The irreducible representations of the Witt algebra $W$ are completely known. A classification of the irreducible $U_χ(W)$--modules was first established by Chang and later simplified by Strade. The aim of this article is to give a classification of the extensions of the simple $U_χ(W)$--modules having $p$--character $χ$ of height $-1,0,1$ and $p-1$ where $p$ denotes the characteristic of the ground field.
In this paper, we establish an explicit classification of length two extensions of tensor modules for the Witt algebra using the cohomology of the Witt algebra with coefficients in the module of the space of homomorphisms between the two modules of interest. To do this we extended our module to a module that has a compatible action of the commutative algebra of Laurent polynomials in one variable. In this setting, we are be able to directly compute all possible 1-cocycles.
In this paper we study irreducible modules for loop of $A\rtimes DerA$ with finite dimensional weight spaces. In particular, we show that Larsson's constructed modules of tensor fields exhausted all irreducible modules.
In this paper, we classify irreducible modules for loop extended Witt algebras with finite dimensional weight spaces. They turn out to be either modules with uniformly bounded weight spaces or highest weight modules. We further prove that all these modules are single point evaluation modules ($n \geq 2$). So they are actually irreducible modules for extended Witt algebras.
Let $W_π$ be the lattice Lie algebra of Witt type associated with an additive inclusion $π: \mathbb{Z}^N \hookrightarrow \mathbb{C}^2$ with $N>1$. In this article, the classification of simple $\mathbb{Z}^N$-graded $W_π$-modules, whose multiplicities are uniformly bounded, is given.
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 $A_{m,n}$ be the tensor product of the Laurient polynomial algebra in $m$ even variables and the exterior algebra in $n$ odd variables over the complex field $\bC$, and the Witt superalgebra $W_{m,n}$ be the Lie superalgebra of superderivations of $A_{m,n}$. In this paper, we classify the simple weight $W_{m,n}$ modules with finite-dimensional weight spaces with respect to the standard Cartan algebra of $W_{m,0}$. Every such module is either a simple quotient of a tensor module or a module of highest weight type.
Let $d>1$ be an integer. In 1986, Shen defined a class of weight modules $F^α_b(V)$ over the Witt algebra $\mathcal{W}_d$ for $\a\in\C^d$, $b\in\C$, and an irreducible module $ V$ over the special linear Lie algebra $\sl_d$. In 1996, Eswara Rao determined the necessary and sufficient conditions for these modules to be irreducible when $V$ is finite dimensional. In this note, we will determine the necessary and sufficient conditions for all these modules $F^α_b(V)$ to be irreducible where $V$ is not necessarily finite dimensional. Therefore we obtain a lot of irreducible $\mathcal{W}_d$-modules…
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 $W_n$).
The Whittaker module $M_ψ$ and its quotient Whittaker module $L_{ψ, ξ}$ for the twisted affine Nappi-Witten Lie algebra $\widehat{H}_{4}[τ]$ are studied. For nonsingular type, it is proved that if $ξ\neq 0$, then $L_{ψ,ξ}$ is irreducible and any irreducible Whittaker $\widehat{H}_{4}[τ]$-module of type $ψ$ with ${\bf k}$ acting as a non-zero scalar $ξ$ is isomorphic to $L_{ψ,ξ}$. Furthermore, for $ξ=0$, all Whittaker vectors of $L_{ψ, 0}$ are completely determined. For singular type, the Whittaker vectors of $L_{ψ, ξ}$ with $ξ\neq 0$ are fully characterized.
In this paper, we study the tensor module $P\otimes M$ over the Witt superalgebra $W_{m,n}^+$ (resp. $W_{m,n}$), where $P$ is a simple module over the Weyl superalgebra $K_{m,n}^+$ (resp. $K_{m,n}$) and $M$ is simple weight module over the general linear Lie superalgebra $\mathfrak{gl}(m,n)$. We obtain the necessary and sufficient conditions for $P\otimes M$ to be simple, and determine all simple subquotient of $P\otimes M$ when it is not simple. All the work leads to completion of some classification problems on the weight representation theory of $W_{m,n}^+$ and $W_{m,n}$.
