Let $(\mathfrak{g},τ)$ be a real simple symmetric Lie algebra and let $W \subset \mathfrak{g}$ be an invariant closed convex cone which is pointed and generating with $τ(W) = -W$. For elements $h \in \mathfrak{g}$ with $τ(h) = h$, we classify the Lie algebras $\mathfrak{g}(W,τ,h)$ which are generated by the closed convex cones \[C_{\pm}(W,τ,h) := (\pm W) \cap \mathfrak{g}_{\pm 1}^{-τ}(h),\] where $\mathfrak{g}^{-τ}_{\pm 1}(h) := \{x \in \mathfrak{g} : τ(x) = -x, [h,x] = \pm x\}$. These cones occur naturally as the skew-symmetric parts of the Lie wedges of endomorphism semigroups of certain standard subspaces. We prove in particular that, if $\mathfrak{g}(W,τ,h)$ is non-trivial, then it is either a hermitian simple Lie algebra of tube type or a direct sum of two Lie algebras of this type. Moreover, we give for each hermitian simple Lie algebra and each equivalence class of involutive automorphisms $τ$ of $\mathfrak{g}$ with $τ(W) = -W$ a list of possible subalgebras $\mathfrak{g}(W,τ,h)$ up to isomorphy.
In this paper, we study certain ad-nilpotent subalgebras contained in the non-zero graded portion of a simple Z_n-graded Lie algebra. These subalgebras respect the grading on the Lie algebra and are modules for a Borel subalgebra for the grade-zero Lie subalgebra. We show that semisimple elements in such subalgebras lie in the center of the subalgebra, and we provide a classification of these subalgebras whose weight space decompositions have only non-zero weights.
Let g=g_0+ g_1 be a simple Z_2-graded Lie algebra and let b_0 be a fixed Borel subalgebra of g_0. We describe and enumerate the abelian b_0-stable subalgebras of g_1.
For a semisimple Lie algebra over the complex numbers, Dynkin (1952) developed an algorithm to classify the regular semisimple subalgebras, up to conjugacy by the inner automorphism group. For a graded semisimple Lie algebra over the complex numbers, Vinberg (1979) showed that a classification of a certain type of regular subalgebras (called carrier algebras) yields a classification of the nilpotent orbits in a homogeneous component of that Lie algebra. Here we consider these problems for (graded) semisimple Lie algebras over the real numbers. First, we describe an algorithm to classify the re…
In this paper we define a new presentation for the Dunkl-Opdam subalgebra of the rational Cherednik algebra. This shows that the Dunkl-Opdam subalgebra is a Drinfeld algebra. We use this fact to define Dirac cohomology for the DO subalgebra. We also formalise generalised graded Hecke algebras and define a Langlands classification to generalised graded Hecke algebras.
We give uniform formulas for the branching rules of level 1 modules over orthogonal affine Lie algebras for all conformal pairs associated to symmetric spaces. We also provide a combinatorial intepretation of these formulas in terms of certain abelian subalgebras of simple Lie algebras.
Given a quasi-hereditary algebra $B$, we present conditions which guarantee that the algebra $\gr B$ obtained by grading $B$ by its radical filtration is Koszul and at the same time inherits the quasi-hereditary property and other good Lie-theoretic properties that $B$ might possess. The method involves working with a pair $(A,{\mathfrak a})$ consisting of a quasi-hereditary algebra $A$ and a (positively) graded subalgebra $\mathfrak a$. The algebra $B$ arises as a quotient $B=A/J$ of $A$ by a defining ideal $J$ of $A$. Along the way, we also show that the standard (Weyl) modules for $B$ have …
Let $k$ be an algebraically closed field and $A$ be a (left and right) Noetherian associative $k$-algebra. Assume further that $A$ is either positively graded or semiperfect (this includes the class of finite dimensional $k$-algebras, and $k$-algebras that are finitely generated modules over a Noetherian central Henselian ring). Let $e$ be a primitive idempotent of $A$, which we assume is of degree $0$ if $A$ is positively graded. We consider the idempotent subalgebra $Γ= (1-e)A(1-e)$ and $S_e$ the simple right $A$-module $S_e = eA/e{\rm rad}A$, where ${\rm rad}A$ is the Jacobson radical of $A…
We classify subalgebras of a ring of differential operators which are big in the sense that the extension of associated graded rings is finite. We show that these subalgebras correspond, up to automorphisms, to uniformly ramified finite morphisms. This generalizes a theorem of Levasseur-Stafford on the generators of the invariants of a Weyl algebra under a finite group.
