Finite-dimensional Simple Modules Research Articles

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.

Extensions of modules for twisted current algebras

Twisted current algebras are fixed point subalgebras of current algebras under a finite group action. Special cases include equivariant map algebras and twisted forms of current algebras. Their finite-dimensional simple modules fall into two categories, those which factor through an evaluation map and those which do not. We show that there are no nontrivial extensions between finite-dimensional simple evaluation and non-evaluation modules. We then compute extensions between any pair of finite-dimensional simple modules for twisted current algebras, and use this information to study the block decomposition for the category. In the special case of twisted forms, this decomposition can be described in terms of maps to the fundamental group of the underlying root system.

Cyclicity and R-matrices

Let $S_1, \cdots, S_N$ simple finite-dimensional modules of a quantum affine algebra. We prove that if $S_i\otimes S_j$ is cyclic for any $i < j$ (i.e. generated by the tensor product of the highest weight vectors), then $S_1\otimes \cdots \otimes S_N$ is cyclic. The proof is based on the study of $R$-matrices.

Triple crystal action in Fock spaces

We make explicit a triple crystal structure on higher level Fock spaces, by investigating at the combinatorial level the actions of two affine quantum groups and of a Heisenberg algebra. To this end, we first determine a new indexation of the basis elements that makes the two quantum group crystals commute. Then, we define a so-called Heisenberg crystal, commuting with the other two. This gives new information about the representation theory of cyclotomic rational Cherednik algebras, relying on some recent results of Shan and Vasserot and of Losev. In particular, we give an explicit labelling of their finite-dimensional simple modules.

Finite dimensional simple modules of deformed current Lie algebras

The deformed current Lie algebra was introduced in [4] to study the representation theory of cyclotomic q-Schur algebras at q=1. In this paper, we classify finite dimensional simple modules of deformed current Lie algebras.

Formulae of ı-divided powers in [formula omitted

The existence of the ı-canonical basis (also known as the ı-divided powers) for the coideal subalgebra of the quantum sl2 were established by Bao and Wang, with conjectural explicit formulae. In this paper we prove the conjectured formulae of these ı-divided powers. This is achieved by first establishing closed formulae of the ı-divided powers in basis for quantum sl2 and then formulae for the ı-canonical basis in terms of Lusztig's divided powers in each finite-dimensional simple module of quantum sl2. These formulae exhibit integrality and positivity properties.

Simple modules of the quantum double of the Nichols algebra of unidentified diagonal type 𝔲𝔣𝔬(7)

ABSTRACTThe finite-dimensional simple modules over the Drinfeld double of the bosonization of the Nichols algebra 𝔲𝔣𝔬(7) are classified.

Non-Noetherian generalized Heisenberg algebras

In this note, we classify the non-Noetherian generalized Heisenberg algebras [Formula: see text] introduced in [R. Lü and K. Zhao, Finite-dimensional simple modules over generalized Heisenberg algebras, Linear Algebra Appl. 475 (2015) 276–291]. In case deg [Formula: see text] &gt; 1, we determine all locally finite and also all locally nilpotent derivations of [Formula: see text] and describe the automorphism group of these algebras.

Gradings on Modules Over Lie Algebras of E Types

For any grading by an abelian group $G$ on the exceptional simple Lie algebra $\mathcal{L}$ of type $E_6$ or $E_7$ over an algebraically closed field of characteristic zero, we compute the graded Brauer invariants of simple finite-dimensional modules, thus completing the computation of these invariants for simple finite-dimensional Lie algebras. This yields the classification of $G$-graded simple $\mathcal{L}$-modules, as well as necessary and sufficient conditions for an $\mathcal{L}$-module to admit a $G$-grading compatible with the given $G$-grading on $\mathcal{L}$.

Generalised Jantzen filtration of exceptional Lie superalgebras

Let g be an exceptional Lie superalgebra, and let p be the maximal parabolic subalgebra which contains the distinguished Borel subalgebra and has a purely even Levi subalgebra. For any parabolic Verma module in the parabolic category Op, it is shown that the Jantzen filtration is the unique Loewy filtration, and the decomposition numbers of the layers of the filtration are determined by the coefficients of inverse Kazhdan–Lusztig polynomials. An explicit description of the submodule lattices of the parabolic Verma modules is given, and formulae for characters and dimensions of the finite-dimensional simple modules are obtained.

PBW-type filtration on quantum groups of type An

We will introduce an N-filtration on the negative part of a quantum group of type An, such that the associated graded algebra is a q-commutative polynomial algebra. This filtration is given in terms of the representation theory of quivers, by realizing the quantum group as the Hall algebra of a quiver. We show that the induced associated graded module of any simple finite-dimensional module (of type 1) is isomorphic to a quotient of this polynomial algebra by a monomial ideal, and we provide a monomial basis for this associated graded module.This construction can be viewed as a quantum analog of the classical PBW framework, and in fact, by considering the classical limit, this basis is the monomial basis provided by Feigin, Littelmann and the second author in the classical setup.

