Hoyt, Crystal

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Title On good $\mathbb{Z}$-gradings of basic Lie superalgebras(Problems in Representation Theory and Harmonic Analysis)Author(s)Hoyt, CrystalCitation数理解析研究所講究録 (2011), 1770: 114-118Issue Date2011-11URL http://hdl.handle.net/2433/171660RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityOn good $\mathbb{Z}$-gradings of basic Lie superalgebrasCrystal Hoyt*\daggerAbstractWe discuss the classification of good $Z\cdot gradings$ of basic Lie superalgebras. This problemarose in connection to W-algebras, where good Z-gradings play a role in their $\infty nstmction$ .1 IntroductionOne component of the definition of a finite or affine super W-algebra is a good $\mathbb{Z}$-grading fora nilpotent element. Affine super W-algebras are (super) vertex algebras obtained from affineLie superalgebras by quantum Hamiltonian reduction [8], whereas flnite super W-algebras areassociative superalgebras which can be defined via the universal enveloping algebra of a finite-dimensional simple Lie superalgebra [3].Good Z-gradings of simple finite-dimensional Lie algebras where classified by A. Elashvili andV.G. Kac in 2005 [4]. K. Baur and N. Wallach classified nice parabolic subalgebras of reductiveLie algebras in [1], which correspond to good even Z-gradings by [4, Theorem 2.1]. J. Brundanand S. Goodwin classified good R-gradings for semisimple finite-dimensional Lie algebras usingcertain polytopes, and proved that two finite W-algebras defined by the same nilpotent element$e\in g$ are isomorphic [2]. This often allows one to reduce to the case that the good Z-grading iseven.Here we discuss the classification of good Z-gradings of basic Lie superalgebras [5]. In thecase that $g$ is $\mathfrak{g}t(m|n)$ or $osp(m|2n)$ the good Z-gradings are parameterized by “good” pyramids,generalizing the definition of [4]. Whereas, for the exceptional Lie superalgebras, all good &gradings are shown to be Dynkin. Using this classification, one can determine which nilpotentelements have a good even Z-grading. For example, every nilpotent even element of $\mathfrak{g}\mathfrak{l}(m|n)$ hasa good even kyading.2 Basic Lie superalgebrasFinite-dimensional simple Lie superalgebras were classified by V.G. Kac in [7]. These can beseparated lnto three types: basic, strange and Cartan. A finite-dimensional simple Lie superal-gebra $\mathfrak{g}=l6\oplus \mathfrak{g}_{\overline{1}}$ is called basic if th is a reductive Lie algebra and $\mathfrak{g}$ has an even nondegenerate*Department of Mathematics, Bar-Ilan University, Ramat Gan 52900, Israel; hoyt@math.biu.ac.il.