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==introduction== | ==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? | * what are good bases? | ||
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| − | == | + | ==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) | |
| − | + | * <math>A</math> : algebra (or vector space) over <math>\mathbb{C}</math> (for any field <math>\mathbb{F}</math> of characteristic 0) | |
| − | * | + | ;def |
| − | + | An ''integral form'' (or a <math>\mathbb{Z}</math>-form) <math>A_\mathbb{Z}</math> of <math>A</math> to be a <math>\mathbb{Z}</math>-algebra (<math>\mathbb{Z}</math>-module) such that <math>\mathbb{C}\otimes_\mathbb{Z}A_\mathbb{Z}=A</math>. | |
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| + | An ''integral basis'' for <math>A</math> is a <math>\mathbb{Z}</math>-basis for <math>A_\mathbb{Z}</math>. | ||
| − | ===sl( | + | ==review of basics on <math>\mathfrak{sl}_2</math>== |
| + | ===Lie algebra <math>\mathfrak{sl}(2)</math>=== | ||
| + | * <math>\mathfrak{g}=\mathbb{C}\langle E,F,H \rangle</math> | ||
| + | * commutator | ||
| + | :<math> | ||
| + | [E,F]=H \\ | ||
| + | [H,E]=2E \\ | ||
| + | [H,F]=-2F | ||
| + | </math> | ||
| + | * <math>\mathfrak{g}_{\mathbb{Z}}=\mathbb{\mathbb{Z}}\langle E,F,H \rangle</math> is an integral form (<math>\mathfrak{g}_{\mathbb{Z}}</math> is a Lie algebra over <math>\mathbb{Z}</math>) | ||
| − | * | + | ===UEA=== |
| − | + | * universal enveloping algebra <math>U(\mathfrak{g})</math> PBW basis | |
| − | * | + | :<math>\{F^kH^lE^m|k,l,m\geq 0\}</math> |
| + | * Hopf algebra with coproduct <math>\Delta : U(\mathfrak{g})\to U(\mathfrak{g})</math> defined by <math>\Delta(x)=x\otimes 1+1\otimes x</math> for <math>x\in \mathfrak{g}</math> | ||
| + | * integral form and integral basis ? answer later | ||
| + | ===finite dimensional representations=== | ||
| + | * <math>V</math> : irreducible finite dimensional module | ||
| + | * <math>V=\oplus_{\mu\in\mathbb{C}}V_{\mu}</math>, <math>V_{\mu}=\{v\in V|Hv=\mu v\}</math> | ||
| + | * there exists <math>v_0\neq 0</math> such that | ||
| + | :<math>Ev_0=0</math> | ||
| + | :<math>Hv_0=\lambda v_0</math> | ||
| + | * let <math>F^{(j)}:=\frac{F^j}{j!}</math>, <math>E^{(j)}:=\frac{E^j}{j!}</math> | ||
| + | * define <math>v_j:=F^{(j)}v_0, j \in\mathbb{Z}_{\geq 0}</math>, we have | ||
| + | :<math>H v_j=(\lambda -2j)v_j</math> | ||
| + | :<math>F v_j=(j+1)v_{j+1}</math> | ||
| + | :<math>E v_j=(\lambda -j+1)v_{j-1}</math> | ||
| + | * as <math>V</math> is finite dimensional, there exists <math>l\in \in\mathbb{Z}_{\geq 0}</math> such that <math>v_m\neq 0</math> and <math>v_{m+1}=0</math> | ||
| + | * then <math>Ev_{m+1}=(\lambda-m)v_{m}=0</math> and so <math>\lambda-m=0</math>. So <math>\lambda \in\mathbb{Z}_{\geq 0}</math> | ||
| + | * Let <math>V_{\mathbb{Z}}</math> be the <math>\mathbb{Z}</math>-span of <math>\{v_j|j\geq 0\}</math> | ||
| + | * as <math>V</math> is irreducible, <math>V=\mathbb{C}\otimes_{\mathbb{Z}}V_{\mathbb{Z}}</math> | ||
| + | * so <math>V_{\mathbb{Z}}</math> is an integral form for <math>V</math> with integral basis <math>\{v_0,\cdots, v_m\}</math> | ||
| + | ;Question. | ||
| + | where do <math>F^{(j)}</math> come from? | ||
| + | ;prop | ||
| + | <math>V_{\mathbb{Z}}</math> is stable under the action of <math>F^{(j)}</math> and <math>E^{(j)}</math> and thus stable also under the action of <math>\exp (tE)</math> and <math>\exp (tF)</math> (matrices with coefficients in <math>\mathbb{Z}[t]</math>, key fact to define the Chevalley groups) | ||
