Z}}\)
• so \(V_{\mathbb{Z}}\) is an integral form for \(V\) with integral basis \(\{v_0,\cdots, v_m\}\)

Question.

where do \(F^{(j)}\) come from?

prop

\(V_{\mathbb{Z}}\) is stable under the action of \(F^{(j)}\) and \(E^{(j)}\) and thus stable also under the action of \(\exp (tE)\) and \(\exp (tF)\) (matrices with coefficients in \(\mathbb{Z}[t]\), key fact to define the Chevalley groups)

basis of \(\mathfrak{g}\) and structure constants

basis

• simple Lie algebra \(\mathfrak{g}\) over \(\mathbb{C}\), we have a non-deg invariant bilinear form \((\cdot,\cdot)\).
• fix a Cartan subalgebra \(\mathfrak{h}\)
• \(\Delta\) : root system
• \(\Pi\) : fundamental system
• Cartan decomposition

\[ \mathfrak{g}=\mathfrak{h}\oplus \left(\oplus_{\alpha\in \Delta} \mathfrak{g}_{\alpha}\right) \]

• fix \(H_{\alpha}\in \mathfrak{h}\) uniquely for each \(\alpha\in \Delta\) by

\[ \beta(H_{\alpha})=2\frac{(\alpha,\beta)}{(\alpha,\alpha)}\,\quad \beta\in \mathfrak{h}^{*} \]

exercise

\(H_{\alpha}\) can be written as a \(\mathbb{Z}\)-linear combination of \(H_{\alpha_i}, \alpha_i\in \Pi\).

• we can choose \(x_{\alpha}\in \mathfrak{g}_{\alpha}\) so that

\[[x_{\alpha},x_{-\alpha}]=H_{\alpha}\]

• The elements \(\{H_{\alpha_i} : \alpha_i\in \Pi\}\) together with elements \(x_{\alpha}\in \mathfrak{g}_{\alpha}\) (\(\alpha\in \Delta\)) form a basis of \(\mathfrak{g}\)

structure constants

• multiplication in \(\mathfrak{g}\)

\[ [h,x_{\alpha}]=\alpha(h)x_{\alpha}\\ [x_{\alpha},x_{-\alpha}]=H_{\alpha}\\ [x_{\alpha},x_{\beta}]=n_{\alpha,\beta}x_{\alpha+\beta} \]

• structure constants \(n_{\alpha,\beta}\)
• \(n_{\alpha,\beta}\neq 0\) only if \(\alpha+\beta\in \Delta\)
• \(n_{\alpha,\beta}\) is not fixed by the above condition. how much freedom do we have?
• The structure constants \(n_{\alpha,\beta}\) for extraspecial pairs \((\alpha,\beta)\) can be chosen as arbitrary non-zero elements of \(\mathbb{C}\), by appropriate choice of the elements \(x_{\alpha}\).
• All the structure constants \(n_{\alpha,\beta}\) are determined by the structure constants for extraspecial pairs.
• see Lie Algebras of Finite and Affine Type by Carter for more

Chevalley

• a synthesis between the theory of Lie groups and the theory of finite groups

observation

• if we make another choice \(x_{\alpha}'=u_{\alpha}x_{\alpha}\) with \(u_{\alpha}u_{-\alpha}=1\), then structure constants satisfy the following property

\[ n_{\alpha,\beta}'n_{-\alpha,-\beta}'=n_{\alpha,\beta}n_{-\alpha,-\beta} \]

lemma

The number \(n_{\alpha,\beta}n_{-\alpha,-\beta}\) is given by \(-(p+1)^2\) where \(p\) is the largest integer \(\geq 0\) such that \(\beta-p\alpha\in \Delta\). (\(\alpha\) string through \(\beta\))

remark

it is the minimum \(p\in \mathbb{Z}_{\geq 0}\) such that \[ \left(\text{ad} x_{-\alpha}\right)^{p}\left(x_{\beta}\right)=0 \]

lemma

It is possible to choose basis elements \(x_{\alpha}'\in \mathfrak{g}_{\alpha}\) such that \([x_{\alpha}',x_{-\alpha}']=H_{\alpha}\), and \(n_{-\alpha,-\beta}=-n_{\alpha,\beta}\) for all \(\alpha\) and \(\beta\). For this choice of \(x_{\alpha}'\), we have \(n_{\alpha,\beta}=\pm (p+1)\)

Hint : Use the Chevalley involution \(\sigma :\mathfrak{g}\to \mathfrak{g}\). It is an involution with \(\sigma(h)=-h\) for any \(h\in \mathfrak{h}\) and \(\sigma(x_{\alpha})=x_{-\alpha}\).

Chevalley basis

thm (Chevalley 1955)

The elements \(\{H_{\alpha_i} : \alpha_i\in \Pi\}\) together with elements \(X_{\alpha}\in \mathfrak{g}_{\alpha}\) (\(\alpha\in \Delta\)) chosen to satisfy \([X_{\alpha},X_{-\alpha}]=H_{\alpha}\) and \([X_{\alpha},X_{\beta}]=\pm (p+1) X_{\alpha+\beta}\) (if \(\alpha+\beta\in \Delta)\) form a basis for a \(\mathbb{Z}\)-form \(\mathfrak{g}_{\mathbb{Z}}\) of \(\mathfrak{g}\).

