Finite-dimensional Simple Lie Algebra Research Articles

Prime decomposition of modular tensor categories of local modules of type D

Let \mathcal{C}(\mathfrak{g},k) be the unitary modular tensor categories arising from the representation theory of quantum groups at roots of unity for arbitrary simple finite-dimensional complex Lie algebra \mathfrak{g} and positive integer levels k . Here we classify nondegenerate fusion subcategories of the modular tensor categories of local modules \mathcal{C}(\mathfrak{g},k)_R^0 where R is the regular algebra of Tannakian Rep (H)\subset\mathcal{C}(\mathfrak{g},k)_\text{pt} . We describe the decomposition of \mathcal{C}(\mathfrak{g},k)_R^0 into prime factors, and as an application we classify relations in the Witt group of nondegenerately braided fusion categories generated by the equivalency classes of \mathcal{C}(\mathfrak{so}_5,k) and \mathcal{C}(\mathfrak{g}_2,k) for k\in\mathbb{Z}_{\geq1} .

Weyl modules and Weyl functors for hyper-map algebras

We investigate the representations of the hyperalgebras associated to the map algebras [Formula: see text], where [Formula: see text] is any finite-dimensional complex simple Lie algebra and [Formula: see text] is any associative commutative unitary algebra with a multiplicatively closed basis. We consider the natural definition of the local and global Weyl modules, and the Weyl functor for these algebras. Under certain conditions, we prove that these modules satisfy certain universal properties, and we also give conditions for the local or global Weyl modules to be finite-dimensional or finitely generated, respectively.

Biderivations and strong commutativity-preserving maps on parabolic subalgebras of simple Lie algebras

ABSTRACT A linear map ψ on a Lie algebra over a field F with char is called to be commuting (resp., skew-commuting) if (resp., ) for all , and to be strong commutativity-preserving if for all . Let L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic 0, P a parabolic subalgebra of L. In this paper, firstly, we improve existing results about skew-symmetric biderivations on P by determining related linear commuting maps. Secondly, we classify the linear skew-commuting maps and the related symmetric biderivations on P, and so the biderivations of P are characterized. Finally, we classify the invertible linear strong commutativity-preserving maps of P.

Yangians versus minimal W-algebras: A surprising coincidence

We prove that the singularities of the [Formula: see text]-matrix [Formula: see text] of the minimal quantization of the adjoint representation of the Yangian [Formula: see text] of a finite dimensional simple Lie algebra [Formula: see text] are the opposite of the roots of the monic polynomial [Formula: see text] entering in the OPE expansions of quantum fields of conformal weight [Formula: see text] of the universal minimal affine [Formula: see text]-algebra at level [Formula: see text] attached to [Formula: see text].

Yangians and Baxter’s relations

We study a category O of representations of the Yangian associated to an arbitrary finite-dimensional complex simple Lie algebra. We obtain asymptotic modules as analytic continuation of a family of finite-dimensional modules, the Kirillov--Reshetikhin modules. In the Grothendieck ring we establish the three-term Baxter's TQ relations for the asymptotic modules. We indicate that Hernandez--Jimbo's limit construction can also be applied, resulting in modules over anti-dominantly shifted Yangians.

Finite-Dimensional Representations of Hyper Multicurrent and Multiloop Algebras

We investigate the categories of finite-dimensional representations of multicurrent and multiloop hyperalgebras in positive characteristic, i.e., the hyperalgebras associated to the multicurrent algebras $\mathfrak {g}\otimes \mathbb {C}[t_{1},\ldots ,t_{n}]$ and to the multiloop algebras $\mathfrak {g}\otimes \mathbb {C}[t_{1}^{\pm 1},\ldots ,t_{n}^{\pm 1}]$ , where $\mathfrak {g}$ is any finite-dimensional complex simple Lie algebra. The main results are the construction of the universal finite-dimensional highest-weight modules and a classification of the irreducible modules in each category. In the characteristic zero setting we also provide a relationship between these modules.

