Standard parabolic and Levi subalgebras from a subset of simple roots

Assume the Axiom of Choice (The Axiom of Choice), inherited from the published root-space and ${\mathfrak{s}\mathfrak{l}}_2$ suppliers. Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$, root system $\mathrm{\Phi }$ (The root set is a reduced crystallographic root system, Root-space decomposition, Root spaces of a complex semisimple Lie algebra are one-dimensional), positive system ${\mathrm{\Phi }}^{+}$ with base of simple roots $\mathrm{\Delta }=\{{\alpha }_1,\dots ,{\alpha }_r\}$ (Positive systems and simple roots, Reduced crystallographic Euclidean root system), and let $J$ be a subset of the simple indices $\{1,\dots ,r\}$. Write ${\mathrm{\Phi }}_J$ for the root subsystem spanned by the simple roots indexed by $J$, ${\mathrm{\Phi }}_J=\mathrm{\Phi }\cap {\mathrm{span}}_{\mathbb{R}}\{{\alpha }_j:j\in J\},$ and let $H_j=h_{{\alpha }_j}\in \mathfrak{h}$ be the coroot of ${\alpha }_j$ (Coroot of a Lie-algebra root via Killing-dual vector of a root). Put ${\mathfrak{h}}_J={\mathrm{span}}_{\mathbb{C}}\{H_j:j\in J\},{\mathfrak{z}}_J=\{h\in \mathfrak{h}:{\alpha }_j(h)=0\text{ for all }j\in J\},$ and define the following subspaces of $\mathfrak{g}$, where ${\mathfrak{g}}_{\alpha }$ is the root space of $\alpha$ (Root-space decomposition):

${\mathfrak{s}}_J={\mathfrak{h}}_J\oplus {⨁}_{\alpha \in {\mathrm{\Phi }}_J}{\mathfrak{g}}_{\alpha },{\mathfrak{l}}_J={\mathfrak{s}}_J\oplus {\mathfrak{z}}_J=\mathfrak{h}\oplus {⨁}_{\alpha \in {\mathrm{\Phi }}_J}{\mathfrak{g}}_{\alpha },$ ${\mathfrak{u}}_J={⨁}_{\beta \in {\mathrm{\Phi }}^{+}\setminus {\mathrm{\Phi }}_J^{+}}{\mathfrak{g}}_{\beta },{\mathfrak{v}}_J={⨁}_{\beta \in {\mathrm{\Phi }}^{-}\setminus {\mathrm{\Phi }}_J^{-}}{\mathfrak{g}}_{\beta },{\mathfrak{p}}_J={\mathfrak{l}}_J\oplus {\mathfrak{u}}_J\mathrm{.}$

Here ${\mathrm{\Phi }}_J^{\pm }={\mathrm{\Phi }}_J\cap {\mathrm{\Phi }}^{\pm }$ and ${\mathrm{\Phi }}^{\pm }$ are the positive and negative roots with ${\mathfrak{n}}^{\pm }$, $\mathfrak{b}$ as in Positive and negative nilpotent subalgebras and the Borel. The construction depends only on the positive system and on $J$. The following structural facts are part of the definition (proofs and references in the Remarks; $\mathrm{rad}$ and $\mathrm{nilrad}$ are those of Solvable radical and Nilradical).

(1) Decomposition of the Cartan. $\mathfrak{h}={\mathfrak{h}}_J\oplus {\mathfrak{z}}_J$ and $\dim {\mathfrak{h}}_J=\mathrm{\mid }J\mathrm{\mid }$, so $\dim {\mathfrak{z}}_J=r-\mathrm{\mid }J\mathrm{\mid }$.

(2) The semisimple part. ${\mathfrak{s}}_J$ is a Lie subalgebra of $\mathfrak{g}$; it is semisimple, and its root system with respect to the Cartan subalgebra ${\mathfrak{h}}_J$ is ${\mathrm{\Phi }}_J$, with positive system ${\mathrm{\Phi }}_J^{+}$ and Weyl group $W_J=⟨s_j:j\in J⟩\le W$ (Weyl group, Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups).

(3) The reductive Levi algebra. ${\mathfrak{l}}_J$ is a Lie subalgebra with root system ${\mathrm{\Phi }}_J$ relative to $\mathfrak{h}$; it is reductive as in Simple, semisimple, and reductive Lie algebras, with center ${\mathfrak{z}}_J$ and derived algebra $[{\mathfrak{l}}_J,{\mathfrak{l}}_J]={\mathfrak{s}}_J$; $\mathfrak{h}$ is a Cartan subalgebra of ${\mathfrak{l}}_J$ and the Weyl group of $({\mathfrak{l}}_J,\mathfrak{h})$ is $W_J$. The algebra ${\mathfrak{l}}_J$ need not be semisimple: its center ${\mathfrak{z}}_J$ is nonzero as soon as $J\mathrm{\ne }\{1,\dots ,r\}$.