We study the interaction between the block decompositions of reduced universal enveloping algebras in positive characteristic, the PBW filtration, and the nilpotent cone. We provide two natural versions of the PBW filtration on the block subalgebra $A_λ$ of the restricted universal enveloping algebra $\mathcal{U}_χ(\mathfrak{g})$ and show these are dual to each other. We also consider a shifted PBW filtration for which we relate the associated graded algebra to the algebra of functions on the Frobenius neighbourhood of $0$ in the nilpotent cone and the coinvariants algebra corresponding to $λ$…
Let g be a complex simple Lie algebra and e a nilpotent element in g. To a certain nilpotent subalgebra m attached to e, called an admissible subalgebra of g, we associate an endomorphism algebra H. When m is constructed from a good grading for e, we recover the finite W-algebra associated to e and it is well-known that gr(H) is isomorphic to \C[S] as a graded Poisson algebra where S is the Slodowy slice of e and gr(H) is the graded algebra associated to the Kazhdan filtration. In this paper, we consider the case where g =\sl_{pn}(\C) and e consists of p Jordan blocks all of the same size n. H…
Let $Γ$ be a finite group and $V$ a finite-dimensional $Γ$-graded space over an algebraically closed field of characteristic not equal to 2. In the sense of conjugation, we classify all the so-called pre-nil or nil maximal abelian subalgebras for the general linear Lie color algebra $\frak{gl}(V,Γ)$. In the situation of $Γ$ being a cyclic group, we determine the minimal dimensions of pre-nil or nil faithful representations for any finite-dimensional abelian Lie color algebra.
We show that if $T$ is a simple non-negatively graded regular vertex operator algebra with a nonsingular invariant bilinear form and $σ$ is a finite order automorphism of $T$, then the fixed-point vertex operator subalgebra $T^σ$ is also regular. This yields regularity for fixed point vertex operator subalgebras under the action of any finite solvable group. As an application, we obtain an $SL_2(\mathbb{Z})$-compatibility between twisted twining characters for commuting finite order automorphisms of holomorphic vertex operator algebras. This resolves one of the principal claims in the Generali…
Any graded restricted simple Lie algebra of Cartan type contains a subalgebra isomorphic to the Witt algebra over a field of prime characteristic. As some analogue of study on branching rules for restricted non-classical Lie algebras, it is shown that each graded restricted simple Lie algebra of Cartan type can be decomposed into a direct sum of restricted baby Verma modules and simple modules as an adjoint module over the Witt algebra. In particular, the composition factors are precisely determined.
Let g=g_0+g_1 be a Z_2-graded Lie algebra. We study the posets of abelian subalgebras of g_1 which are stable w.r.t. a Borel subalgebra of g_0. In particular, we find out a natural parametrization of maximal elements and dimension formulas for them. We recover as special cases several results of Kostant, Panyushev, Suter.
We study the representation theory of graded Hecke algebras, starting from scratch and focusing on representations that are obtained with induction from a discrete series representation of a parabolic subalgebra. We determine all intertwining operators between such parabolically induced representations, and use them to parametrize the irreducible representations.
We classify the simple restricted modules for the minimal $p$-envelope of the non-graded, non-restricted Hamiltonian Lie algebra $H(2; (1,1); Φ(1))$ over an algebraically closed field $k$ of characteristic $p \geq 5$. We also give the restrictions of these modules to a subalgebra isomorphic to the first Witt Algebra, a result stated in [S. Herpel and D. Stewart, \emph{Selecta Mathematica} 22:2 (2016) 765--799] with an incomplete proof.
We investigate the perfect derived category dgPer(A) of a positively graded differential graded (dg) algebra A whose degree zero part is a dg subalgebra and semisimple as a ring. We introduce an equivalent subcategory of dgPer(A) whose objects are easy to describe, define a t-structure on dgPer(A) and study its heart. We show that dgPer(A) is a Krull-Remak-Schmidt category. Then we consider the heart in the case that A is a Koszul ring with differential zero satisfying some finiteness conditions.