Let g be the finite dimensional Witt algebra W(1,1) over an algebraically closed field of characteristic p > 3. It is well known that all simple W(1,1)-modules are finite dimensional. Each simple module admits a character χin g^*. Given such a χone can form the (finite dimensional) reduced enveloping algebra u(g,χ). The simple modules for u(g,χ) are precisely those simple W(1,1)-modules admitting the character χ. In this paper the authors compute Ext^1 between pairs of simple modules for u(g,χ).
We study when an sl(2)-representation extends to a representation of the Witt and Virasoro algebras. We give a criterion for extendability and apply it to certain classes of weight sl(2)-modules. For all simple weight sl(2)-modules and those in any of the abelian Krull-Schmidt categories of weight modules whose unique simple object is a dense module, we fully characterize which ones admit extensions, and we obtain explicit expressions for all of them. We also give partial results in the same direction for the abelian categories of weight modules which have two and three simple objects.
We study ${\mathbb Z}$-graded thread $W^+$-modules $$V=\oplus_i V_i, \; \dim{V_i}=1, -\infty \le k< i < N\le +\infty, \; \dim{V_i}=0, \; {\rm \; otherwise},$$ over the positive part $W^+$ of the Witt (Virasoro) algebra $W$. There is well-known example of infinite-dimensional ($k=-\infty, N=\infty$) two-parametric family $V_{λ, μ}$ of $W^+$-modules induced by the twisted $W$-action on tensor densities $P(x)x^μ(dx)^{-λ}, μ, λ\in {\mathbb K}, P(x) \in {\mathbb K}[t]$. Another family $C_{α, β}$ of $W^+$-modules is defined by the action of two multiplicative generators $e_1, e_2$ of $W^+$ as $e_1f_…
Let $A(1):=k[X]/(X^p)$ be the natural representation of the Witt algebra $W(1)$ over an algebraically closed field of prime characteristic $p>3$. In this note, we decompose the $W(1)$-module $A(1)\otimes A(1)$ into two invariant subspaces, and precisely construct their Jordan-Hölder composition series. As a consequence, we obtain all decomposition factors of the tensor product of the simple restricted $W(1)$-module with "highest" weight $p-1$.
For an irreducible module $P$ over the Weyl algebra $\mathcal{K}_n^+$ (resp. $\mathcal{K}_n$) and an irreducible module $M$ over the general liner Lie algebra $\mathfrak{gl}_n$, using Shen's monomorphism, we make $P\otimes M$ into a module over the Witt algebra $W_n^+$ (resp. over $W_n$). We obtain the necessary and sufficient conditions for $P\otimes M$ to be an irreducible module over $W_n^+$ (resp. $W_n$), and determine all submodules of $P\otimes M$ when it is reducible. Thus we have constructed a large family of irreducible weight modules with many different weight supports and many irred…
We extend Witt's theorem to several kinds of simultaneous isometries of subspaces. We determine sufficient and necessary conditions for the extension of an isometry of subspaces $ϕ:E\to E'$ to an isometry $ϕ_V:V\to V'$ that also sends a given subspace to another, or a given self-dual flag to another, or a Witt's decomposition to another and a special self-dual flag to another. We also determine sufficient and necessary conditions for the isometry of generic flags or the simultaneous isometry of (subspace, self-dual flag) pairs.