(4) Parabolic and radicals. ${\mathfrak{l}}_J$, ${\mathfrak{u}}_J$, ${\mathfrak{v}}_J$ and ${\mathfrak{p}}_J$ are Lie subalgebras (Brackets of root spaces); ${\mathfrak{u}}_J$ is an ideal of ${\mathfrak{p}}_J$, ${\mathfrak{l}}_J$ stabilizes ${\mathfrak{u}}_J$ and ${\mathfrak{v}}_J$, and ${\mathfrak{p}}_J={\mathfrak{l}}_J⋉{\mathfrak{u}}_J$ is a semidirect product (Levi subalgebras and Levi decompositions). Moreover $\mathrm{nilrad}({\mathfrak{p}}_J)={\mathfrak{u}}_J$ and $\mathrm{rad}({\mathfrak{p}}_J)={\mathfrak{z}}_J\oplus {\mathfrak{u}}_J\mathrm{.}$ Consequently $[{\mathfrak{p}}_J,{\mathfrak{p}}_J]={\mathfrak{s}}_J\oplus {\mathfrak{u}}_J$ and the quotient ${\mathfrak{p}}_J\mathrm{/}\mathrm{rad}({\mathfrak{p}}_J)\cong {\mathfrak{s}}_J$ is semisimple: in the sense of Levi subalgebras and Levi decompositions, ${\mathfrak{s}}_J$ is a semisimple Levi factor of ${\mathfrak{p}}_J$ and ${\mathfrak{l}}_J$ is the reductive complement to the nilradical. The parabolic ${\mathfrak{p}}_J$ contains $\mathfrak{b}$ and ${\mathfrak{p}}_J=\mathfrak{b}\oplus {⨁}_{\alpha \in {\mathrm{\Phi }}_J^{-}}{\mathfrak{g}}_{\alpha }$, the span of the negative root spaces of ${\mathrm{\Phi }}_J$; for $J=\mathrm{\varnothing }$ one has ${\mathfrak{p}}_J=\mathfrak{b}$, and for $J=\{1,\dots ,r\}$ one has ${\mathfrak{l}}_J={\mathfrak{s}}_J=\mathfrak{g}$ and ${\mathfrak{u}}_J={\mathfrak{v}}_J=0$.

(5) The positive grading. There exists $\zeta \in {\mathfrak{z}}_J$ with ${\alpha }_j(\zeta )=0$ for $j\in J$ and ${\alpha }_i(\zeta )=1$ for $i\mathrm{\notin }J$. With respect to $\mathrm{ad}(\zeta )$ every root space ${\mathfrak{g}}_{\beta }$ has weight $\beta (\zeta )$, so:

• ${\mathfrak{l}}_J$ has degree $0$ (all of ${\mathrm{\Phi }}_J$ vanishes on $\zeta$);
• ${\mathfrak{u}}_J$ has strictly positive integral degrees $\beta (\zeta )\in {\mathbb{Z}}_{>0}$;
• ${\mathfrak{v}}_J$ has strictly negative integral degrees.

In particular ${\mathfrak{u}}_J$ is nilpotent by this positive grading, and both ${\mathfrak{u}}_J$ and ${\mathfrak{v}}_J$ have finite-dimensional fixed-degree slices of their PBW algebras: $\dim U({\mathfrak{u}}_J{)}_d<\mathrm{\infty }$ and $\dim U({\mathfrak{v}}_J{)}_d<\mathrm{\infty }$ for every $d\in \mathbb{Z}$ (PBW gives an ordered monomial basis for the enveloping algebra).