\dagger Supportd by JSPS at Nara Women’s University, Japan, and by ISF center of excellence 1691/10 at Bar-IlanUniversity, Israel. Supported by the Minerva foundation with funding from the Federal German Ministry forEducation and Research.1770 2011 114-118 114invariant bilinear form $(\cdot,$ $\cdot)$ . This form is necessarily supersymmetric. The basic Lie superalge-bras are the following: $s1(m|n)$ : $m\neq n$ , pst$(n|n)$ $;=\epsilon 1(n|n)/\langle I_{2n}\},$ $osp(m|2n),$ $D(2,1, \alpha),$ $F(4)$ ,$G(3)$ , and finite dimensional simple Lie algebras.Fix a Cartan subalgebra Then $g$ has a root space decomposition $\mathfrak{g}=$ $\oplus\oplus_{\alpha\in\Delta}g_{\alpha}$ . The$\mathbb{Z}/2\mathbb{Z}$-grading of $\mathfrak{g}$ determines a decomposition of $\Delta$ into the disjoint union of the even roots $\Delta_{6}$and the odd roots $\Delta_{\overline{1}}$ . Corresponding to a set of simple roots $\Pi=\{\alpha_{1}, \ldots, \alpha_{n}\}\subset\Delta$ of $g$ , wehave the triangular decomposition $\mathfrak{g}=\mathfrak{n}^{-}\oplus$ $\oplus \mathfrak{n}^{+}$ .Most basic Lie superalgebras have more than one distinct Dynkin diagram. This is due tothe fact that the Weyl group does not act simply transitively on the set of bases. However, wecan extend the Weyl group to a Weyl groupoid by including “odd reflections”, which allow usto move between the different bases. In particular, if $\alpha_{k}\in\Pi$ is a simple isotropic root, then wecan define the odd reflection at $\alpha_{k}$ to obtain a new set of simple roots $\Pi’$ for $\Delta[9]$ .3 Good Z-gradingsA Zgrading $\mathfrak{g}=\oplus_{j\in Z}\mathfrak{g}(j)$ is called good if there exists $e\in g_{0}(2)$ such that the map ad $e$ :$\mathfrak{g}(j)arrow \mathfrak{g}(j+2)$ is injective for $j\leq-1$ and surjective for $j\geq-1$ . If a Z-grading of $\mathfrak{g}$ is defined bya semisimple element $h\in g_{6}$ , then this condition is equivalent to all of the eigenvalues of ad$(h)$on the centralizer $g^{\epsilon}$ of $e$ in $g$ being non-negative.An example of a good Z-grading for a nilpotent element $e\in g_{6}$ is the Dynkin grading. Bythe Jacobson-Morosov Theorem, $e$ belongs to an $zl_{2}$-triple $ff=\{e, f, h\}\subset g_{0}$ , where $[e, f]=h$,$[h, e]=2e$ and $[h, f]=-2f$ . By $\epsilon \mathfrak{l}_{2}$-theory, the grading of $g$ defined by ad $h$ is a good Z-gradingfor $e$ .For each nilpotent even element $x\in \mathfrak{g}$ (up to conjugacy) we describe all Z-gradings for whichthis element is good. For the exceptional Lie superalgebras, we have the followingTheorem 3.1 (Hoyt [5]). All good Z-gradings of the exceptional Lie superalgebras, $F(4),$ $G(3)$ ,and $D(2,1, \alpha)$ , are Dynkin gradings.To describe the good Z-gradings of $gl(m|n)$ we generalize the definition of a pyramid given in[2, 4]. A pyramid $P$ is a finite collection of boxes of size 2 $x2$ in the upper half plane whichare centered at integer coordinates, such that for each $j=1,$ $\ldots,$ $N$ , the second coordinates ofthe $j^{th}$ row equal $2j-1$ and the first coordinates of the $j^{th}$ row form an arithmetic progression$f_{j},$ $f_{j}+2,$ $\ldots$ , $l_{j}$ with difference 2, such that the first row is centered at $(0,0)$ , i.e. $f_{1}=-l_{1}$ , and$f_{j}\leq f_{j+1}\leq l_{j+1}\leq l_{j}$ for all $j$ . (1)Each box of $P$ has even or odd parity. We say that $P$ has size $(m|n)$ if $P$ has exactly $m$ evenboxes and $n$ odd boxes.Fix $m,$ $n\in \mathbb{Z}+$ and let $(p, q)$ be a partition of $(m|n)$ . Let $r=\psi(p, q)\in Par(m+n)$ be the totalordering of the partitions $p$ and $q$ which satisfies: if $p_{i}=q_{j}$ for some $i,j$ then $\psi(p_{i})<\psi(q_{j})$ . Wedefine $Pyr(p, q)$ to be the set of pyramids which satisfy the following two conditions: (1) the $j^{th}$row of a pyramid $P\in Pyr(p, q)$ has length $r_{j};(2)$ if $\psi^{-1}(r_{j})\in p$ $($resp. $\psi^{-1}(r_{j})\in q)$ then all115boxes in the $j^{th}$ row have even (resp. odd parity) and we mark these boxes with a $+$” (resp.