Let $\mathcal U$ be the 2-category associated with $\mathfrak{sl}_2$. We prove that a complex of 1-morphisms of $\mathcal U$ is null-homotopic if and only if its image in every simple 2-representation is null-homotopic. Under mild boundedness assumptions, we prove that it actually suffices for the image in the simple 2-representations to be acyclic. We apply this result to the study of the Rickard complex $Θ$ categorifying the action of the simple reflection of $\mathrm{SL}_2$. We prove that $Θ$ is invertible in the homotopy category of $\U$, and that there is a homotopy equivalence $ΘE \simeq FΘ[-1]$.
We classify the simple bounded weight modules of ${\mathfrak{sl}(\infty})$, ${\mathfrak{o}(\infty)}$ and ${\mathfrak{sp}(\infty)}$, and compute their annihilators in $U({\mathfrak{sl}(\infty}))$, $U({\mathfrak{o}(\infty))}$, $U({\mathfrak{sp}(\infty))}$, respectively.
We study the structure of bounded simple weight $\mathfrak{sl}(\infty)$-, $\mathfrak{o}(\infty)$-, $\mathfrak{sp}(\infty)$-modules, which have been recently classified in [6]. Given a splitting parabolic subalgebra $\mathfrak{p}$ of $\mathfrak{sl}(\infty)$, $\mathfrak{o}(\infty)$, $\mathfrak{sp}(\infty)$, we introduce the concepts of $\mathfrak{p}$-aligned and pseudo $\mathfrak{p}$-aligned $\mathfrak{sl}(\infty)$-, $\mathfrak{o}(\infty)$-, $\mathfrak{sp}(\infty)$-modules, and give necessary and sufficient conditions for bounded simple weight modules to be $\mathfrak{p}$-aligned or pseudo $\mat…
We classify completely prime primitive ideals whose associated varieties are the closure of the minimal nilpotent orbit of $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$, and classify irreducible $(\mathfrak{g},\mathfrak{k})$-modules which have those ideals as annihilators. Moreover, we irreducibly decompose them as $\mathfrak{k}$-modules.
This paper gives a classification of all pairs $(\mathfrak g, \mathfrak h)$ with $\mathfrak g$ a simple real Lie algebra and $\mathfrak h < \mathfrak g$ a reductive subalgebra for which there exists a minimal parabolic subalgebra $\mathfrak p < \mathfrak g$ such that $\mathfrak g = \mathfrak h + \mathfrak p$ as vector sum.
Let $\mathfrak{p}$ be a parabolic subalgebra of $\mathfrak{sl}(V)$ of maximal dimension and let $\mathfrak{n} \subset \mathfrak{p}$ be the corresponding nilradical. In this paper we classify the set of $\mathfrak{sl}(V)$-modules whose restriction to $U(\mathfrak{n})$ is free of rank $1$. It turns out that isomorphism classes of such modules are parametrized by polynomials in $\dim V-1$ variables. We determine the submodule structure for these modules and we show that they generically are simple.
We construct a new family of simple $\mathfrak{gl}_{2n}$-modules which depends on $n^2$ generic parameters. Each such module is isomorphic to the regular $U(\mathfrak{gl}_{n})$-module when restricted the $\mathfrak{gl}_{n}$-subalgebra naturally embedded into the top-left corner.
Let $\mathfrak g$ be a simple Lie algebra, $\mathfrak h$ a Levi subalgebra, and $C_{\mathfrak h}\in U(\mathfrak h)$ the Casimir element defined via the restriction of the Killing form on $\mathfrak g$ to $\mathfrak h$. We study $C_{\mathfrak h}$-eigenvalues in $\mathfrak g/\mathfrak h$ and related $\mathfrak h$-modules. Without loss of generality, one may assume that $\mathfrak h$ is a maximal Levi. Then $\mathfrak g$ is equipped with the natural $\mathbb Z$-grading $\mathfrak g=\bigoplus_{i\in\mathbb Z}\mathfrak g(i)$ such that $\mathfrak g(0)=\mathfrak h$ and $\mathfrak g(i)$ is a simple $\m…
We prove faithfulness of infinite-dimensional generalised Verma modules for Iwasawa algebras corresponding to split simple Lie algebras with a Chevalley basis. We use this to prove faithfulness of all infinite-dimensional highest-weight modules in the case of type $A_{2}$. In this case we also show that all prime ideals of the corresponding Iwasawa algebras are annihilators of finite-dimensional simple modules.