• The Cartan splitting (1). The pairing matrix $A=({\alpha }_i(H_j){)}_{i,j\in J}$ is the principal submatrix of the Cartan matrix of $(\mathrm{\Phi },\mathrm{\Delta })$ with respect to the base $\{{\alpha }_j:j\in J\}$ of ${\mathrm{\Phi }}_J$ (Simple roots form a signed integral basis). By Properties of finite-type Cartan matrices (4) the symmetrization $DA{D}^{-1}$ is positive definite, so $A$ is nonsingular. Hence if $h={\sum }_j{c}_j{H}_j$ satisfies ${\alpha }_i(h)=0$ for all $i\in J$, then $Ac=0$ and $c=0$; so ${\mathfrak{h}}_J\cap {\mathfrak{z}}_J=0$ and $\dim {\mathfrak{h}}_J=\mathrm{\mid }J\mathrm{\mid }$. Counting dimensions gives $\mathfrak{h}={\mathfrak{h}}_J\oplus {\mathfrak{z}}_J$ and $\dim {\mathfrak{z}}_J=r-\mathrm{\mid }J\mathrm{\mid }$.
• Semisimplicity of ${\mathfrak{s}}_J$ (2). The Killing-dual map is linear, so $\alpha ={\sum }_{j\in J}{m}_j{\alpha }_j$ gives $H_{\alpha }={\sum }_{j\in J}{m}_j{H}_{{\alpha }_j}\in {\mathfrak{h}}_J$ (Killing-dual vector of a root, Coroot of a Lie-algebra root). Opposite-root brackets therefore lie in ${\mathfrak{h}}_J$, and root brackets give subalgebra closure. Use its intrinsic Killing form $\kappa$, which need not equal the ambient Killing restriction. Its root-space decomposition gives $\kappa (h,h^{\mathrm{\prime }})={\sum }_{\alpha \in {\mathrm{\Phi }}_J}\alpha (h)\alpha (h^{\mathrm{\prime }})$ on ${\mathfrak{h}}_J$. On the real coroot span this is positive definite: the simple-root pairing matrix is nonsingular by (1), and the sum of squares vanishes only when every simple-root value is zero. Thus its complexification is nondegenerate there. Trace-form invariance (Trace forms are symmetric and invariant) gives orthogonality of weights whose sum is nonzero: applying a Cartan element gives $(\alpha +\beta )(h)\kappa (x_{\alpha },x_{\beta })=0$. For a normalized ${\mathfrak{s}\mathfrak{l}}_2$ triple $e_{\alpha },f_{\alpha },h_{\alpha }$ (The root sl_2 triple), invariance gives $\kappa (h_{\alpha },h_{\alpha })=2\kappa (e_{\alpha },f_{\alpha })\mathrm{\ne }0$. One-dimensionality of the root spaces makes each opposite pair nondegenerate. Hence $\kappa$ is nondegenerate on ${\mathfrak{s}}_J$, and Cartan's semisimplicity criterion proves semisimplicity. Every root restricts nontrivially to ${\mathfrak{h}}_J$, since it vanishes on ${\mathfrak{z}}_J$ and $\mathfrak{h}={\mathfrak{h}}_J\oplus {\mathfrak{z}}_J$. The weight decomposition then shows that the centralizer and normalizer of ${\mathfrak{h}}_J$ in ${\mathfrak{s}}_J$ are both ${\mathfrak{h}}_J$; thus it is a Cartan subalgebra. Restriction identifies its roots with ${\mathrm{\Phi }}_J$ and its simple reflections with $s_j$, $j\in J$, extended by the identity on ${\mathfrak{z}}_J$.
• Reductive ${\mathfrak{l}}_J$ (3). Since ${\mathfrak{z}}_J$ centralizes ${\mathfrak{h}}_J$ and each root space of ${\mathrm{\Phi }}_J$ (${\alpha }_j(h)=0$ for $h\in {\mathfrak{z}}_J$), it commutes with ${\mathfrak{s}}_J$, so ${\mathfrak{l}}_J={\mathfrak{z}}_J\oplus {\mathfrak{s}}_J$ is a Lie subalgebra with central summand; $[{\mathfrak{l}}_J,{\mathfrak{l}}_J]={\mathfrak{s}}_J$ because every root vector is a bracket with a Cartan element, and the simple coroots spanning ${\mathfrak{h}}_J$ are the brackets $[e_j,f_j]$. The center contributes nothing. Its center is ${\mathfrak{z}}_J$ together with the center of ${\mathfrak{s}}_J$, which is zero, and the displayed sum is direct. Hence ${\mathfrak{l}}_J$ is reductive in the sense of Simple, semisimple, and reductive Lie algebras, with root system ${\mathrm{\Phi }}_J$ relative to $\mathfrak{h}$ and Weyl group $W_J$. For $J\mathrm{\ne }$ all indices, ${\mathfrak{z}}_J\mathrm{\ne }0$ by (1), so ${\mathfrak{l}}_J$ is not semisimple.