“–,, sign).Corresponding to each pyramid $P\in Pyr(p, q)$ we define a nilpotent element $e(P)\in g\circ$and semisimple element $h(P)\in g\circ$ , as follows. Recall $gt(m|n)=$ End$(V_{0}\oplus V_{1})$ . Fix a basis$\{v_{1}, \ldots, v_{m}\}$ of $V_{0}$ and $\{v_{m+1}, \ldots, v_{m+n}\}$ of $V_{1}$ . Label the even (resp. odd) boxes of $P$ by thebasis vectors of $V_{0}$ (resp. $V_{1}$ ). Define an endomorphism $e(P)$ of $V_{0}\oplus V_{1}$ as acting along the rowsof the pyramid, i.e. by sending a basis vector $v_{i}$ to the basis vector which labels the box to theright of the box labeled by $v_{i}$ or to zero if it has no right neighbor. Then $e(P)$ is nilpotent andcorresponds to the partition $(p,q)$ . Sinoe $e(P)$ does not depend the choice of $P$ in $Pyr(p,q)$ , wemay denote it by $e_{p,q}$ . Moreover, $e_{p,q}\in l0$ because boxes in the same row have the same parity.Define $h(P)$ to be the $(m+n)$-diagonal matrix where the $i^{th}$ diagonal entry is the first coor-dinate of the box labeled by the basis vector $v_{i}$ . Then $h(P)$ defines a Z-grading of $\mathfrak{g}$ for which$e_{p,q}\in g(2)$ . Let $P_{p,q}$ denote the symmetric pyramid from $Pyr(p, q)$ . Then $h(P_{p,q})$ defines aDynkin grading for $e_{p,q}$ , and $P_{p,q}$ is called the Dynkin pyramid for the partition $(p|q)$ .Theorem 3.2 (Hoyt [5]). Let $\mathfrak{g}=gl(m|n)$ , and let $(p,q)$ be a partition of $(m|n)$ . If $P$ is apymmid ffom $Pyr(p, q)$ , then the pair $(h(P), e_{p,q})$ is good. Moreover, every good grading for $e_{p,q}$is of the form $(h(P), e_{p,q})$ for some pyramid $P\in Pyr(p,q)$ .This theorem is proven by studying the centralizer of a nilpotent element and of an $\epsilon I_{2}$ triple in$gl(m|n)$ . In a similar manner, we classify the good Z-gradings for the Lie superalgebra $osp(m|2n)$(see [5]).A good Z-grading of the Lie superalgebra 1 for a nilpotent element $e\in g_{6}$ restricts to a goodZ-grading for the Lie algebra $l\overline{0}$ . So it is natural to ask which good Z-gradings of $f6$ extend toa good Z-grading of $g$ , and to what extent is an extension unique.Example 3.3. Let $g=gl(4|6)$ and consider the partitions $p=(3,1)$ and $q=(4,2)$ . The Dynhngrading of $g_{\overline{0}}=gl(4)xgl(6)$ for the partition $(p, q)$ corresponds to the folloutng symmetricpyramids.