Regular Hom-Lie structures on Borel subalgebras of finite-dimensional simple Lie algebras

A Hom-structure on a Lie algebra is a linear map which satisfies the Hom-Jacobi identity for all A Hom-structure is called regular if is also a Lie algebra isomorphism. Let be a Borel subalgebra of a finite-dimensional simple Lie algebra L over an algebraically closed field of characteristic zero. In this paper, we prove that any regular Hom-structure on is an extremal inner automorphism induced by a highest root vector of Communicated by Sudarshan Kumar Sehgal

Graded modules over simple Lie algebras

The paper is devoted to the study of graded-simple modules and gradings on simple modules over finite-dimensional simple Lie algebras. In general, a connection between these two objects is given by the so-called loop construction. We review the main features of this construction as well as necessary and sufficient conditions under which finite-dimensional simple modules can be graded. Over the Lie algebra sl2(C), we consider specific gradings on simple modules of arbitrary dimension.

Diagram automorphisms and quantum groups

Let $\mathbf{U}^-_q = \mathbf{U}^-_q(\mathfrak{g})$ be the negative part of the quantum group associated to a finite dimensional simple Lie algebra $\mathfrak{g}$, and $\sigma : \mathfrak{g} \to \mathfrak{g}$ be the automorphism obtained from the diagram automorphism. Let $\mathfrak{g}^{\sigma}$ be the fixed point subalgebra of $\mathfrak{g}$, and put $\underline{\mathbf{U}}^-_q = \mathbf{U}^-_q(\mathfrak{g}^{\sigma})$. Let $\mathbf{B}$ be the canonical basis of $\mathbf{U}_q^-$ and $\underline{\mathbf{B}}$ the canonical basis of $\underline{\mathbf{U}}_q^-$. $\sigma$ induces a natural action on $\mathbf{B}$, and we denote by $\mathbf{B}^{\sigma}$ the set of $\sigma$-fixed elements in $\mathbf{B}$. Lusztig proved that there exists a canonical bijection $\mathbf{B}^{\sigma} \simeq \underline{\mathbf{B}}$ by using geometric considerations. In this paper, we construct such a bijection in an elementary way. We also consider such a bijection in the case of certain affine quantum groups, by making use of PBW-bases constructed by Beck and Nakajima.

Local derivations on Borel subalgebras of finite-dimensional simple Lie algebras

Let be an algebraically closed field of characteristic 0, let be a finite-dimensional simple Lie algebra over , and let be the standard Borel subalgebra of . In this paper we prove that every local derivation of is a derivation.

Codimension growth of central polynomials of Lie algebras

Abstract Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero and let I be the T-ideal of polynomial identities of the adjoint representation of L. We prove that the number of multilinear central polynomials in n variables, linearly independent modulo I, grows exponentially like ( dim ⁡ L ) n {(\dim L)^{n}} .

Simple Singular Whittaker Modules Over the Schrödinger Algebra

There are no simple singular Whittaker modules over most of important algebras, such as simple complex finite-dimensional Lie algebras, affine Kac–Moody Lie algebras, the Virasoro algebra, the Heisenberg–Virasoro algebra and the Schrodinger–Witt algebra. In this paper, however, we construct simple singular Whittaker modules over the Schrodinger algebra. Moreover, simple singular Whittaker modules over the Schrodinger algebra are classified. As a result, simple modules for the Schrodinger algebra which are locally finite over the positive part are completely classified. We also give characterizations of simple highest weight modules and simple singular Whittaker modules.

Generators and relations for (generalised) Cartan type superalgebras

In Kac’s classification of finite-dimensional Lie superalgebras, the contragredient ones can be constructed from Dynkin diagrams similar to those of the simple finite-dimensional Lie algebras, but with additional types of nodes. For example, can be constructed by adding a “gray” node to the Dynkin diagram of , corresponding to an odd null root. The Cartan superalgebras constitute a different class, where the simplest example is W (n), the derivation algebra of the Grassmann algebra on n generators. Here we present a novel construction of W(n), from the same Dynkin diagram as A(n − 1, 0), but with additional generators and relations.

Trace functions of the parafermion vertex operator algebras

The trace functions for the Parafermion vertex operator algebra associated to any finite dimensional simple Lie algebra g and any positive integer k are studied and an explicit modular transformation formula of the trace functions is obtained.

Geometric construction of Gelfand–Tsetlin modules over simple Lie algebras

In the present paper we describe a new class of Gelfand–Tsetlin modules for an arbitrary complex simple finite-dimensional Lie algebra g and give their geometric realization as the space of ‘δ-functions’ on the flag manifold G/B supported at the 1-dimensional submanifold. When g=sl(n,C) (or gl(n,C)) these modules form a subclass of Gelfand–Tsetlin modules with infinite-dimensional weight subspaces. We discuss their properties and describe the simplicity criterion for these modules in the case of the Lie algebra sl(3,C).