| − | === | + | ==basis of <math>\mathfrak{g}</math> and structure constants== |
| + | ===basis=== | ||
| + | * simple Lie algebra <math>\mathfrak{g}</math> over <math>\mathbb{C}</math>, we have a non-deg invariant bilinear form <math>(\cdot,\cdot)</math>. | ||
| + | * fix a Cartan subalgebra <math>\mathfrak{h}</math> | ||
| + | * <math>\Delta</math> : root system | ||
| + | * <math>\Pi</math> : fundamental system | ||
| + | * Cartan decomposition | ||
| + | :<math> | ||
| + | \mathfrak{g}=\mathfrak{h}\oplus \left(\oplus_{\alpha\in \Delta} \mathfrak{g}_{\alpha}\right) | ||
| + | </math> | ||
| + | * fix <math>H_{\alpha}\in \mathfrak{h}</math> uniquely for each <math>\alpha\in \Delta</math> by | ||
| + | :<math> | ||
| + | \beta(H_{\alpha})=2\frac{(\alpha,\beta)}{(\alpha,\alpha)}\,\quad \beta\in \mathfrak{h}^{*} | ||
| + | </math> | ||
| + | ;exercise | ||
| + | <math>H_{\alpha}</math> can be written as a <math>\mathbb{Z}</math>-linear combination of <math>H_{\alpha_i}, \alpha_i\in \Pi</math>. | ||
| + | * we can choose <math>x_{\alpha}\in \mathfrak{g}_{\alpha}</math> so that | ||
| + | :<math>[x_{\alpha},x_{-\alpha}]=H_{\alpha}</math> | ||
| + | * The elements <math>\{H_{\alpha_i} : \alpha_i\in \Pi\}</math> together with elements <math>x_{\alpha}\in \mathfrak{g}_{\alpha}</math> (<math>\alpha\in \Delta</math>) form a basis of <math>\mathfrak{g}</math> | ||
| − | * | + | ===structure constants=== |
| − | + | * multiplication in <math>\mathfrak{g}</math> | |
| − | + | :<math> | |
| − | * | + | [h,x_{\alpha}]=\alpha(h)x_{\alpha}\\ |
| − | + | [x_{\alpha},x_{-\alpha}]=H_{\alpha}\\ | |
| + | [x_{\alpha},x_{\beta}]=n_{\alpha,\beta}x_{\alpha+\beta} | ||
| + | </math> | ||
| + | * structure constants <math>n_{\alpha,\beta}</math> | ||
| + | * <math>n_{\alpha,\beta}\neq 0</math> only if <math>\alpha+\beta\in \Delta</math> | ||
| + | * <math>n_{\alpha,\beta}</math> is not fixed by the above condition. how much freedom do we have? | ||
| + | * The structure constants <math>n_{\alpha,\beta}</math> for extraspecial pairs <math>(\alpha,\beta)</math> can be chosen as arbitrary non-zero elements of <math>\mathbb{C}</math>, by appropriate choice of the elements <math>x_{\alpha}</math>. | ||
| + | * All the structure constants <math>n_{\alpha,\beta}</math> 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 <math>x_{\alpha}'=u_{\alpha}x_{\alpha}</math> with <math>u_{\alpha}u_{-\alpha}=1</math>, then structure constants satisfy the following property |
| − | = | + | :<math> |
| − | + | n_{\alpha,\beta}'n_{-\alpha,-\beta}'=n_{\alpha,\beta}n_{-\alpha,-\beta} | |
| − | + | </math> | |
| + | ;lemma | ||
| + | The number <math>n_{\alpha,\beta}n_{-\alpha,-\beta}</math> is given by <math>-(p+1)^2</math> where <math>p</math> is the largest integer <math>\geq 0</math> such that <math>\beta-p\alpha\in \Delta</math>. (<math>\alpha</math> string through <math>\beta</math>) | ||
| + | ;remark | ||
| + | it is the minimum <math>p\in \mathbb{Z}_{\geq 0}</math> such that | ||
:<math> | :<math> | ||
| − | + | \left(\text{ad} x_{-\alpha}\right)^{p}\left(x_{\beta}\right)=0 | |
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</math> | </math> | ||
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;lemma | ;lemma | ||
| − | + | It is possible to choose basis elements <math>x_{\alpha}'\in \mathfrak{g}_{\alpha}</math> such that <math>[x_{\alpha}',x_{-\alpha}']=H_{\alpha}</math>, and <math>n_{-\alpha,-\beta}=-n_{\alpha,\beta}</math> for all <math>\alpha</math> and <math>\beta</math>. For this choice of <math>x_{\alpha}'</math>, we have <math>n_{\alpha,\beta}=\pm (p+1)</math> | |