$\frac{+++\underline{\cap+}}{}$There exist pyramids in $Pyr(p, q)$ for which the induced grading of go is the one given above,and these correspond to good Z-gradings. They are represented by the following pymmids:Example 3.4. Let $g=g\mathfrak{l}(4|6)$ and consider the parktions $p=(3,1)$ and $q=(4,2)$ . Thefollowing pyramids represent a good Z-gmding of $\emptyset 0$ for which there is no good Z-grading of $\mathfrak{g}$utth this induced good Z-grading of $m$ .1164 Centralizers of $\epsilon 1_{2}$-triplesThe centralizers of $\epsilon 1_{2}$-triples in $gl(m|n)$ and osp $(m|2n)$ can be described following the ideas of[6] for the Lie algebras gl$(m)$ , so $(m)$ and $sp(2n)$ . There is a one-to-one correspondence betweenG-orbits of nilpotent even elements in $\mathfrak{g}\mathfrak{l}(m|n)$ and partitions of $(m|n)$ . Let $p=(r_{1}^{m_{1}}, \ldots, r_{N^{N}}^{m})$be a partition of $m$ and $q=(r_{1}^{n_{1}}, \ldots, r_{N^{N}}^{n})$ a partition of $n$ , that is $r_{i}$ has multiplicity $m_{i}$ in $p$and multiplicity $n_{i}\ln q$ . We note that $m_{i}$ or $n_{i}$ may be zero.Theorem 4.1 (Hoyt [5]). Let $g=gl(m|n)$ . Let $e$ be a nilpotent even element corresponding toa partition $(p, q)$ of $(m|n)$ , and let $s=\{e_{\}}f, h\}\subset g_{\overline{0}}’$ be an $\epsilon l_{2}$ -triple for $e$ . Then we have anisomorphism$g^{\epsilon}arrow\sim gl(m_{1}, n_{1})\cross\cdots\cross gI(m_{N}, n_{N})$of Lie superalgebras.A partition is called symplectic (resp. orthogonal) if $m_{p:}$ is even for odd $p_{i}$ (resp. even $p_{i}$ ).We say that a partition $(p, q)$ of $(m|2n)$ is orthosymplectic if $p$ is an orthogonal partition of $m$and $q$ is a symplectic partition of $2n$ . There is a one-to-one correspondence between G-orbits ofnilpotent even elements in osp $(m|2n)$ and orthosymplectic partitions of $(m|2n)$ . Let $(p,q)$ be anorthosymplectic partition of $(m|2n)$ , and represent it as $p=(r_{1}^{m_{1}}, \ldots, r_{N^{N}}^{m}, s_{1}^{2c_{1}}, \ldots, s_{T}^{2c_{T}})$ and$q=(r_{1}^{2n_{1}}, \ldots, r_{N}^{2nN}, s_{1}^{d_{1}}, \ldots, s_{T^{T}}^{d\prime})$ , where $r_{i}$ are the even parts and $s_{i}$ are the odd parts.Theorem 4.2 (Hoyt [5]). Let $\mathfrak{g}=0\epsilon \mathfrak{p}(m|2n)$ . Let $e$ be a nilpotent even element correspondingto an orthosymplectic partition $(p, q)$ of $(m|n)$ , and let $s=\{e, f, h\}\subset \mathfrak{g}_{\overline{0}}’$ be an $sl_{2}$ -triple for $e$ .Then we have an isomorphism$\mathfrak{g}^{\epsilon}arrow osp(m_{1},2n_{1})\sim\cross\cdots xo\epsilon p(m_{N}, 2n_{N})\cross osp(d_{1},2c_{1})\cross\cdots xozp(d_{T}, 2c_{T})$of Lie superalgebras.5 Z-gradings and the Weyl groupoidLet $g$ be a basic Lie superalgebra, $\mathfrak{g}\neq$ ps$1(2|2)$ , or let $g$ be $\mathfrak{g}\mathfrak{l}(m|n)$ or $\epsilon \mathfrak{l}(n|n)$ . Fix a Z-grading$0=\oplus_{j\in zg(j)}$ satisfying $3(g6)cg_{0}(0)$ . Fix a Cartan subalgebra $\subset\emptyset o(0)$ , and let $\Delta$ be the setof roots. The root space decomposition with respect to is compatible with the Z-grading, sowe can define a map Deg : $\Delta\cup\{0\}arrow \mathbb{Z}$ by Deg$(\alpha)=k$ if $\alpha\in\Delta_{k}$ and Deg(O) $=0$ .Now for each base $\Pi\subset\Delta$ , the degree map of a Z-grading is determined by its restriction to$\Pi$ , that is, by $D$ : $\Piarrow$ Z. A reflection at a simple root of $\Pi$ yields a new map $D^{l}$ : $\Pi’arrow \mathbb{Z}$ ,where $\Pi$‘ is the reflected base and $D’$ is defined on $\Pi^{l}$ by linearity. The