Lie algebras formed via semi-direct sums of the Witt algebra $\text{Der}(\mathbb{C}[t,t^{-1}])$ and its modules have become increasingly prominent in both physics and mathematics in recent years. In this paper, we complete the study of (Leibniz) central extensions, derivations and automorphisms of the Lie algebras formed from the semi-direct sum of the Witt algebra and its indecomposable intermediate series modules (that is, graded modules with one-dimensional graded components). Our techniques exploit the internal grading of the Witt algebra, which can be applied to a wider class of graded Li…
Wawamoto generalized the Witt algebra using Laurent extension of polynomial ring. We construct the generalized Witt algebra $W(g_p,n)$ by using an additive map $g_p$ from a set of integers into a field of characteristic zero where $1\leq p \leq n.$
Let $d\ge1$ be an integer, $W_d$ and $\mathcal{K}_d$ be the Witt algebra and the weyl algebra over the Laurent polynomial algebra $A_d=\mathbb{C} [x_1^{\pm1}, x_2^{\pm1}, ..., x_d^{\pm1}]$, respectively. For any $\mathfrak{gl}_d$-module $M$ and any admissible module $P$ over the extended Witt algebra $\widetilde W_d$, we define a $W_d$-module structure on the tensor product $P\otimes M$. We prove in this paper that any simple $W_d$-module that is finitely generated over the cartan subalgebra is a quotient module of the $W_d$-module $P \otimes M$ for a finite dimensional simple $\mathfrak{gl}_d…
In this paper we define a notion of Witt group for sesquilinear forms in hermitian categories, which in turn provides a notion of Witt group for sesquilinear forms over rings with involution. We also study the extension of scalars for K-linear hermitian categories, where K is a field of characteristic not 2. We finally extend several results concerning sesquilinear forms to the setting of systems of such forms.
In this paper we construct two kinds of infinite-dimensional 3-Lie algebras from a given commutative associative algebra, and show that they are all canonical Nambu 3-Lie algebras. We relate their inner derivation algebras to Witt algebras, and then study the regular representations of these 3-Lie algebras and the natural representations of the inner derivation algebras. In particular, for the second kind of 3-Lie algebras, we find that their regular representations are Harish-Chandra modules, and the inner derivation algebras give rise to intermediate series modules of the Witt algebras and c…
Let $G$ be a finite group, and $F$ a field of characteristic $0$ or prime to the order of $G$. In $1952$, Witt and in $1956$, Berman independently proved that the number of inequivalent irreducible $F$-representations of $G$ is equal to the number of $F$-conjugacy classes of the elements of $G$, where "$F$-conjugacy" was defined in a certain way. In this paper, we define $F$-conjugacy on $G$ in a natural way and give a proof of the above Witt-Berman theorem. In addition, we give an explicit formula for computing a primitive central idempotent (pci) of the group algebra $F[G]$ corresponding to …
By elementary and direct calculations the vanishing of the (algebraic) second Lie algebra cohomology of the Witt and the Virasoro algebra with values in the adjoint module is shown. This yields infinitesimal and formal rigidity or these algebras. The first (and up to now only) proof of this important result was given 1989 by Fialowski in an unpublished note. It is based on cumbersome calculations. Compared to the original proof the presented one is quite elegant and considerably simpler.
We provide formulas for computing the cocycles on a 3-point Witt algebra Der(R), using an isomorphism between two 3-point algebras Der(R) and Der(S), where the cocycle is already defined. These cocycles can be used to construct universal central extensions and the 3-point Virasoro, which are useful for the representation theory of a 3-point current algebra. The computations determining the cocycles on Der(R) involve elegant applications of the Chu-Vandermonde convolution and other identities for sums of binomial coefficients.
In this thesis we classify modules over a Witt-type Lie algebra and superalgebra such that when considered as modules of $\mathcal{U}(\mathfrak{h})$ they are free of rank 1. We provide sufficient conditions for simplicity, and compute the action of an automorphism on the module. Lastly, we apply the weighting functor and show that the resulting modules are intermediate series modules.