| − | === | + | Hint : Use the Chevalley involution <math>\sigma :\mathfrak{g}\to \mathfrak{g}</math>. It is an involution with <math>\sigma(h)=-h</math> for any <math>h\in \mathfrak{h}</math> and <math>\sigma(x_{\alpha})=x_{-\alpha}</math>. |
| − | ;thm | + | |
| − | Chevalley | + | ===Chevalley basis=== |
| − | + | ;thm (Chevalley 1955) | |
| − | + | The elements <math>\{H_{\alpha_i} : \alpha_i\in \Pi\}</math> together with elements <math>X_{\alpha}\in \mathfrak{g}_{\alpha}</math> (<math>\alpha\in \Delta</math>) chosen to satisfy <math>[X_{\alpha},X_{-\alpha}]=H_{\alpha}</math> and <math>[X_{\alpha},X_{\beta}]=\pm (p+1) X_{\alpha+\beta}</math> (if <math>\alpha+\beta\in \Delta)</math> form a basis for a <math>\mathbb{Z}</math>-form <math>\mathfrak{g}_{\mathbb{Z}}</math> of <math>\mathfrak{g}</math>. | |
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==Kostant== | ==Kostant== | ||
| − | * | + | ===integral form=== |
| − | + | * Let <math>\{X_{\alpha}\}</math> and <math>\{H_{\alpha_i}\}</math> be a Chevalley basis for <math>\mathfrak{g}</math> | |
| − | + | * Let <math>U(\mathfrak{g})_{\mathbb{Z}}</math> be the <math>\mathbb{Z}</math>-subalgebra of <math>U(\mathfrak{g})</math> generated by <math>X_{\alpha}^{(n)}=X_{\alpha}^{n}/n!</math> for all <math>\alpha\in \Delta</math> and <math>n\in \mathbb{Z}_{\geq 0}</math>. | |
| − | + | * it is an integral form for <math>U(\mathfrak{g})</math> | |
| − | + | * can we describe its integral basis? | |
| − | \ | + | |
| − | + | ===basis=== | |
| − | + | * let <math>\Delta^{+}=\{\alpha_1,\cdots, \alpha_N\}</math> be an ordered set of positive roots | |
| − | + | * for <math>Q=(q_1,\cdots, q_N)</math> with <math>q_i\in \mathbb{Z}_{\geq 0}</math>, put | |
| − | \ | + | :<math> |
| − | + | e_{Q}=\prod_{i=1}^N X_{\alpha_i}^{(q_i)} | |
| − | for | + | </math> |
| − | * See | + | * for <math>S=(s_1,\cdots, s_N)</math> with <math>q_i\in \mathbb{Z}_{\geq 0}</math>, put |
| − | * for | + | :<math> |
| + | f_{S}=\prod_{i=1}^N X_{-\alpha_i}^{(s_i)} | ||
| + | </math> | ||
| + | * for <math>x\in \mathfrak{g}</math> and <math>s\in \mathbb{Z}_{\geq 0}</math>, put | ||
| + | :<math> | ||
| + | \binom{x}{s}=\frac{x(x-1)\cdots (x-s+1)}{s!}\in U(\mathfrak{g}) | ||
| + | </math> | ||
| + | * let <math>l</math> be the rank of <math>\mathfrak{g}</math> for each <math>l</math>-tuple <math>P=(p_1,\cdots, p_l)</math>, define | ||
| + | :<math> | ||
| + | h_{P}=\prod_{i=1}^{l}\binom{H_{\alpha_i}}{p_i} | ||
| + | </math> | ||
| + | ;thm (Kostant 1966) | ||
| + | The elements | ||
| + | :<math> | ||
| + | \{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}\} | ||
| + | </math> | ||
| + | form an integral basis for <math>U(\mathfrak{g})_{\mathbb{Z}}</math>. | ||
| + | * See Humphreys chapter 26 for a proof | ||
| + | |||
| + | ===example=== | ||
| + | * for <math>\mathfrak{g}=\mathfrak{sl}_2</math>, | ||
:<math>\{\frac{F^k}{k!}\binom{H}{l}\frac{E^m}{m!}|k,l,m\geq 0\}</math> | :<math>\{\frac{F^k}{k!}\binom{H}{l}\frac{E^m}{m!}|k,l,m\geq 0\}</math> | ||
| − | * <math>\exp(tE)</math> and <math>\exp(tF)</math> exist | + | * let us compute <math>E^2F^2</math> |
| − | * <math>\ | + | :<math> |
| + | E^2F^2=2 H^2-8 FE-2 H+F^2E^2+4 FHE | ||
| + | </math> | ||
| + | * thus | ||
| + | :<math> | ||
| + | E^{(2)}F^{(2)}=\frac{H^2}{2}-2FE-\frac{H}{2}+F^{(2)}E^{(2)}+FHE | ||
| + | </math> | ||