RTT Realization of Quantum Affine Superalgebras and Tensor Products

We use the RTT realization of the quantum affine superalgebra associated with the Lie superalgebra |$\mathfrak {gl}(M,N)$| to study its finite-dimensional representations and their tensor products. In the case |$\mathfrak {gl}(1,1)$|⁠, the cyclicity condition of tensor products of finite-dimensional simple modules is determined completely in terms of zeros and poles of rational functions. This in turn induces cyclicity of some particular tensor products of Kirillov–Reshetikhin modules related to |$\mathfrak {gl}(M,N)$|⁠.

Hecke modules and supersingular representations of U(2,1)

Let F F be a nonarchimedean local field of odd residual characteristic p p . We classify finite-dimensional simple right modules for the pro- p p -Iwahori-Hecke algebra H C ( G , I ( 1 ) ) \mathcal {H}_C(G,I(1)) , where G G is the unramified unitary group U ( 2 , 1 ) ( E / F ) \textrm {U}(2,1)(E/F) in three variables. Using this description when C = F ¯ p C = \overline {\mathbb {F}}_p , we define supersingular Hecke modules and show that the functor of I ( 1 ) I(1) -invariants induces a bijection between irreducible nonsupersingular mod- p p representations of G G and nonsupersingular simple right H C ( G , I ( 1 ) ) \mathcal {H}_C(G,I(1)) -modules. We then use an argument of Paškūnas to construct supersingular representations of G G .

Finite-dimensional simple modules over generalized Heisenberg algebras

Generalized Heisenberg algebras H(f) for any polynomial f(h)∈C[h] have been used to explain various physical systems and many physical phenomena for the last 20 years. In this paper, we first obtain the center of H(f), and the necessary and sufficient conditions on f for two H(f) to be isomorphic. Then we determine all finite dimensional simple modules over H(f) for any polynomial f(h)∈C[h]. More precisely, there are three classes of them, AH(f)′(λ,z˙,a), BH(f)′(λ,z˙,a), and CH(f)(z˙,n). If f=wh+c for any c∈C and n-th (n>1) primitive root w of unity we actually obtain a complete classification of all irreducible modules over H(f).

Character and Dimension Formulae for Queer Lie Superalgebra

Closed formulae are constructed for the characters and dimensions of the finite dimensional simple modules of the queer Lie superalgebra q(n). This is achieved by refining Brundan's algorithm for computing simple q(n)-characters.

Simple [formula omitted]-module structures on [formula omitted

We study the category M consisting of U(sln+1)-modules whose restriction to U(h) is free of rank 1, in particular we classify isomorphism classes of objects in M and determine their submodule structure. This leads to new sln+1-modules. For n=1 we also find the central characters and derive an explicit formula for taking tensor product with a simple finite dimensional module.

Representations of twisted current algebras

We use evaluation representations to give a complete classification of the finite-dimensional simple modules of twisted current algebras. This generalizes and unifies recent work on multiloop algebras, current algebras, equivariant map algebras, and twisted forms.

MODULES FOR RELATIVE YANGIANS (FAMILY ALGEBRAS) AND KAZHDAN-LUSZTIG POLYNOMIALS

Let g be a complex simple Lie algebra and let mod U (g) be the category of finite dimensional U (g)-modules. The relative Yangian Y V (g) with respect to the pair g, V: V ∈ mod U (g) is defined to be the g invariant subalgebra of End V ⊗ U (g) with respect to the natural “diagonal” action. According to recent work (see [12, Sect. 5] and references therein), the finite dimensional simple modules of the Yangians for g = sl(n) or the twisted Yangians for g = sp(2n), so(n) are described by the simple modules of relative Yangians Y V (g) : V ∈ mod U (g). Here a classification of the simple modules of a relative Yangian is obtained simply and briefly as an advanced exercise in Frobenius reciprocity inspired by a Bernstein–Gelfand equivalence of categories [2]. An unexpected fact is that the dimension of these modules are determined by the Kazhdan-Lusztig polynomials, and conversely the latter are described in terms of dimensions of certain extension groups associated to finite dimensional modules of relative Yangians.

SIMPLE MODULES AND NON-COMMUTATIVE QUADRICS

In this paper we classify non-commutative quadrics and study their homological properties. In fact we find all non-commutative algebras of degree 2 up to isomorphism and we study these algebras via their homomorphic images onto the polynomial algebra k[x,y] as well as the Ext 1(k(p), k(q))-groups, where k(p) and k(q) are one-dimensional simple modules. Moreover some general results on simple finite-dimensional modules are obtained. Some of these results are applied to the special cases of non-commutative quadrics.

Maximal ideals and representations of twisted forms of algebras

Given a central simple algebra $\mathfrak{g}$ and a Galois extension of base rings $S/R$, we show that the maximal ideals of twisted $S/R$-forms of the algebra of currents $\mathfrak{g}(R)$ are in natural bijection with the maximal ideals of $R$. When $\mathfrak{g}$ is a Lie algebra, we use this to give a complete classification of the finite-dimensional simple modules over twisted forms of $\mathfrak{g}(R)$.