For a finite-dimensional Lie algebra $\mathfrak{L}$ over $\mathbb{C}$ with a fixed Levi decomposition $\mathfrak{L} = \mathfrak{g} \oplus \mathfrak{r}$ where $\mathfrak{g}$ is semi-simple, we investigate $\mathfrak{L}$-modules which decompose, as $\mathfrak{g}$-modules, into a direct sum of simple finite-dimensional $\mathfrak{g}$-modules with finite multiplicities. We call such modules $\mathfrak{g}$-Harish-Chandra modules. We give a complete classification of simple $\mathfrak{g}$-Harish-Chandra modules for the Takiff Lie algebra associated to $\mathfrak{g} = \mathfrak{sl}_2$, and for the Sc…
Let $\mathfrak{g}=\mathfrak{sl}(\infty)$. We compute the annihilators of a class of simple integrable weight $\mathfrak{g}$-modules with finite-dimensional weight spaces. It is a claim of I. Dimitrov, that this class exhausts all simple integrable weight $\mathfrak{g}$-modules with finite-dimensional weight spaces. The main feature of interest is that Dimitrov's class of modules contains non highest weight modules. Here we provide another construction for these modules, which allows to apply results of [PP18] to compute such annihilators.
Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$. A $\operatorname{Y}(\mathfrak{g})$-module is said to be weight if it is a weight $\mathfrak{g}$-module. We give a complete classification of simple weight modules for $\operatorname{Y}(\mathfrak{sl}_2)$ which admits a one-dimensional weight space. We prove that there are four classes of such modules: finite, highest weight, lowest weight and dense modules. Different from the classical $\mathfrak{sl}_{2}$ representation theory, we show that there exist a class of $\operatorname{Y}(\mathfrak{sl}_{2})$ irreducible mo…
For a simple complex Lie algebra $\mathfrak g$ we study the space of invariants $A=\left( \bigwedge \mathfrak g^*\otimes\mathfrak g^*\right)^{\mathfrak g}$, (which describes the isotypic component of type $\mathfrak g$ in $ \bigwedge \mathfrak g^*$) as a module over the algebra of invariants $\left(\bigwedge \mathfrak g^*\right)^{\mathfrak g}$. As main result we prove that $A$ is a free module, of rank twice the rank of $\mathfrak g$, over the exterior algebra generated by all primitive invariants in $(\bigwedge \mathfrak g^*)^{\mathfrak g}$, with the exception of the one of highest degree.
We prove that the tensor product of a simple and a finite dimensional $\mathfrak{sl}_n$-module has finite type socle. This is applied to reduce classification of simple $\mathfrak{q}(n)$-supermodules to that of simple $\mathfrak{sl}_n$-modules. Rough structure of simple $\mathfrak{q}(n)$-supermodules, considered as $\mathfrak{sl}_n$-modules, is described in terms of the combinatorics of category $\mathcal{O}$.
Let $\mathfrak g$ be a simple Lie algebra and $\mathfrak{Ab}$ the poset of all abelian ideals of a fixed Borel subalgebra of $\mathfrak g$. If $\mathfrak a\in\mathfrak{Ab}$, then the normaliser of $\mathfrak a$ is a standard parabolic subalgebra of $\mathfrak g$. We give an explicit description of the normaliser for a class of abelian ideals that includes all maximal abelian ideals. We also elaborate on a relationship between abelian ideals and $\mathbb Z$-gradings of $\mathfrak g$ associated with their normalisers.
The complexity of a module is the rate of growth of the minimal projective resolution of the module while the $z$-complexity is the rate of growth of the number of indecomposable summands at each step in the resolution. Let $\mathfrak{g}=\mathfrak{osp}(k|2)$ ($k>2$) be the type II orthosymplectic Lie superalgebra of types $B$ or $D$. In this paper, we compute the complexity and the $z$-complexity of the simple finite-dimensional $\mathfrak{g}$-supermodules. We then give geometric interpretations using support and associated varieties for these complexities.
