Finite-dimensional Irreducible Modules Research Articles

Gaudin Hamiltonians on unitarizable modules over classical Lie (super)algebras

Let M be a tensor product of unitarizable irreducible highest weight modules over the Lie (super)algebra G, where G is gl(m|n), osp(2m|2n) or spo(2m|2n). We show, using super duality, that the singular eigenvectors of the (super) Gaudin Hamiltonians for G on M can be obtained from the singular eigenvectors of the Gaudin Hamiltonians for the corresponding Lie algebras on some tensor products of finite-dimensional irreducible modules. As a consequence, the (super) Gaudin Hamiltonians for G are diagonalizable on the space spanned by singular vectors of M and hence on M. In particular, we establish the diagonalization of the Gaudin Hamiltonians, associated to any of the orthogonal Lie algebra so(2n) and the symplectic Lie algebra sp(2n), on the tensor product of infinite-dimensional unitarizable irreducible highest weight modules.

Finite-dimensional representations of map superalgebras

We describe the block decomposition of the categories of finite-dimensional modules for classical map superalgebras and affine superalgebras in terms of spectral characters. En route to showing these block decompositions, we obtain a new description of finite-dimensional irreducible modules for classical map and affine superalgebras, provide formulas for their (super)characters and describe their extension groups. As an application, we establish a relation between the highest weights of a given finite-dimensional irreducible module for a classical map superalgebra with respect to non-conjugate Borel subalgebras.

Strong minuscule elements in the finite Weyl groups

We introduce the notion of a strong minuscule element, which is a dominant minuscule element $w$ in the Weyl group for which there exists a unique (dominant) integral weight $\Lambda$ such that $w$ is $\Lambda$-minuscule. We prove that the dominant integral weight associated to a strong minuscule element is the fundamental weight corresponding to a short simple root (in this paper, all simple roots in the simply-laced cases are treated as short roots). In addition, we enumerate the strong minuscule elements explicitly. As an application of this enumeration, we determine the dimension of certain Demazure module in the finite-dimensional irreducible module whose highest weight is a minuscule weight.

The finite dimensional irreducible modules for affine walled Brauer algebras

We classify the finite dimensional irreducible modules for affine walled Brauer algebras over an algebraically closed field with arbitrary characteristic.

Classical weight modules over ıquantum groups

ıquantum groups are generalizations of quantum groups which appear as coideal subalgebras of quantum groups in the theory of quantum symmetric pairs. In this paper, we define the notion of classical weight modules over an ıquantum group, and study their properties along the lines of the representation theory of weight modules over a quantum group. In several cases, we classify the finite-dimensional irreducible classical weight modules by a highest weight theory.

Hopf–Hecke algebras, infinitesimal Cherednik algebras, and Dirac cohomology

Hopf-Hecke algebras and Barbasch-Sahi algebras were defined by the first named author (2016) in order to provide a general framework for the study of Dirac cohomology. The aim of this paper is to explore new examples of these definitions and to contribute to their classification. Hopf-Hecke algebras are distinguished by an orthogonality condition and a PBW property. The PBW property for algebras such as the ones considered here has been of great interest in the literature and we extend this discussion by further results on the classification of such deformations and by a class of hitherto unexplored examples. We study infinitesimal Cherednik algebras of $GL_n$ as defined by Etingof, Gan, and Ginzburg in [Transform. Groups, 2005] as new examples of Hopf-Hecke algebras with a generalized Dirac cohomology. We show that they are in fact Barbasch-Sahi algebras, that is, a version of Vogan's conjecture analogous to the results of Huang and Pandžic in [J. Amer. Math. Soc., 2002] is available for them. We derive an explicit formula for the square of the Dirac operator and use it to study the finite-dimensional irreducible modules. We find that the Dirac cohomology of these modules is non-zero and that it, in fact, determines the modules uniquely.

Representations of ω-Lie algebras and tailed derivations of Lie algebras

We study the representation theory of finite-dimensional [Formula: see text]-Lie algebras over the complex field. We derive an [Formula: see text]-Lie version of the classical Lie’s theorem, i.e., any finite-dimensional irreducible module of a soluble [Formula: see text]-Lie algebra is 1-dimensional (1D). We also prove that indecomposable modules of some 3D [Formula: see text]-Lie algebras could be parametrized by the complex field and nilpotent matrices. We introduce the notion of a tailed derivation of a nonassociative algebra [Formula: see text] and prove that if [Formula: see text] is a Lie algebra, then there exists a one-to-one correspondence between tailed derivations of [Formula: see text] and 1D [Formula: see text]-extensions of [Formula: see text].