Let $V=\oC^n$ and let $T:=T(V)\otimes T(V^*)$ be the mixed tensor algebra over $V$. We characterize those subsets $A$ of $T$ for which there is a subgroup $G$ of the unitary group $\UU(n)$ such that $A=T^G$. They are precisely the nondegenerate contraction-closed graded $*$-subalgebras of $T$. While the proof makes use of the First Fundamental Theorem for $\GL(n,\oC)$ (in the sense of Weyl), the characterization has as direct consequences First Fundamental Theorems for several subgroups of $\GL(n,\oC)$. Moreover, a Galois connection between linear algebraic $*$-subgroups of $\GL(n,\oC)$ and no…
We consider the subalgebras of split real, non-twisted affine Kac-Moody Lie algebras that are fixed by the Chevalley involution. These infinite-dimensional Lie algebras are not of Kac-Moody type and admit finite-dimensional unfaithful representations. We exhibit a formulation of these algebras in terms of $\mathbb{N}$-graded Lie algebras that allows the construction of a large class of representations using the techniques of induced representations. We study how these representations relate to previously established spinor representations as they arise in the theory of supergravity.
We study the analogue of the Harish-Chandra homomorphism where the universal enveloping algebra is replaced by the Clifford algebra, $Cl(g)$, of a semisimple Lie algebra $g$. Two main goals are achieved. First, we prove that there is a Harish-Chandra type isomorphism between the subalgebra of $g$-invariants in $Cl(g)$ and the Clifford algebra of the Cartan subalgebra of $g$. Second, the Cartan subalgebra is identified, via this isomorphism, with a graded space of the so-called primitive skew-symmetric invariants of $g$. This leads to a distinguished orthogonal basis of the Cartan subalgebra, w…
We present a new conformal algebra. It is Z2 x Z2 graded and generated by three N=1 superconformal algebras coupled to each other by nontrivial relations of parafermionic type. The representation theory and unitary models of the algebra are briefly discussed. We also conjecture the existence of infinite series of parafermionic algebras containing many N=1 or N=2 superconformal subalgebras.
Let G be a reductive group over C. Assume that the Lie algebra g of G has a given grading (g_j) indexed by a cyclic group Z/m such that g_0 contains a Cartan subalgebra of g. The subgroup G_0 of G corresponding to g_0 acts on the variety of nilpotent elements in g_1 with finitely many orbits. We are interested in computing the local intersection cohomology of the closures of these orbits with coefficients in irreducible G-equivariant local systems which are in the "principal block". We show that these can be computed by a purely combinatorial algorithm.
We look to gradations of Kac-Moody Lie algebras by Kac-Moody root systems with finite dimensional weight spaces. We extend, to general Kac-Moody Lie algebras, the notion of C-admissible pair as introduced by H. Rubenthaler and J. Nervi for semi-simple and affine Lie algebras. If g is a Kac-Moody Lie algebra (with Dynkin diagram indexed by I) and (I,J) is such a C-admissible pair, we construct a C-admissible subalgebra g^J, which is a Kac-Moody Lie algebra of the same type as g, and whose root system Σgrades finitely the Lie algebra g. For an admissible quotient ρ: I \rightarrow I we build also…
Let $A_n(K)$ be the Kostant form of $\mathfrak{U}(sl_n^+)$ and $Γ$ the monoid generated by the positive roots of $sl_n$. For each $λ\in Λ(n,r)$ we construct a functor $F_λ$ from the category of finitely generated $Γ$-graded $A_n(K)$-modules to the category of finite dimensional $S^+(n,r)$-modules, with the property that $F_λ$ maps (minimal) projective resolutions of the one-dimensional $A_n(K)$-module $K_{A}$ to (minimal) projective resolutions of the simple $S^+(n,r)$-module $K_λ$.
We explore the structure of the derived spherical Hecke algebra of a $p$-adic group, a graded associative algebra whose degree $0$ subalgebra is the classical spherical Hecke algebra. Working with $\mathbb Z/ p^a$ coefficients, we establish a Satake homomorphism relating this graded algebra to the corresponding graded algebra for the torus. We investigate the image of this homomorphism in degree $1$, as well as other properties, such as transitivity with respect to inclusion of Levi subgroups.