We study the effects of the branching $\mathfrak{osp}(1|2n)\supset \mathfrak{gl}(n)$ on a particular class of simple infinite-dimensional $\mathfrak{osp}(1|2n)$-modules $L(p)$ characterized by a positive integer $p$. In the first part we use combinatorial methods such as Young tableaux and Young subgroups to construct a new basis for $L(p)$ that respects this branching and we express the basis elements explicitly in two distinct ways. First as monomials of negative root vectors of $\mathfrak{gl}(n)$ acting on the $\mathfrak{gl}(n)$-highest weight vectors in $L(p)$ and then as polynomials in th…
In this paper, we determine all simple restricted modules over the mirror Heisenberg-Virasoro algebra ${\mathfrak{D}}$, and the twisted Heisenberg-Virasoro algebra $\bar\mathfrak{D}$ with nonzero level. As applications, we characterize simple Whittaker modules and simple highest weight modules over ${\mathfrak{D}}$. A vertex-algebraic interpretation of our result is the classification of simple weak twisted and untwisted modules over the Heisenberg-Virasoro vertex operator algebras $\mathcal V^{c} \cong V_{Vir}^{c}\otimes M(1)$. We also present a few examples of simple restricted ${\mathfrak{D…
If ${\mathfrak g}$ is a real reductive Lie algebra and ${\mathfrak h} < {\mathfrak g}$ is a subalgebra, then $({\mathfrak g}, {\mathfrak h})$ is called real spherical provided that ${\mathfrak g} = {\mathfrak h} + {\mathfrak p}$ for some choice of a minimal parabolic subalgebra ${\mathfrak p} \subset {\mathfrak g}$. In this paper we classify all real spherical pairs $({\mathfrak g}, {\mathfrak h})$ where ${\mathfrak g}$ is semi-simple but not simple and ${\mathfrak h}$ is a reductive real algebraic subalgebra. The paper is based on the classification of the case where ${\mathfrak g}$ is simple…
In this paper, we construct, investigate and, in some cases, classify several new classes of (simple) modules over the Takiff $\mathfrak{sl}_{2}$. More precisely, we first explicitly construct and classify, up to isomorphism, all modules over the Takiff $\mathfrak{sl}_{2}$ that are $U\left(\overline{\mathfrak{h}}\right)$-free of rank one. These split into three general families of modules. The sufficient and necessary conditions for simplicity of these modules are presented, and their isomorphism classes are determined. Using the vector space duality and Mathieu's twisting functors, these thre…
Let G be a simply connected simple algebraic group over an algebraically closed field K of characteristic p>0 with root system R, and let ${\mathfrak g}={\cal L}(G)$ be its restricted Lie algebra. Let V be a finite dimensional ${\mathfrak g}$-module over K. For any point $v\inV$, the {\it isotropy subalgebra} of $v$ in $\mathfrak g$ is ${\mathfrak g}_v=\{x\in{\mathfrak g}/x\cdot v=0\}$. A restricted ${\mathfrak g}$-module V is called exceptional if for each $v\in V$ the isotropy subalgebra ${\mathfrak g}_v$ contains a non-central element (that is, ${\mathfrak g}_v\not\subseteq {\mathfrak z(\ma…
We prove the rationality of the exceptional W-algebras associated with the simple Lie algebra $\mathfrak{sp}_4$ and subregular nilpotent elements, proving a new particular case of a conjecture of Kac-Wakimoto. Moreover, we describe the simple $\mathcal{W}_k(\mathfrak{sp}_4,f_{subreg})$-modules and compute their characters. We also explicit the nontrivial action of the component group on the set of these simple modules.
Let $\mathfrak g$ be a simple Lie algebra over an algebraically closed field $\bf k$ of characteristic zero and $\bf G$ its adjoint group. Let $\mathfrak q$ be a biparabolic subalgebra of $\mathfrak g$. The algebra $Sy(\mathfrak q)$ of semi-invariants on $\mathfrak q^*$ is polynomial in most cases, in particular when $\mathfrak g$ is simple of type $A$ or $C$. On the other hand $\mathfrak q$ admits a canonical truncation $\mathfrak q_Λ$ such that $Sy(\mathfrak q)=Sy(\mathfrak q_Λ)=Y(\mathfrak q_Λ)$ where $Y(\mathfrak q_Λ)$ denotes the algebra of invariant functions on $\mathfrak q_Λ^*$. An ada…
We calculate the cohomology of $\mathfrak{sl}_3(k)$ over an algebraically closed field $k$ of characteristic $p>3$ with coefficients in simple modules and Weyl modules. We also give descriptions of the corresponding cohomology of $\mathfrak{gl}_3(k)$.