Finite-dimensional irreducible modules of the universal Askey–Wilson algebra at roots of unity

Let F denote an algebraically closed field and assume that q∈F is a primitive dth root of unity with d≠1,2,4. The universal Askey–Wilson algebra △q is a unital associative F-algebra defined by generators and relations. The generators are A,B,C and the relations assert that each ofA+qBC−q−1CBq2−q−2,B+qCA−q−1ACq2−q−2,C+qAB−q−1BAq2−q−2 commutes with A,B,C. We show that every finite-dimensional irreducible △q-module is of dimension less than or equal to{difdis odd;d/2ifdis even. Moreover we provide an example to show that the bound is tight.

Etingof’s conjecture for quantized quiver varieties

We compute the number of finite dimensional irreducible modules for the algebras quantizing Nakajima quiver varieties. We get a lower bound for all quivers and vectors of framing. We provide an exact count in the case when the quiver is of finite type or is of affine type and the framing is the coordinate vector at the extending vertex. The latter case precisely covers Etingof’s conjecture on the number of finite dimensional irreducible representations for Symplectic reflection algebras associated to wreath-product groups. We use several different techniques, the two principal ones are categorical Kac–Moody actions and wall-crossing functors.

Finite-dimensional irreducible modules of the universal DAHA of type

Assume that is an algebraically closed field and let q denote a nonzero scalar in that is not a root of unity. The universal DAHA (double affine Hecke algebra) of type is an unital associative -algebra defined by generators and relations. The generators are and the relations assert that In this paper we describe the finite-dimensional irreducible -modules from many viewpoints and classify the finite-dimensional irreducible -modules up to isomorphism. The proofs are carried out in the language of linear algebra.

Finite-dimensional irreducible modules of the Bannai–Ito algebra at characteristic zero

Assume that $\mathbb F$ is an algebraically closed with characteristic $0$. The Bannai--Ito algebra $\mathfrak{BI}$ is a unital associative $\mathbb F$-algebra generated by $X,Y,Z$ and the relations assert that each of \begin{gather*} \{X,Y\}-Z, \qquad \{Y,Z\}-X, \qquad \{Z,X\}-Y \end{gather*} is central in $\mathfrak{BI}$. In this paper we classify the finite-dimensional irreducible $\mathfrak{BI}$-modules up to isomorphism. As we will see the elements $X,Y,Z$ are not always diagonalizable on finite-dimensional irreducible $\mathfrak{BI}$-modules.

Supergroup OSP(2,2n) and super Jacobi polynomials

The coefficients of the super Jacobi polynomials of BC1,n-type are rational functions in three parameters k,p,q. At the point (−1,0,0) these coefficients may have poles. Let us set q=0 and consider a pair (k,p) as the point of A2. If we apply blow up procedure at the point (−1,0) then we get a new family of polynomials. This family depends on one parameter t∈P. If t=∞ then we get Euler supercharacters for Lie supergroup OSP(2,2n). The supercharacters of finite dimensional irreducible modules can be also obtained by a specialization of the parameter t. But in such a case the specialization depends on the highest weight of the corresponding irreducible module.

Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero

Assume that ${\mathbb F}$ is an algebraically closed field with characteristic zero. The Racah algebra $\Re$ is the unital associative ${\mathbb F}$-algebra defined by generators and relations in the following way. The generators are $A$, $B$, $C$, $D$ and the relations assert that $[A,B]=[B,C]=[C,A]=2D$ and that each of $[A,D]+AC-BA$, $[B,D]+BA-CB$, $[C,D]+CB-AC$ is central in $\Re$. In this paper we discuss the finite-dimensional irreducible $\Re$-modules in detail and classify them up to isomorphism. To do this, we apply an infinite-dimensional $\Re$-module and its universal property. We additionally give the necessary and sufficient conditions for $A$, $B$, $C$ to be diagonalizable on finite-dimensional irreducible $\Re$-modules.

Super formal Darboux-Weinstein theorem and finite W-superalgebras

Let v=v0¯+v1¯ be a Z2-graded (super) vector space with an even C×-action and χ∈v0¯⁎ be a fixed point of the induced action. In this paper we prove an equivariant Darboux-Weinstein theorem for the formal polynomial algebras Aˆ=S[v0¯]∧χ⊗⋀(v1¯). We also give a quantum version of it. Let g=g0¯+g1¯ be a basic Lie superalgebra and e∈g0¯ be a nilpotent element. We use the equivariant quantum Darboux-Weinstein theorem to give a Poisson geometric realization of the finite W-superalgebra U(g,e) in the sense of Losev. An indirect relation between finite W-(super)algebras U(g,e) and U(g0¯,e) is presented. Finally we use such a realization to study the finite-dimensional irreducible modules over U(g,e).