Parabolic subalgebras $\frak{p}$ of semisimple Lie algebras define a $\Bbb{Z}$-grading of the Lie algebra. If there exists a nilpotent element in the first graded part of $\frak{g}$ on which the adjoint group of $\frak{p}$ acts with a dense orbit, the parabolic subalgebra is said to be nice. The corresponding nilpotent element is also called admissible. Nice parabolic subalgebras of simple Lie algebras have been classified. In the case of Borel subalgebras a Richardson element of $\frak{g}_1$ is exactly one that involves all simple root spaces. It is however difficult to write down such nilpot…
For any simple Lie algebra $\mathfrak{g}$ and an element $μ\in\mathfrak{g}^*$, the corresponding commutative subalgebra $\mathcal{A}_μ$ of $\mathcal{U}(\mathfrak{g})$ is defined as a homomorphic image of the Feigin-Frenkel centre associated with $\mathfrak{g}$. It is known that when $μ$ is regular this subalgebra solves Vinberg's quantisation problem, as the graded image of $\mathcal{A}_μ$ coincides with the Mishchenko-Fomenko subalgebra $\overline{\mathcal{A}}_μ$ of $\mathcal{S}(\mathfrak{g})$. By a conjecture of Feigin, Frenkel and Toledano Laredo, this property extends to an arbitrary eleme…
Let $\g$ be a simple Lie algebra of type A or C. We show that the coadjoint representation of any seaweed subalgebra of $\g$ has some properties similar to that of the adjoint representation of a reductive Lie algebra. Namely, a) the field of invariants is rational and b) there exists a generic stabiliser whose identity component is a torus. Our main tool for this is a result about coadjoint representations of some N-graded Lie algebras, which can be regarded as an extension of Rais' theorem for the index of semi-direct products. For all other simple types, we give a uniform description of a p…
The main result of this paper shows that, over large enough fields of characteristic different from $2$, the alternating Hecke algebras are $\mathbb{Z}$-graded algebras that are isomorphic to fixed-point subalgebras of the quiver Hecke algebra of the symmetric group $\mathfrak{S}_n$. As a special case, this shows that the group algebra of the alternating group, over large enough fields of characteristic different from $2$, is a $\mathbb{Z}$-graded algebra. We give a homogeneous presentation for these algebras, compute their graded dimension and show that the blocks of the quiver Hecke algebras…
We develop the theory of global and local Weyl modules for the hyperspecial maximal parabolic subalgebra of type $A_{2n}^{(2)}$. We prove that the dimension of a local Weyl module depends only on its highest weight, thus establishing a freeness result for global Weyl modules. Furthermore, we show that the graded local Weyl modules are level one Demazure modules for the corresponding affine Lie algebra. In the last section we derive the same results for the special maximal parabolic subalgebras of the twisted affine Lie algebras not of type $A_{2n}^{(2)}$.
We consider symmetric pairs of Lie superalgebras which are strongly reductive and of even type, and introduce a graded Harish-Chandra homomorphism. We prove that its image is a certain explicit filtered subalgebra of the Weyl invariants on a Cartan subspace whose associated graded is the image of Chevalley's restriction map on symmetric invariants. This generalises results of Harish-Chandra and V. Kac, M. Gorelik.
We will construct standard pentads which are analogues of Cartan subalgebras, and moreover, we will study graded Lie algebras corresponding to these standard pentads. We call such pentads pentads of Cartan type and describe them by two positive integers and three matrices. The structures of their corresponding Lie algebras are related with contragredient Lie algebras.
We recover the classification of the maximally supersymmetric bosonic backgrounds of eleven-dimensional supergravity by Lie algebraic means. We classify all filtered deformations of the $\mathbb Z$-graded subalgebras $\mathfrak{h}=\mathfrak{h}_{-2}\oplus\mathfrak{h}_{-1}\oplus\mathfrak{h}_{0}$ of the Poincaré superalgebra $\mathfrak{g}=\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}=V\oplus S\oplus \mathfrak{so}(V)$ which differ only in zero degree, that is $\mathfrak{h}_0\subset\mathfrak{g}_0$ and $\mathfrak{h}_j=\mathfrak{g}_j$ for $j<0$. Aside from the Poincaré superalgebra i…
Let $\mathbb{F}$ be a field of characteristic zero and let $\mathfrak{g}$ be a non-zero finite-dimensional split semisimple Lie algebra with root system $Δ$. Let $Γ$ be a finite set of integral weights of $\mathfrak{g}$ containing $Δ$ and $\{0\}$. Following [2,10], we say that a Lie algebra $L$ over $\mathbb{F}$ is \emph{generalized root graded}, or more exactly $(Γ,\mathfrak{g})$-\emph{graded}, if $L$ contains a semisimple subalgebra isomorphic to $\mathfrak{g}$, the $\mathfrak{g}$-module $L$ is the direct sum of its weight subspaces $L_α$ ($α\inΓ$) and $L$ is generated by all $L_α$ with $α\n…
The essential feature of a root-graded Lie algebra L is the existence of a split semisimple subalgebra g with respect to which L is an integrable module with weights in a possibly non-reduced root system S of the same rank as the root system R of g. Examples include map algebras (maps from an affine scheme to g, S = R), matrix algebras like sl_n(A) for a unital associative algebra A (S = R = A_{n-1}), finite-dimensional isotropic central-simple Lie algebras (S properly contains R in general), and some equivariant map algebras. In this paper we study the category of representations of a root-gr…
We introduce a new class of simple Lie algebras $W(n,m)$ that generalize the Witt algebra by using "exponential" functions, and also a subalgebra $W^*(n,m)$ thereof; and we show each derivation of $W^*(1,0)$ can be written as a sum of an inner derivation and a scalar derivation. The Lie algebra $W(n,m)$ is $Z$-graded and is infinite growth.