• Brackets, ideals and grading (4)-(5). For roots $\alpha ,\beta \in {\mathrm{\Phi }}_J$ (i.e. sums of simple roots in $J$) one has $\alpha +\beta \in {\mathrm{\Phi }}_J$ whenever the sum is a root; if the sum is neither a root nor zero, the bracket is zero. This gives the bracket closure for ${\mathfrak{l}}_J$; the mixed brackets with ${\mathfrak{u}}_J$ likewise stay in ${\mathfrak{u}}_J\cup {\mathfrak{l}}_J\cup \{0\}$ by Brackets of root spaces and the signs of the simple-root coefficients (Simple roots form a signed integral basis), so ${\mathfrak{u}}_J$ is an ideal of ${\mathfrak{p}}_J$ and ${\mathfrak{l}}_J$ preserves ${\mathfrak{u}}_J$ and ${\mathfrak{v}}_J$; this gives the semidirect product ${\mathfrak{p}}_J={\mathfrak{l}}_J⋉{\mathfrak{u}}_J$, and ${\mathfrak{p}}_J=\mathfrak{b}\oplus {⨁}_{\alpha \in {\mathrm{\Phi }}_J^{-}}{\mathfrak{g}}_{\alpha }$ is a restatement of ${\mathfrak{l}}_J=\mathfrak{h}\oplus {⨁}_{\alpha \in {\mathrm{\Phi }}_J}{\mathfrak{g}}_{\alpha }$. The simple roots form a basis of ${\mathfrak{h}}^{\ast }$, so there is a unique $\zeta \in \mathfrak{h}$ with the prescribed simple-root values. Its zero values on $J$ put it in ${\mathfrak{z}}_J$. Every positive root outside ${\mathrm{\Phi }}_J$ has a positive sum of coefficients outside $J$; negative roots have the opposite sign. This proves (5), and brackets add degrees, proving nilpotency of ${\mathfrak{u}}_J$. For maximality, let $\mathfrak{I}$ be a nilpotent ideal of ${\mathfrak{p}}_J$. Its image in ${\mathfrak{l}}_J$ is nilpotent and has zero projection to the semisimple ${\mathfrak{s}}_J$, since nilpotent algebras are solvable. If its central projection contained $h\mathrm{\ne }0$, then some $i\mathrm{\notin }J$ would have ${\alpha }_i(h)\mathrm{\ne }0$. Choose $x=h+a\in \mathfrak{I}$ with $a\in {\mathfrak{u}}_J$ and $0\mathrm{\ne }{e}_i\in {\mathfrak{g}}_{{\alpha }_i}$. The element $[x,e_i]\in \mathfrak{I}$ has degree-one component ${\alpha }_i(h)e_i$; the contributions from $a$ have degree at least two. For every $k\ge 0$, $(\mathrm{ad}x{)}^k[x,e_i]$ has degree-one component ${\alpha }_i(h{)}^{k+1}{e}_i\mathrm{\ne }0$. This contradicts nilpotency of $\mathfrak{I}$, whose lower central series makes $\mathrm{ad}x$ nilpotent on $\mathfrak{I}$. Thus $\mathfrak{I}\subseteq {\mathfrak{u}}_J$, proving the nilradical assertion. Finally ${\mathfrak{z}}_J\oplus {\mathfrak{u}}_J$ is a solvable ideal with semisimple quotient ${\mathfrak{s}}_J$, so it is exactly $\mathrm{rad}({\mathfrak{p}}_J)$; and by Levi subalgebras and Levi decompositions the semisimple complement ${\mathfrak{s}}_J$ is a Levi factor of ${\mathfrak{p}}_J$. The fixed-degree PBW finiteness in (5) uses finite-dimensionality of each root space, the finite height bound on positive roots occurring in a fixed degree, and the PBW basis theorem (PBW gives an ordered monomial basis for the enveloping algebra).
• The cases $J=\mathrm{\varnothing }$ and $J$ all. For $J=\mathrm{\varnothing }$ one has ${\mathfrak{h}}_J=0$, ${\mathfrak{z}}_J=\mathfrak{h}$, ${\mathrm{\Phi }}_J=\mathrm{\varnothing }$, ${\mathfrak{l}}_J=\mathfrak{h}$ and ${\mathfrak{u}}_J={\mathfrak{n}}^{+}$, so ${\mathfrak{p}}_J=\mathfrak{b}$. For $J=\{1,\dots ,r\}$ one has ${\mathfrak{z}}_J=0$, ${\mathrm{\Phi }}_J=\mathrm{\Phi }$, ${\mathfrak{u}}_J={\mathfrak{v}}_J=0$ and ${\mathfrak{p}}_J=\mathfrak{g}$. These are the two extreme cases of the standard parabolic construction.
• Choice. The Axiom of Choice is inherited from the published root-space and ${\mathfrak{s}\mathfrak{l}}_2$ suppliers used in the justifications; no further choice principle enters this construction.