On truncated Weyl modules

We study structural properties of truncated Weyl modules. A truncated Weyl module is a local Weyl module for , where is a finite-dimensional simple Lie algebra. It has been conjectured that, if N is sufficiently small with respect to λ, the truncated Weyl module is isomorphic to a fusion product of certain irreducible modules. Our main result proves this conjecture when λ is a multiple of certain fundamental weights, including all minuscule ones for simply laced . We also take a further step towards proving the conjecture for all multiples of fundamental weights by proving that the corresponding truncated Weyl module is isomorphic to a natural quotient of a fusion product of Kirillov–Reshetikhin modules. One important part of the proof of the main result shows that any truncated Weyl module is isomorphic to a Chari–Venkatesh module and explicitly describes the corresponding family of partitions. This leads to further results in the case that related to Demazure flags and chains of inclusions of truncated Weyl modules.

Finite-dimensional representations of minimal nilpotent W-algebras and zigzag algebras

Let g \frak g be a simple finite-dimensional Lie algebra over an algebraically closed field F \mathbb F of characteristic 0. We denote by U ( g ) \mathrm {U}(\frak g) the universal enveloping algebra of g \frak g . To any nilpotent element e ∈ g e\in \frak g one can attach an associative (and noncommutative as a general rule) algebra U ( g , e ) \mathrm {U}(\frak g, e) which is in a proper sense a “tensor factor” of U ( g ) \mathrm {U}(\frak g) . In this article we consider the case in which e e belongs to the minimal nonzero nilpotent orbit of g \frak g . Under these assumptions U ( g , e ) \mathrm {U}(\frak g, e) was described explicitly in terms of generators and relations. One can expect that the representation theory of U ( g , e ) \mathrm {U}(\frak g, e) would be very similar to the representation theory of U ( g ) \mathrm {U}(\frak g) . For example one can guess that the category of finite-dimensional U ( g , e ) \mathrm {U}(\frak g, e) -modules is semisimple. The goal of this article is to show that this is the case if g \frak g is not simply-laced. We also show that, if g \frak g is simply-laced and is not of type A n A_n , then the regular block of finite-dimensional U ⁡ ( g , e ) \operatorname {U}(\frak g, e) -modules is equivalent to the category of finite-dimensional modules of a zigzag algebra.

Analysis of the symmetry group and exact solutions of the dispersionless KP equation in n + 1 dimensions

The Lie algebra of the symmetry group of the (n + 1)-dimensional generalization of the dispersionless Kadomtsev–Petviashvili equation is obtained and identified as a semi-direct sum of a finite dimensional simple Lie algebra and an infinite dimensional nilpotent subalgebra. Group transformation properties of solutions under the subalgebra sl(2,R) are presented. Known explicit analytic solutions in the literature are shown to be actually group-invariant solutions corresponding to certain specific infinitesimal generators of the symmetry group.

Local automorphisms of finite dimensional simple Lie algebras

Let g be a finite dimensional simple Lie algebra over an algebraically closed field K of characteristic 0. A linear map φ:g→g is called a local automorphism if for every x in g there is an automorphism φx of g such that φ(x)=φx(x). We prove that a linear map φ:g→g is local automorphism if and only if it is an automorphism or an anti-automorphism.

Non-minimality of certain irregular coherent preminimal affinizations

Let $\mathfrak g$ be a finite-dimensional simple Lie algebra of type $D$ or $E$ and $\lambda$ be a dominant integral weight whose support bounds the subdiagram of type $D_4$. We study certain quantum affinizations of the simple $\mathfrak g$-module of highest weight $\lambda$ which we term preminimal affinizations of order two (this is the maximal order for such $\lambda$). This class can be split in two: the coherent and the incoherent affinizations. If $\lambda$ is regular, Chari and Pressley proved that the associated minimal affinizations belong to one of the three equivalent classes of coherent preminimal affinizations. In this paper we show that, if $\lambda$ is irregular, the coherent preminimal affinizations are not minimal under certain hypotheses. Since these hypotheses are always satisfied if $\mathfrak g$ is of type $D_4$, this completes the classification of minimal affinizations for type $D_4$ by giving a negative answer to a conjecture of Chari-Pressley stating that the coherent and the incoherent affinizations were equivalent in type $D_4$ (this corrects the opposite claim made by the first author in a previous publication).