Finite-dimensional Simple Modules Research Articles

On representations of the super-Yangian of the queer Lie superalgebra

Let Q(n) be the queer Lie superalgebra. We determine conditions under which two 1-dimensional modules over the super-Yangian of Q(n) can be extended nontrivially. We describe the dual modules of the simple finite-dimensional modules over YQ(1). We use these results to describe blocks in the subcategory of finite-dimensional YQ(1)-modules admitting the zero generalized central character.

Whittaker modules over the N = 2 Neveu–Schwarz algebra

In this paper, we study Whittaker modules over the [Formula: see text] Neveu–Schwarz algebra [Formula: see text]. We first classify all simple finite-dimensional modules over the positive part of [Formula: see text]. Motivated by this classification, we define Whittaker modules and investigate their simplicity. Finally, we also study the simplicity of the quotient modules of the universal Whittaker module if it is not simple.

On modified extension graphs of a fixed atypicality

In this paper we study extensions between finite-dimensional simple modules over classical Lie superalgebras gl(m|n), osp(M|2n) and qm. We consider a simplified version of the extension graph which is produced from the Ext1-graph by identifying representations obtained by parity change and removal of the loops. We give a necessary condition for a pair of vertices to be connected and show that this condition is sufficient in most of the cases. This condition implies that the image of a finite-dimensional simple module under the Duflo-Serganova functor has indecomposable isotypical components. This yields semisimplicity of Duflo-Serganova functor for Fin(gl(m|n)) and for Fin(osp(M|2n)).

On finite-dimensional representations of finite W-superalgebras

Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}+\mathfrak{g}_{\bar{1}}$ be a basic Lie superalgebra, $\mathcal{W}_0$ (resp.$\mathcal{W}$) be the finite W-(resp.super-) algebras constructed from a fixed nilpotent element in $\mathfrak{g}_{\bar{0}}$. Based on a relation between finite W-algebra $\mathcal{W}_0$ and W-superalgebra $\mathcal{W}$ found recently by the author and Shu, we study the finite dimensional representations of finite W-superalgebras in this paper. We first formulate and prove a version of Premet's conjecture for the finite W-superalgebras from basic simple Lie superalgebras. As in the W-algebra case, the Premet's conjecture is very close to give a classification to the finite dimensional simple $\mathcal{W}$-modules. In the case of $\ggg$ is Lie superalgebras of basic type \Rmnum{1}, we prove the set of simple $\mathcal{W}$-supermodules is bijective with that of simple $\mathcal{W}_0$-modules; presenting a triangular decomposition to the tensor product of $\mathcal{W}$ with a Clifford algebra, we also give an algorithm to compute the character of the finite dimensional simple $\mathcal{W}$-supermodules with integral central character.

Construction of color Lie algebras from homomorphisms of modules of Lie algebras

A close relation between categories of color Lie algebras and sets of pairs of homomorphisms of modules of Lie algebras is discussed, which is applicable to construct color Lie algebras from Lie algebras and their modules. It is also shown that such construction yields equivalent categories of color Lie algebras under an equivalent relation between symmetric bicharacters. As an application we construct and classify all simple color Lie algebras from the three dimensional simple complex Lie algebra and its finite-dimensional simple modules.

On the Laistrygonian Nichols algebras that are domains

We consider a class of Nichols algebras $$\mathscr {B}(\mathfrak L_q( 1, \mathscr {G}))$$ introduced in Andruskiewitsch et al. which are domains and have many favourable properties like AS-regular and strongly noetherian. We classify their finite-dimensional simple modules and their point modules.

Generic Gelfand-Tsetlin modules of quantized and classical orthogonal algebras

We construct infinite-dimensional analogues of finite-dimensional simple modules of the nonstandard q-deformed enveloping algebra Uq′(son) defined by Gavrilik and Klimyk, and we do the same for the classical universal enveloping algebra U(son). In this paper we only consider the case when q is not a root of unity, and q→1 for the classical case. Extending work by Mazorchuk on son, we provide rational matrix coefficients for these infinite-dimensional modules of both Uq′(son) and U(son). We use these modules with rationalized formulas to embed the respective algebras into skew group algebras of shift operators. Casimir elements of Uq′(son) were given by Gavrilik and Iorgov, and we consider the commutative subalgebra Γ⊂Uq′(son) generated by these elements and the corresponding subalgebra Γ1⊂U(son). The images of Γ and Γ1 under their respective embeddings into skew group algebras are equal to invariant algebras under certain group actions. We use these facts to show that Γ is a Harish-Chandra subalgebra of Uq′(son) and Γ1 is a Harish-Chandra subalgebra of U(son).

Faithfulness of generalised Verma modules for Iwasawa algebras

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 A2. In this case we also show that all prime ideals of the corresponding Iwasawa algebras are annihilators of finite-dimensional simple modules.