We classify simple bounded weight modules over the complex simple Lie superalgebras $\mathfrak{sl}(\infty |\infty)$ and $\mathfrak{osp} (m | 2n)$, when at least one of $m$ and $n$ equals $\infty$. For $\mathfrak{osp} (m | 2n)$ such modules are of spinor-oscillator type, i.e., they combine into one the known classes of spinor $\mathfrak{o} (m)$-modules and oscillator-type $\mathfrak{sp} (2n)$-modules. In addition, we characterize the category of bounded weight modules over $\mathfrak{osp} (m | 2n)$ (under the assumption $\dim \, \mathfrak{osp} (m | 2n) = \infty$) by reducing its study to alread…
Given a complex orthosymplectic superspace $V$, the orthosymplectic Lie superalgebra $\mathfrak {osp}(V)$ and general linear algebra ${\mathfrak {gl}}_N$ both act naturally on the coordinate super-ring $\mathcal{S}(N)$ of the dual space of $V\otimes{\mathbb C}^N$, and their actions commute. Hence the subalgebra $\mathcal{S}(N)^{\mathfrak {osp}(V)}$ of $\mathfrak {osp}(V)$-invariants in $\mathcal{S}(N)$ has a ${\mathfrak {gl}}_N$-module structure. We introduce the space of super Pfaffians as a simple ${\mathfrak {gl}}_N$-submodule of $\mathcal{S}(N)^{\mathfrak {osp}(V)}$, give an explicit formu…
We study simple non-weight ${\mathfrak{sl}}(2)$-modules which are finitely generated as ${\mathbb C}[z]$-modules. We show that they are described in terms of semilinear endomorphisms and prove that the Smith type induces a stratification on the set of these ${\mathfrak{sl}}(2)$-modules, providing thus new invariants. Moreover, we show that there is a notion of duality for these type of ${\mathfrak{sl}}(2)$-modules. Finally, we show that there are simple non-weight ${\mathfrak{sl}}(2)$-modules of arbitrary rank by constructing a whole new family of them.
Dynkin's classification of maximal subalgebras of simple finite dimensional complex Lie algebras is generalized to linear Lie superalgebras. Namely, the maximal non-simple irreducible subalgebras of $\mathfrak{gl}(p|q), \mathfrak{q}(n), \mathfrak{sl}(p|q), \mathfrak{osp}(m|2n), \mathfrak{pe}(n)$, and $\mathfrak{spe}(n)$ are classified.
Let $k$ be a field of characteristic not two or three. We classify up to isomorphism all finite-dimensional Lie superalgebras $\mathfrak{g}=\mathfrak{g}_0\oplus \mathfrak{g}_1$ over $k$, where $\mathfrak{g}_0$ is a three-dimensional simple Lie algebra. If $\mathcal{Z}(\mathfrak{g})$ denotes the centre of $\mathfrak{g}$, the result is the following: either $\lbrace \mathfrak{g}_1,\mathfrak{g}_1 \rbrace=\lbrace 0 \rbrace$ or $\mathfrak{g}_1=(\mathfrak{g}_0 \oplus k)\oplus \mathcal{Z}(\mathfrak{g})$ or $\mathfrak{g}\cong \mathfrak{osp}_k(1|2)\oplus \mathcal{Z}(\mathfrak{g})$.
Given a complex semisimple Lie algebra ${\mathfrak g}$ and a commutative ${\mathbb C}$-algebra $A$, let ${\mathfrak g}[A] = {\mathfrak g} \otimes A$ be the corresponding generalized current algebra. In this paper we explore questions involving the computation and finite-dimensionality of extension groups for finite-dimensional ${\mathfrak g}[A]$-modules. Formulas for computing $\operatorname{Ext}^{1}$ and $\operatorname{Ext}^{2}$ between simple ${\mathfrak g}[A]$-modules are presented. As an application of these methods and of the use of the first cyclic homology, we completely describe $\oper…
We determine the Lie subalgebra $\mathfrak{g}_{nil}$ of a Borcherds symmetrizable generalized Kac-Moody Lie algebra $\mathfrak{g}$ generated by $ad$-locally nilpotent elements and show that it is `essentially' the same as the Levi subalgebra of $\mathfrak{g}$ with its simple roots precisely the real simple roots of $\mathfrak{g}$.