Let $G$ be an almost simple simply connected group over complex numbers. For a positive element $α$ of the coroot lattice of $G$ let $Z^α$ denote the space of based maps from the projective line to the flag variety of $G$ of degree $α$. This space is known to be isomorphic to the space of framed euclidean $G$-monopoles with maximal symmetry breaking at infinity of charge $α$. In [Finkelberg-Kuznetsov-Markarian-Mirković] a system of (étale, rational) coordinates on $Z^α$ is introduced. In this note we compute various known structures on $Z^α$ in terms of the above coordinates. As a byproduct we…
In this paper, by using the "twisting technique" we obtain a class of new modules $A_b$ over the Witt algebras $\mathcal{W}_n$ from modules $A$ over the Weyl algebras $\mathcal{K}_n$ (of Laurent polynomials) for any $b\in\mathbb{C}$. We give the necessary and sufficient conditions for $A_b$ to be irreducible, and determine the necessary and sufficient conditions for two such irreducible $\mathcal{W}_n$-modules to be isomorphic. Since $\sl_{n+1}(\mathbb{C})$ is a subalgebra of $\mathcal{W}_n$, all the above irreducible $\mathcal{W}_n$-modules $A_b$ can be considered as $\sl_{n+1}(\mathbb{C})$-m…
Let $A$ be the one point extension of an algebra $B$ by a projective $B$-module. We prove that the extension of a given support $τ$-tilting $B$-module is a support $τ$-tilting $A$-module; and, conversely, the restriction of a given support $τ$-tilting $A$-module is a support $τ$-tilting $B$-module. Moreover, we prove that there exists a full embedding of quivers between the corresponding poset of support $τ$-tilting modules.
We give a Poincare-Birkhoff-Witt type basis for local Weyl modules of the current algebra of type $C$. As a consequence, we get a fermionic character formula for these modules.
The purpose of this paper is to define the representation and the cohomology of Hom-Lie superalgebras. Moreover we study Central extensions and provide as application the computations of the derivations and second cohomology group of q-deformed Witt superalgebra.
For q generic or a primitive l-th root of unity, q-Witt algebras are described by means of q-divided power algebras. The structure of the universal q-central extension of the q-Witt algebra, the q-Virasoro algebra, is also determined. q-Lie algebras are investigated and the q-PBW theorem for the universal enveloping algebras of q-Lie algebras is proved. A realization of a class of representations of the q-Witt algebras is given. Based on it, the q-holomorph structure for the q-Witt algebras is constructed, which interprets the realization in the context of representation theory.
Let H be a Hopf algebra and let B be a Koszul H-module algebra. We provide necessary and sufficient conditions for a filtered algebra to be a Poincare-Birkhoff-Witt (PBW) deformation of the smash product algebra B#H. Many examples of these deformations are given.
The Witt group of nondegenerate braided fusion categories $\mathcal{W}$ contains a subgroup $\mathcal{W}_\text{un}$ consisting of Witt equivalence classes of pseudo-unitary nondegenerate braided fusion categories. For each finite-dimensional simple Lie algebra $\mathfrak{g}$ and positive integer $k$ there exists a pseudo-unitary category $\mathcal{C}(\mathfrak{g},k)$ consisting of highest weight integerable $\hat{g}$-modules of level $k$ where $\hat{\mathfrak{g}}$ is the corresponding affine Lie algebra. Relations between the classes $[\mathcal{C}(\mathfrak{sl}_2,k)]$, $k\geq1$ have been compl…
The concept of $λ$-differential operators is a natural generalization of differential operators and difference operators. In this paper, we determine the $λ$-differential Lie algebraic structure on the Witt algebra and the Virasoro algebra for invertible $λ$. Then we consider several families of modules over the Virasoro algebra with explicit module actions and determine the $λ$-differential module structures on them.
We classify Jet modules for the Lie (super)algebras $\mathfrak{L}=W\ltimes(\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}])$, where $W$ is the Witt algebra and $\mathfrak{g}$ is a Lie superalgebra with an even diagonlizable derivation. Then we give a concept method to classify all simple cuspidal modules for $\mathfrak{L}$ and the map superalgebras, which are of the form $\mathfrak{L}\otimes R$, where $R$ is a Noetherian unital supercommutative associative superalgebra.