| + | * so we cannot use <math>\frac{H^k}{k!}</math> as elements of integral basis | ||
| + | * that's where <math>\binom{H}{2}=\frac{H^2}{2}-\frac{H}{2}</math> comes from. In general, we have | ||
| + | :<math> | ||
| + | E^{(m)}F^{(n)}=\sum_{j=0}^k F^{(n-j)}\binom{H-m-n+2j}{j}E^{(m-j)} | ||
| + | </math> | ||
| + | where <math>k=\min(m,n)</math> | ||
| + | ;exercise | ||
| + | Let <math>j,k\in\mathbb{Z}_{\geq 0}</math>. The polynomial <math>\binom{x-j}{k}</math> can be written as a <math>\mathbb{Z}</math>-linear combination of <math>\binom{x}{i}</math>'s. | ||
| + | |||
| + | ===properties=== | ||
| + | * <math>\exp(tE)</math> and <math>\exp(tF)</math> exist in <math>U(\mathfrak{g})_{\mathbb{Z}}[[t]]</math> | ||
| + | * a nice property of this integral form is | ||
| + | :<math> | ||
| + | \Delta(Z_{\alpha}) = \sum_{0\leq\beta\leq\alpha}Z_{\beta} \otimes Z_{\alpha−\beta}. | ||
| + | </math> | ||
| + | where <math>Z_{\alpha}=f_{Q}h_Pe_{S}</math>, <math>\alpha=(Q,P,S)\in\mathbb{Z}_{\geq 0}^{N+l+N}</math> and <math>\Delta : U(\mathfrak{g})\to U(\mathfrak{g})</math> is the coproduct defined by | ||
| + | :<math> | ||
| + | \Delta(x)=x\otimes 1+1\otimes x | ||
| + | </math> | ||
| + | for <math>x\in \mathfrak{g}</math> | ||
| + | * partial ordering on <math>\mathbb{Z}_{\geq 0}^{N+l+N}</math> | ||
| + | ==remarks on Chevalley groups== | ||
| + | ;def | ||
| + | An ''admissible'' integral form of a <math>\mathfrak{g}</math>-module <math>V</math> is an integral form <math>M</math> such that <math>U(\mathfrak{g})_{\mathbb{Z}}\cdot M\subseteq M</math> | ||
| + | ;prop | ||
| + | Let <math>V</math> be a finite dimensional <math>\mathfrak{g}</math>-module. Then <math>V</math> has an admissible integral form. If <math>V</math> is irreducible and <math>v_0</math> is a highest weight vector, then <math>U(\mathfrak{g})_{\mathbb{Z}}.v_0</math> is an admissible integral form of <math>V</math>. | ||
| − | == | + | * Let <math>(\rho,V)</math> a faithful representation of <math>\mathfrak{g}</math> and <math>M</math> an admissible integral form |
| − | + | * Choose an integral basis <math>\{m_1,\cdots, m_d\}</math>of <math>M</math>. Then <math>e_{\alpha}(t):=\exp \left(t\rho(X_{\alpha})\right)\in GL_{d}(\mathbb{Z}[t])</math> | |
| + | * now let <math>k</math> arbitrary field and <math>M^k=k\otimes_{\mathbb{Z}}M</math>, a vector space over <math>k</math> | ||
| + | ;def | ||
| + | The ''Chevalley group'' <math>G_{V,k}</math> is the subgroup of <math>GL(M^k)</math> generated by all <math>e_{\alpha}(u),\, \alpha\in \Delta, u\in k</math> regarded as a <math>k</math>-linear transformation of <math>M^k</math> | ||
| + | * it only depends on <math>k</math> and the lattice of weights <math>\Gamma_{V}</math> of <math>\mathfrak{g}</math>-module <math>V</math> (not on <math>M</math>) | ||
| + | * When <math>\Gamma_V=Q</math>, <math>Q</math> the root lattice, we call it an adjoint Chevalley group | ||
| + | ;thm (Chevalley-Dickson theorem) | ||
| + | Let <math>G</math> be an adjoint Chevalley group. If <math>|k|=2</math>, suppose <math>\Delta</math> is not of type <math>A_1,B_2</math> or <math>G_2</math>. If <math>|k|=3</math>, suppose that <math>\Delta</math> is not of type <math>A_1</math>. Then <math>G</math> 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]] | ||
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[[분류:talks and lecture notes]] | [[분류:talks and lecture notes]] | ||
| + | [[분류:Lie theory]] | ||
| + | [[분류:migrate]] | ||
2020년 11월 13일 (금) 20:04 기준 최신판
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