We study multiplicity space signatures in tensor products of representations of $\mathfrak{sl}_2$ and $U_q(\mathfrak{sl}_2)$, and give some applications. We completely classify definite multiplicity spaces for generic tensor products of $\mathfrak{sl}_2$ Verma modules. This provides a classification of a family of unitary representations of a basic quantized quiver variety, one of the first such classifications for any quantized quiver variety. We use multiplicity space signatures to provide the first real critical point lower bound for generic $\mathfrak{sl}_2$ master functions. As a corollar…
The affine Yangian of $\mathfrak{gl}_1$ has recently appeared simultaneously in the work of Maulik-Okounkov and Schiffmann-Vasserot in connection with the Alday-Gaiotto-Tachikawa conjecture. While the former presentation is purely geometric, the latter algebraic presentation is quite involved. In this article, we provide a simple loop realization of this algebra which can be viewed as an "additivization" of the quantum toroidal algebra of $\mathfrak{gl}_1$ in the same way as the Yangian $Y_h(\mathfrak{g})$ is an "additivization" of the quantum loop algebra $U_q(L\mathfrak{g})$ for a simple Lie…
In this paper, as the first step towards classification of simple weight modules with finite dimensional weight spaces over Witt algebras $W_n$, we explicitly describe supports of such modules. We also obtain some descriptions on the support of an arbitrary simple weight module over a $\Z^n$-graded Lie algebra $\mathfrak{g}$ having a root space decomposition $\oplus_{α\in\Z^n}\mathfrak {g}_α$ with respect to the abelian subalgebra $\mathfrak {g}_0$, with the property $[\mathfrak{g}_α,\mathfrak {g}_β]= \mathfrak {g}_{α+β}$ for all $α,β\in\Z^n$, $α\neq β$ (this class contains the algebra $W_n$).
Let $\mathfrak{g}$ be a simple complex Lie algebra and let $\mathfrak{t} \subset \mathfrak{g}$ be a toral subalgebra of $\mathfrak{g}$. As a $\mathfrak{t}$-module $\mathfrak{g}$ decomposes as \[\mathfrak{g} = \mathfrak{s} \oplus \big(\oplus_{ν\in \mathcal{R}} \mathfrak{g}^ν\big)\] where $\mathfrak{s} \subset \mathfrak{g}$ is the reductive part of a parabolic subalgebra of $\mathfrak{g}$ and $\mathcal{R}$ is the Kostant root system associated to $\mathfrak{t}$. When $\mathfrak{t}$ is a Cartan subalgebra of $\mathfrak{g}$ the decomposition above is nothing but the root decomposition of $\mathfra…
In this paper, we show that the Dirac cohomology $H_{D}(L(λ))$ of a simple highest weight module $L(λ)$ in $\mathcal{O}^\mathfrak{p}$ can be parameterized by a specific set of weights: a subset $\mathcal{W}_I(λ)$ of the orbit of the Weyl group $W$ acting on $λ+ρ$. As an application, we show that any simple module in $\mathcal{O}^\mathfrak{p}$ is determined up to isomorphism by its Dirac cohomology. We describe four parameterizations of $H_D(L(λ))$ when $λ$ is regular. Two of these parameterizations are geometric in terms of a partial ordering on the dual of the Cartan subalgebra and a generali…
For a simple Lie algebra, Shapovalov elements give rise to highest weight vectors in Verma modules. The usual construction of these elements uses induction on the length of a certain Weyl group element. If $\mathfrak{g}= \mathfrak{sl}(N+1)$ explicit expressions for Shapovalov elements were given in [Mus22a]. Here we adapt the argument to the quantized enveloping algebra of $\mathfrak{g}$.
Let $G$ be a noncompact connected simple Lie group, and $(G,G^Γ)$ a Klein four symmetric pair. In this paper, the author shows a necessary condition for the discrete decomposability of unitarizable simple $(\mathfrak{g},K)$-modules for Klein for symmetric pairs. Precisely, if certain conditions hold for $(G,G^Γ)$, there does not exist any unitarizable simple $(\mathfrak{g},K)$-module that is discretely decomposable as a $(\mathfrak{g}^Γ,K^Γ)$-module. As an application, for $G=\mathrm{E}_{6(-14)}$, the author obtains a complete classification of Klein four symmetric pairs $(G,G^Γ)$ with $G^Γ$ n…
In this paper, we study the representations of the periplectic Lie superalgebra using the Duflo-Serganova functor. Given a simple $\mathfrak{p}(n)$-module $L$ and a certain element $x\in \mathfrak{p}(n)$ of rank $1$, we give an explicit description of the composition factors of the $\mathfrak{p}(n-1)$-module $DS_x(L)$, which is defined as the homology of the complex $$ΠM \xrightarrow{x} M \xrightarrow{x} ΠM.$$ In particular, we show that this $\mathfrak{p}(n-1)$-module is multiplicity-free. We then use this result to give a simple explicit combinatorial formula for the superdimension of a simp…
Let $\mathfrak g$ be a simple Lie algebra with Cartan subalgebra $\mathfrak h$ and Weyl group $W$. We build up a graded map $(\mathcal H\otimes \bigwedge\mathfrak h\otimes \mathfrak h)^W\to (\bigwedge \mathfrak g\otimes \mathfrak g)^\mathfrak g$ of $(\bigwedge \mathfrak g)^\mathfrak g\cong S(\mathfrak h)^W$-modules, where $\mathcal H$ is the space of $W$-harmonics. In this way we prove an enhanced form of a conjecture of Reeder for the adjoint representation. New version with different title. Various improvements. New section 7.