Finite Dimensional Simple Modules over Some GIM Lie Algebras

GIM Lie algebras are the generalizations of Kac–Moody Lie algebras. However, the structures of GIM Lie algebras are more complex than the latter, so they have encountered many new difficulties to study their representation theory. In this paper, we classify all finite dimensional simple modules over the GIM Lie algebra Qn+1(2,1) as well as those over Θ2n+1.

Isomorphisms among quantum Grothendieck rings and propagation of positivity

Abstract Let ( 𝔤 , 𝗀 ) {\mathfrak{g},\mathsf{g})} be a pair of complex finite-dimensional simple Lie algebras whose Dynkin diagrams are related by (un)folding, with 𝗀 {\mathsf{g}} being of simply-laced type. We construct a collection of ring isomorphisms between the quantum Grothendieck rings of monoidal categories 𝒞 𝔤 {\mathscr{C}_{\mathfrak{g}}} and 𝒞 𝗀 {\mathscr{C}_{\mathsf{g}}} of finite-dimensional representations over the quantum loop algebras of 𝔤 {\mathfrak{g}} and 𝗀 {\mathsf{g}} , respectively. As a consequence, we solve long-standing problems: the positivity of the analogs of Kazhdan–Lusztig polynomials and the positivity of the structure constants of the quantum Grothendieck rings for any non-simply-laced 𝔤 {\mathfrak{g}} . In addition, comparing our isomorphisms with the categorical relations arising from the generalized quantum affine Schur–Weyl dualities, we prove the analog of Kazhdan–Lusztig conjecture (formulated in [D. Hernandez, Algebraic approach to q , t q,t -characters, Adv. Math. 187 2004, 1, 1–52]) for simple modules in remarkable monoidal subcategories of 𝒞 𝔤 {\mathscr{C}_{\mathfrak{g}}} for any non-simply-laced 𝔤 {\mathfrak{g}} , and for any simple finite-dimensional modules in 𝒞 𝔤 {\mathscr{C}_{\mathfrak{g}}} for 𝔤 {\mathfrak{g}} of type B n {\mathrm{B}_{n}} . In the course of the proof we obtain and combine several new ingredients. In particular, we establish a quantum analog of T-systems, and also we generalize the isomorphisms of [D. Hernandez and B. Leclerc, Quantum Grothendieck rings and derived Hall algebras, J. reine angew. Math. 701 2015, 77–126, D. Hernandez and H. Oya, Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan–Lusztig algorithm, Adv. Math. 347 2019, 192–272] to all 𝔤 {\mathfrak{g}} in a unified way, that is, isomorphisms between subalgebras of the quantum group of 𝗀 {\mathsf{g}} and subalgebras of the quantum Grothendieck ring of 𝒞 𝔤 {\mathscr{C}_{\mathfrak{g}}} .

BGG categories in prime characteristics

Let \({\mathfrak {g}}\) be a simple complex Lie algebra. In this paper we study the BGG category \({\mathcal {O}}_q\) for the quantum group \(U_q({\mathfrak {g}})\) with q being a root of unity in a field K of characteristic \(p >0\). We first consider the simple modules in \({\mathcal {O}}_q\) and prove a Steinberg tensor product theorem for them. This result reduces the problem of determining the corresponding irreducible characters to the same problem for a finite subset of finite dimensional simple modules. Then we investigate more closely the Verma modules in \({\mathcal {O}}_q\). Except for the special Verma module, which has highest weight \(-\rho \), they all have infinite length. Nevertheless, we show that each Verma module has a certain finite filtration with an associated strong linkage principle. The special Verma module turns out to be both simple and projective/injective. This leads to a family of projective modules in \({\mathcal {O}}_q\), which are also tilting modules. We prove a reciprocity law, which gives a precise relation between the corresponding family of characters for indecomposable tilting modules and the family of characters of simple modules with antidominant highest weights. All these results are of particular interest when \(q = 1\), and we have paid special attention to this case.

Graded torsion-free 𝔰𝔩2(ℂ)-modules of rank 2

In this paper, we explore the possibility of endowing simple infinite-dimensional [Formula: see text]-modules by the structure of graded modules. The gradings on the finite-dimensional simple modules over simple Lie algebras have been studied in [7,8].

Grothendieck Rings of Towers of Twisted Generalized Weyl Algebras

Twisted generalized Weyl algebras (TGWAs) A(R,σ,t) are defined over a base ring R by parameters σ and t, where σ is an n-tuple of automorphisms, and t is an n-tuple of elements in the center of R. We show that, for fixed R and σ, there is a natural algebra map \(A(R,\sigma ,tt^{\prime })\to A(R,\sigma ,t)\otimes _{R} A(R,\sigma ,t^{\prime })\). This gives a tensor product operation on modules, inducing a ring structure on the direct sum (over all t) of the Grothendieck groups of the categories of weight modules for A(R,σ,t). We give presentations of these Grothendieck rings for n = 1,2, when \(R=\mathbb {C}[z]\). As a consequence, for n = 1, any indecomposable module for a TGWA can be written as a tensor product of indecomposable modules over the usual Weyl algebra. In particular, any finite-dimensional simple module over \(\mathfrak {sl}_{2}\) is a tensor product of two Weyl algebra modules.