For a root system of type $B$ we study an algebra similar to a graded Hecke algebra, isomorphic to a subalgebra of the rational Cherednik algebra. We introduce principal series modules over it and prove an irreducibility criterion for these modules. We deduce similar results for an algebra associated to a root system of type $D$.
In this paper, we explore natural connections among the representations of the extended affine Lie algebra $\widehat{sl_N}(\mathbb{C}_q)$ with $\mathbb{C}_q=\mathbb{C}_q[t_0^{\pm1},t_1^{\pm1}]$ an irrational quantum 2-torus, the simple affine vertex algebra $L_{\widehat{sl_{\infty}}}(\ell,0)$ with $\ell$ a positive integer, and Levi subgroups $G$ of $GL_\ell(\mathbb{C})$. First, we give a canonical isomorphism between the category of integrable restricted $\widehat{sl_N}(\mathbb{C}_q)$-modules of level $\ell$ and that of equivariant quasi $L_{\widehat{sl_{\infty}}}(\ell,0)$-modules. Second, we…
The goal of this paper is to understand the graded limit of a family of irreducible prime representations of the quantum affine algebra associated to a simply-laced simple Lie algebra $\mathfrak{g}$. This family was introduced by David Hernandez and Bernard Leclerc in the context of monoidal categorification of cluster algebras. The graded limit of a member of this family is an indecomposable graded module for the current algebra $\mathfrak{g}[t]$; or equivalently a module for the maximal standard parabolic subalgebra in the affine Lie algebra $\widehat{\mathfrak{g}}$. In this paper we study t…
In [FFR] Feingold, Frenkel and Ries gave a spinor construction of the vertex operator para-algebra (abelian intertwining algebra) V = V^0 \oplus V^1 \oplus V^2 \oplus V^3, whose summands are four level-1 irreducible representations of the affine Kac-Moody algebra D_4^(1). The triality group S_3 = < σ,τ| σ^3 = 1 = τ^2, τστ= σ^{-1} > in Aut(V) was constructed, preserving V^0 and permuting the V^i, for i=1,2,3. V is (1/2)Z-graded where V^i_n denotes the n-graded subspace of V^i. Vertex operators Y(v,z) for v in V^0_1 represent D_4^(1) on V, while those for which σ(v) = v represent G_2^(1). We inv…
We introduce a new set of $q$-difference operators acting as raising operators on a family of symmetric polynomials which are characters of graded tensor products of current algebra ${\mathfrak g}[u]$ KR-modules \cite{FL} for ${\mathfrak g}=A_r$. These operators are generalizations of the Kirillov-Noumi \cite{kinoum} Macdonald raising operators, in the dual $q$-Whittaker limit $t\to\infty$. They form a representation of the quantum $Q$-system of type $A$ \cite{qKR}. This system is a subalgebra of a quantum cluster algebra, and is also a discrete integrable system whose conserved quantities, an…
When the relative commutation relations between a set of m parafermions and n parabosons are of ``relative parafermion type'', the underlying algebraic structure is the classical orthosymplectic Lie superalgebra osp(2m+1|2n). The relative commutation relations can also be chosen differently, of ``relative paraboson type''. In this second case, the underlying algebraic structure is no longer an ordinary Lie superalgebra, but a Z_2 x Z_2$-graded Lie superalgebra, denoted here by pso(2m+1|2n). The identification of this new algebraic structure was performed by Tolstoy, amongst others. In the pres…