Building on work of the first and last author, we prove that an embedding of simple affine vertex algebras $V_{\mathbf{k}}(\mathfrak g^0)\subset V_{k}(\mathfrak g)$, corresponding to an embedding of a maximal equal rank reductive subalgebra $\mathfrak g^0$ into a simple Lie algebra $\mathfrak g$, is conformal if and only if the corresponding central charges are equal. We classify the equal rank conformal embeddings. Furthermore we describe, in almost all cases, when $V_{k}(\mathfrak g)$ decomposes finitely as a $V_{\mathbf{k}}(\mathfrak g^0)$-module.
Let $\mathfrak{g}$ be a simple Lie algebra of exceptional type over an algebraically closed field $k$, and let $G$ be a simple linear algebraic group with Lie algebra $\mathfrak{g}$. For $\mathrm{char} \, k =p >0$, we present a complete classification of the $G$-conjugacy classes of balanced toral elements of $\mathfrak{g}$. As a result, we also obtain the classification of conjugacy classes of balanced inner torsion automorphisms of $\mathfrak{g}$ of order $p$ when $\mathrm{char} \, k =0$.
We study a category of modules over $\mathfrak{gl}(\infty)$ analogous to category $\mathcal O$. We fix adequate Cartan, Borel and Levi-type subalgebras $\mathfrak h, \mathfrak b$ and $\mathfrak l$ with $\mathfrak l \cong \mathfrak{gl}(\infty)^n$, and define $\mathcal O_{\mathsf{LA}}^{\mathfrak l}{\mathfrak{gl}(\infty)}$ to be the category of $\mathfrak h$-semisimple, $\mathfrak n$-nilpotent modules that satisfy a large annihilator condition as $\mathfrak l$-modules. Our main result is that these are highest weight categories in the sense of Cline, Parshall and Scott. We compute the simple mult…
For a reductive Lie algebra $\mathfrak h$ and a simple finite-dimensional $\mathfrak h$-module $V$, the set of weights of $V$, $P(V)$, has a natural poset structure. We consider antichains in the weight poset $P(V)$ and a certain operator $\mathfrak X$ acting on antichains. Eventually, we impose stronger constraints on $(\mathfrak h,V)$ and stick to the case in which $\mathfrak h$ and $V$ are associated with a $Z$-grading of a simple Lie algebra $\mathfrak g$. Then $V$ is a weight multiplicity free $\mathfrak h$-module and $P(V)$ can be regarded as a subposet of $Δ^+$, where $Δ$ is the root sy…
For $n\geq 5$ the natural permutation module for the alternating group $\mathfrak{A}_n$ has a unique non-trivial composition factor, being called its natural simple module. We determine the vertices and sources of the natural simple $\mathfrak{A}_n$-module over fields of characteristic 2.
Let $\mathfrak g$ be a complex simple Lie algebra and $\mathfrak b=\mathfrak t\oplus\mathfrak u^+$ a fixed Borel subalgebra. Let $Δ^+$ be the set of positive roots associated with $\mathfrak u^+$ and $\mathcal K\subsetΔ^+$ the Kostant cascade. We elaborate on some constructions related to $\mathcal K$ and applications of $\mathcal K$. This includes the cascade element $x_{\mathcal K}$ in the Cartan subalgebra $\mathfrak t$ and properties of certain objects naturally associated with $\mathcal K$: an abelian ideal of $\mathfrak b$, a nilpotent $G$-orbit in $\mathfrak g$, and an involution of $\m…
In this paper, we classify all simple jet modules for the Neveu-Schwarz algebra $\widehat{\mathfrak{k}}$ and its contact subalgebra $\mathfrak{k}^+$. Based on these results, we give a classification of simple Harish-Chandra modules for $\widehat{\mathfrak{k}}$ and $\mathfrak{k}^+$.