On the Gaudin model of type G2

We derive a number of results related to the Gaudin model associated to the simple Lie algebra of type G2. We compute explicit formulas for solutions of the Bethe ansatz equations associated to the tensor product of an arbitrary finite-dimensional irreducible module and the vector representation. We use this result to show that the Bethe ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We show that the points of the spectrum of the Gaudin model in type G2 are in a bijective correspondence with self-self-dual spaces of polynomials. We study the set of all self-self-dual spaces — the self-self-dual Grassmannian. We establish a stratification of the self-self-dual Grassmannian with the strata labeled by unordered sets of dominant integral weights and unordered sets of nonnegative integers, satisfying certain explicit conditions. We describe closures of the strata in terms of representation theory.

On Fusion Procedure For The Two-Parameter Quantum Algebra In Type A

Finite dimensional irreducible modules of the two-parameter quantum enveloping algebra $U_{r,s}(\mathfrak{sl}_n)$ are explicitly constructed using the fusion procedure when $rs^{-1}$ is generic. This provides an alternative and combinatorial description of the Schur-Weyl duality for the two-parameter quantum linear algebras of type $A$.

The classification of finite-dimensional irreducible modules of the Racah algebra

Let denote an algebraically closed field of characteristic zero and let be some scalars in . By the Racah algebra associated with , we mean the most general quadratic algebra with two algebraically independent generators , which possesses presentations with ladder relations. In this paper, we classify the finite-dimensional irreducible -modules up to isomorphism by using the theory of the Leonard pairs. For a given irreducible -module V with dimension d + 1, we give its corresponding isomorphism classes of Leonard pairs on V that have Racah type.

Jet modules for the centerless Virasoro-like algebra

In this paper, we studied the jet modules for the centerless Virasoro-like algebra which is the Lie algebra of the Lie group of the area-preserving diffeomorphisms of a [Formula: see text]-torus. The jet modules are certain natural modules over the Lie algebra of semi-direct product of the centerless Virasoro-like algebra and the Laurent polynomial algebra in two variables. We reduce the irreducible jet modules to the finite-dimensional irreducible modules over some infinite-dimensional Lie algebra and then characterize the irreducible jet modules with irreducible finite dimensional modules over [Formula: see text]. To determine the indecomposable jet modules, we use the technique of polynomial modules in the sense of [Irreducible representations for toroidal Lie algebras, J. Algebras 221 (1999) 188–231; Weight modules over exp-polynomial Lie algebras, J. Pure Appl. Algebra 191 (2004) 23–42]. Consequently, indecomposable jet modules are described using modules over the algebra [Formula: see text], which is the “positive part” of a Block type algebra studied first by [Some infinite-dimensional simple Lie algebras in characteristic [Formula: see text] related to those of Block, J. Pure Appl. Algebra 127(2) (1998) 153–165] and recently by [A [Formula: see text]-graded generalization of the Witt-algebra, preprint; Classification of simple Lie algebras on a lattice, Proc. London Math. Soc. 106(3) (2013) 508–564]).

Equivalences between three presentations of orthogonal and symplectic Yangians

We prove the equivalence of two presentations of the Yangian $$Y(\mathfrak {g})$$ of a simple Lie algebra $$\mathfrak {g}$$ , and we also show the equivalence with a third presentation when $$\mathfrak {g}$$ is either an orthogonal or a symplectic Lie algebra. As an application, we obtain an explicit correspondence between two versions of the classification theorem of finite-dimensional irreducible modules for orthogonal and symplectic Yangians.

Gröbner–Shirshov pair of irreducible modules over quantized enveloping algebra of type An

In this paper, first, we give a Gröbner–Shirshov pair of finite-dimensional irreducible module [Formula: see text] over [Formula: see text] the quantized enveloping algebra of type [Formula: see text] by using the double free module method and the known Gröbner–Shirshov basis of [Formula: see text] Then, by specializing a suitable version of [Formula: see text] at [Formula: see text] we get a Gröbner–Shirshov basis of [Formula: see text] and get a Gröbner–Shirshov pair for the finite-dimensional irreducible module [Formula: see text] over [Formula: see text].