Structure and isomorphisms of quantum generalized Heisenberg algebras

In [S. A. Lopes and F. Razavinia, Quantum generalized Heisenberg algebras and their representations, preprint (2020), arXiv:2004.09301] we introduced a new class of algebras, which we named quantum generalized Heisenberg algebras and which depend on a parameter [Formula: see text] and two polynomials [Formula: see text]. We have shown that this class includes all generalized Heisenberg algebras (as defined in [E. M. F. Curado and M. A. Rego-Monteiro, Multi-parametric deformed Heisenberg algebras: A route to complexity, J. Phys. A: Math. Gen. 34(15) (2001) 3253; R. Lü and K. Zhao, Finite-dimensional simple modules over generalized Heisenberg algebras, Linear Algebra Appl. 475 (2015) 276–291, MR 3325233]) as well as generalized down-up algebras (as defined in [G. Benkart and T. Roby, Down-up algebras, J. Algebra 209(1) (1998) 305–344; T. Cassidy and B. Shelton, Basic properties of generalized down-up algebras, J. Algebra 279(1) (2004) 402–421, MR 2078408 (2005f:16051)]), but the parameters of freedom we allow for give rise to many algebras which are in neither one of these two classes. Having classified their finite-dimensional irreducible representations in [S. A. Lopes and F. Razavinia, Quantum generalized Heisenberg algebras and their representations, preprint (2020), arXiv:2004.09301], in this paper, we turn to their classification by isomorphism, the description of their automorphism groups and the study of their ring-theoretical properties.

A class of non-weight modules of 𝑈𝑝(𝖘𝖑2) and Clebsch–Gordan type formulas

Abstract In this paper, we construct a class of new modules for the quantum group U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) which are free of rank 1 when restricted to C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] . The irreducibility of these modules and submodule structure for reducible ones are determined. It is proved that any C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) -module of rank 1 is isomorphic to one of the modules we constructed, and their isomorphism classes are obtained. We also investigate the tensor products of the C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free modules with finite-dimensional simple modules over U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) , and for the generic cases, we obtain direct sum decomposition formulas for them, which are similar to the well-known Clebsch–Gordan formula for tensor products between finite-dimensional weight modules over U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) .

Simple Finite-Dimensional Modules and Monomial Bases from the Gelfand-Testlin Patterns

One of the most important classes of Lie algebras is sl_n, which are the n×n matrices with trace 0. The representation theory for sl_n has been an interesting research area for the past hundred years and in it, the simple finite-dimensional modules have become very important. They were classified and Gelfand and Tsetlin actually gave an explicit construction of a basis for every simple finite-dimensional module. This paper extends their work by providing theorems and proofs and constructs monomial bases of the simple module.

Finite dimensional modules over quantum toroidal algebras

For all generic q ∈ ℂ*, when \(\mathfrak{g}\) is not of type A1, we prove that the quantum toroidal algebra \(U_q(\mathfrak{g}_{\rm{tor}})\) has no nontrivial finite dimensional simple module.

Degenerate quantum general linear groups

Given any pair of positive integers m and n, we construct a new Hopf algebra, which may be regarded as a degenerate version of the quantum group of gl(m+n). We study its structure and develop a highest weight representation theory. The finite dimensional simple modules are classified in terms of highest weights, which are essentially characterised by m+n-2 nonnegative integers and two arbitrary nonzero scalars. In the special case with m=2 and n=1, an explicit basis is constructed for each finite dimensional simple module. For all m and n, the degenerate quantum group has a natural irreducible representation acting on C(q)^(m+n). It admits an R-matrix that satisfies the Yang-Baxter equation and intertwines the co-multiplication and its opposite. This in particular gives rise to isomorphisms between the two module structures of any tensor power of C(q)^(m+n) defined relative to the co-multiplication and its opposite respectively. A topological invariant of knots is constructed from this R-matrix, which reproduces the celebrated HOMFLY polynomial. Degenerate quantum groups of other classical types are briefly discussed.

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.

On the bosonization of the super Jordan plane

Let $H$ and $K$ be the bosonizations of the Jordan and super Jordan plane by the group algebra of a cyclic group; the algebra $K$ projects onto an algebra $L$ that can be thought of as the quantum Borel of $\mathfrak{sl}(2)$ at $-1$. The finite-dimensional simple modules over $H$ and $K$, are classified; they all have dimension $1$, respectively $\le 2$. The indecomposable $L$-modules of dimension $\leq 5$ are also listed. An interesting monoidal subcategory of $\operatorname{rep} L$ is described.