maps $D:\Piarrow \mathbb{Z}$ and$D’$ : $\Pi’arrow \mathbb{Z}$ define the same grading.It is natural to ask the following question: when do two maps $D_{1}$ : $\Pi_{1}arrow N$ and $D_{2}$ : $\Pi_{2}arrow N$define the same Z-grading, i.e. when can they be extended to a linear map Deg; $\Delta\cup\{0\}arrow \mathbb{Z}$?Theorem 5.1 (Hoyt [5]). Let $\Pi_{1}=\{\alpha_{1}, \ldots, \alpha_{n}\},$ $\Pi_{2}=\{\beta_{1}, \ldots, \beta_{n}\}$ be two different bases for$\Delta$ . The maps $D_{1}$ : $\Pi_{1}arrow N,$ $D_{2}$ : $\Pi_{2}arrow \mathbb{N}$ define the same grading if and only if there is a117sequence of even and odd reflections $\mathcal{R}$ at simple roots of degree zero such that (after reordering)$\mathcal{R}(\alpha_{i})=\beta_{i}$ and $D_{1}(\alpha_{i})=D_{2}(\beta_{i})$ for $i=1,$ $\ldots,$ $n$ .Two Dynkin diagrams $\Gamma_{1},\Gamma_{2}$ for a basic Lie supemlgebra $\mathfrak{g}$ utth degree maps $D_{i}$ : $\Gamma_{i}arrow N$ definethe same Z-gmding if and only if there is a sequence of odd oeflections $\mathcal{R}$ at simple isotropic rootsof degree zero such that $\mathcal{R}(\Gamma_{1})=\Gamma_{2}$ and $D_{1}=D_{2}$ with the ordering of the vertices defined by $\mathcal{R}$ .This defines an equivalence relation on Dynkin diagrams with nonnegative integer labels.References[1] K. Baur and N. Wallach, Nice parabolic subalgebrvns of reductive Lie algebras, Represent. Theory 9(2005), 1-29.[2] J. Brundan and S. Goodwin, Good grading polytopes, Proc. London Math. Soc. 94 (2007), 155-180.[3] A. De Sole and V. G. Kac, Finite vs Affine W-algebras, Japan. J. Math. 1 (2006), 137-261.[4] A. G. Elashvili and V. G. Kac, Classification of good gradings of simple Lie algebras, Lie groups andinvariant theory (E. B. Vinberg ed.), Amer. Math. Socl. Transl. Ser. 2213 (2005), 85-104.[5] C. Hoyt, Good gradings of basic Lie superalgebros, http://arxiv.org/abs/1010.3412.[6] J. C. Jantzen, Nilpotent orbits in representation theory, Lie Theory; Lie Algebras and Representations(J.-P. Anker and B. Orsted, eds.), $Progr\infty$ in Math., vol. 228, Birkh th ser Boston (2004), 1-211.[7] V.G. Kac, Lie superalgebras, Advances in Math 26 (1977), S-96.[8] V. G. Kac, S. Roan and M. Wakimoto, Quantum reduction for affine superolgebras, Commun. Math.Phys. 241 (2003), 307-342.[9] D. Leites, M. Savel‘ev and V. Serganova, Embeddings of Lie superdgebm $osp(1|2)$ and nonlinearsupersymmetric $equations_{)}$ Group Theoretical Methods in Physics 1 (1986), 377-394.118

On good $\mathbb{Z}$-gradings of basic Lie superalgebras (Problems in Representation Theory and Harmonic Analysis)

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On good $\mathbb{Z}$ -gradings of basic Lie superalgebras (Problems in Representation Theory and Harmonic Analysis)

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On good Z\mathbb{Z}Z-gradings of basic Lie superalgebras (Problems in Representation Theory and Harmonic Analysis)

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On good $\mathbb{Z}$ -gradings of basic Lie superalgebras (Problems in Representation Theory and Harmonic Analysis)