Kostant

integral form

• Let \(\{X_{\alpha}\}\) and \(\{H_{\alpha_i}\}\) be a Chevalley basis for \(\mathfrak{g}\)
• Let \(U(\mathfrak{g})_{\mathbb{Z}}\) be the \(\mathbb{Z}\)-subalgebra of \(U(\mathfrak{g})\) generated by \(X_{\alpha}^{(n)}=X_{\alpha}^{n}/n!\) for all \(\alpha\in \Delta\) and \(n\in \mathbb{Z}_{\geq 0}\).
• it is an integral form for \(U(\mathfrak{g})\)
• can we describe its integral basis?

basis

• let \(\Delta^{+}=\{\alpha_1,\cdots, \alpha_N\}\) be an ordered set of positive roots
• for \(Q=(q_1,\cdots, q_N)\) with \(q_i\in \mathbb{Z}_{\geq 0}\), put

\[ e_{Q}=\prod_{i=1}^N X_{\alpha_i}^{(q_i)} \]

• for \(S=(s_1,\cdots, s_N)\) with \(q_i\in \mathbb{Z}_{\geq 0}\), put

\[ f_{S}=\prod_{i=1}^N X_{-\alpha_i}^{(s_i)} \]

• for \(x\in \mathfrak{g}\) and \(s\in \mathbb{Z}_{\geq 0}\), put

\[ \binom{x}{s}=\frac{x(x-1)\cdots (x-s+1)}{s!}\in U(\mathfrak{g}) \]

• let \(l\) be the rank of \(\mathfrak{g}\) for each \(l\)-tuple \(P=(p_1,\cdots, p_l)\), define

\[ h_{P}=\prod_{i=1}^{l}\binom{H_{\alpha_i}}{p_i} \]

thm (Kostant 1966)

The elements \[ \{f_{Q}h_Pe_{S}|Q\in\mathbb{Z}_{\geq 0}^{N},P\in\mathbb{Z}_{\geq 0}^{l},S\in\mathbb{Z}_{\geq 0}^{N}\} \] form an integral basis for \(U(\mathfrak{g})_{\mathbb{Z}}\).

• See Humphreys chapter 26 for a proof

example

• for \(\mathfrak{g}=\mathfrak{sl}_2\),

\[\{\frac{F^k}{k!}\binom{H}{l}\frac{E^m}{m!}|k,l,m\geq 0\}\]

• let us compute \(E^2F^2\)

\[ E^2F^2=2 H^2-8 FE-2 H+F^2E^2+4 FHE \]

• thus

\[ E^{(2)}F^{(2)}=\frac{H^2}{2}-2FE-\frac{H}{2}+F^{(2)}E^{(2)}+FHE \]

• so we cannot use \(\frac{H^k}{k!}\) as elements of integral basis
• that's where \(\binom{H}{2}=\frac{H^2}{2}-\frac{H}{2}\) comes from. In general, we have

\[ E^{(m)}F^{(n)}=\sum_{j=0}^k F^{(n-j)}\binom{H-m-n+2j}{j}E^{(m-j)} \] where \(k=\min(m,n)\)

exercise

Let \(j,k\in\mathbb{Z}_{\geq 0}\). The polynomial \(\binom{x-j}{k}\) can be written as a \(\mathbb{Z}\)-linear combination of \(\binom{x}{i}\)'s.

properties

• \(\exp(tE)\) and \(\exp(tF)\) exist in \(U(\mathfrak{g})_{\mathbb{Z}}[[t]]\)
• a nice property of this integral form is

\[ \Delta(Z_{\alpha}) = \sum_{0\leq\beta\leq\alpha}Z_{\beta} \otimes Z_{\alpha−\beta}. \] where \(Z_{\alpha}=f_{Q}h_Pe_{S}\), \(\alpha=(Q,P,S)\in\mathbb{Z}_{\geq 0}^{N+l+N}\) and \(\Delta : U(\mathfrak{g})\to U(\mathfrak{g})\) is the coproduct defined by \[ \Delta(x)=x\otimes 1+1\otimes x \] for \(x\in \mathfrak{g}\)

• partial ordering on \(\mathbb{Z}_{\geq 0}^{N+l+N}\)

remarks on Chevalley groups

def

An admissible integral form of a \(\mathfrak{g}\)-module \(V\) is an integral form \(M\) such that \(U(\mathfrak{g})_{\mathbb{Z}}\cdot M\subseteq M\)

prop

Let \(V\) be a finite dimensional \(\mathfrak{g}\)-module. Then \(V\) has an admissible integral form. If \(V\) is irreducible and \(v_0\) is a highest weight vector, then \(U(\mathfrak{g})_{\mathbb{Z}}.v_0\) is an admissible integral form of \(V\).

• Let \((\rho,V)\) a faithful representation of \(\mathfrak{g}\) and \(M\) an admissible integral form
• Choose an integral basis \(\{m_1,\cdots, m_d\}\)of \(M\). Then \(e_{\alpha}(t):=\exp \left(t\rho(X_{\alpha})\right)\in GL_{d}(\mathbb{Z}[t])\)
• now let \(k\) arbitrary field and \(M^k=k\otimes_{\mathbb{Z}}M\), a vector space over \(k\)

def

The Chevalley group \(G_{V,k}\) is the subgroup of \(GL(M^k)\) generated by all \(e_{\alpha}(u),\, \alpha\in \Delta, u\in k\) regarded as a \(k\)-linear transformation of \(M^k\)

• it only depends on \(k\) and the lattice of weights \(\Gamma_{V}\) of \(\mathfrak{g}\)-module \(V\) (not on \(M\))
• When \(\Gamma_V=Q\), \(Q\) the root lattice, we call it an adjoint Chevalley group

thm (Chevalley-Dickson theorem)

Let \(G\) be an adjoint Chevalley group. If \(|k|=2\), suppose \(\Delta\) is not of type \(A_1,B_2\) or \(G_2\). If \(|k|=3\), suppose that \(\Delta\) is not of type \(A_1\). Then \(G\) is a simple group.

• See (Curtis, 'Chevalley groups and related topics' thm 6.11) for a proof.
• Finite reductive groups and groups of Lie type