Classification Of Simple Modules Research Articles

The 3-cyclic quantum Weyl algebras, their prime spectra and a classification of simple modules ($q$ is not a root of unity)

The 3-cyclic quantum Weyl algebras, their prime spectra and a classification of simple modules ($q$ is not a root of unity)

On ultragraph Leavitt path algebras with finite Gelfand-Kirillov dimension

In this article, we prove that every ultragraph Leavitt path algebra is a direct limit of Leavitt path algebras of finite graphs and determine the Gelfand-Kirillov dimension of an ultragraph Leavitt path algebra. We also characterize ultragraph Leavitt path algebras whose simple modules are finitely presented, and show that these algebras have finite Gelfand-Kirillov dimension. Moreover, we construct new classes of simple modules over ultragraph Leavitt path algebras associated with minimal infinite emitters and minimal sinks, which have not appeared in the context of Leavitt path algebras of graphs.

The Grothendieck and Picard groups of finite rank torsion free mathfrak {sl}(2)-modules

The classification problem for simple {mathfrak {sl}(2)}-modules leads in a natural way to the study of the category of finite rank torsion free {mathfrak {sl}(2)}-modules and its subcategory of rational {mathfrak {sl}(2)}-modules. We prove that the rationalization functor induces an identification between the isomorphism classes of simple modules of these categories. This raises the question of what is the precise relationship between other invariants associated with them. We give a complete solution to this problem for the Grothendieck and Picard groups, obtaining along the way several new results regarding these categories that are interesting in their own right.

Classification of Simple Modules of the Ore Extension K[X][Y; ffrac{d}{dX}

For the algebras Lambda in the title of the paper, a classification of simple modules is given, an explicit description of the prime and completely prime spectra is obtained, the global and the Krull dimensions of Lambda are computed.

On matched pairs defining Hopf algebras in a certain class

In previous work by the author, a class of finite-dimensional semisimple Hopf algebras was considered with respect to the question under what condition all but one isomorphism class of simple modules are one-dimensional. The group theoretical answer given there asks for a classification of certain matched pairs used for the construction of these Hopf algebras. This classification is the content of this paper.

Monoidal categories of modules over quantum affine algebras of type A and B

We construct an exact tensor functor from the category $\mathcal{A}$ of finite-dimensional graded modules over the quiver Hecke algebra of type $A_\infty$ to the category $\mathscr C_{B^{(1)}_n}$ of finite-dimensional integrable modules over the quantum affine algebra of type $B^{(1)}_n$. It factors through the category $\mathcal T_{2n}$, which is a localization of $\mathcal{A}$. As a result, this functor induces a ring isomorphism from the Grothendieck ring of $\mathcal T_{2n}$ (ignoring the gradings) to the Grothendieck ring of a subcategory $\mathscr C^{0}_{B^{(1)}_n}$ of $\mathscr C_{B^{(1)}_n}$. Moreover, it induces a bijection between the classes of simple objects. Because the category $\mathcal T_{2n}$ is related to categories $\mathscr C^{0}_{A^{(t)}_{2n-1}}$ $(t=1,2)$ of the quantum affine algebras of type $A^{(t)}_{2n-1}$, we obtain an interesting connection between those categories of modules over quantum affine algebras of type $A$ and type $B$. Namely, for each $t =1,2$, there exists an isomorphism between the Grothendieck ring of $\mathscr C^{0}_{A^{(t)}_{2n-1}}$ and the Grothendieck ring of $\mathscr C^{0}_{B^{(1)}_n}$, which induces a bijection between the classes of simple modules.

On some radicals and proper classes associated to simple modules

For a unitary right module $M$, there are two known partitions of simple modules in the category $\sigma[M]$: the first one divides them into $M$-injective modules and $M$-small modules, while the second one divides them into $M$-projective modules and $M$-singular modules. We study inclusions between the first two and the last two classes of simple modules in terms of some associated radicals and proper classes.

Existence of Gradings on Associative Algebras

In this article, we study the existence of gradings on finite dimensional associative algebras. We prove that a connected algebra A does not have a nontrivial grading if and only if A is basic, its quiver has one vertex, and its group of outer automorphisms is unipotent. We apply this result to prove that up to graded Morita equivalence there do not exist nontrivial gradings on the blocks of group algebras with quaternion defect groups and one isomorphism class of simple modules.

On a Class of Finite-dimensional Semisimple Hopf Algebras

Let H be a finite-dimensional and semisimple Hopf algebra over an algebraically closed field of characteristic 0 such that H has exactly one isomorphism class of simple modules that have not dimension 1. These Hopf algebras were the object of study in, for instance, [1] and [9]. In this paper we study this property in the context of certain abelian extensions of group algebras and give a group theoretical criterion for such Hopf algebras to be of the above type. We also give a classification result in a special case thereof.

On simple modules over Leavitt path algebras

Given an arbitrary graph E and any field K, a new class of simple modules over the Leavitt path algebra LK(E) is constructed by using vertices that emit infinitely many edges in E. The corresponding annihilating primitive ideals are also described. Given a fixed simple LK(E)-module S, we compute the cardinality of the set of all simple LK(E)-modules isomorphic to S. Using a Boolean subring of idempotents induced by paths in E, bounds for the cardinality of the set of distinct isomorphism classes of simple LK(E)-modules are given. We also obtain a complete structure theorem about the Leavitt path algebra LK(E) of a finite graph E over which every simple module is finitely presented.

Classification of Simple Weight Modules Over the 1-Spatial Ageing Algebra

In this paper we use Block’s classification of simple modules over the first Weyl algebra to obtain a complete classification of simple weight modules, in particular, of Harish-Chandra modules, over the 1-spatial ageing algebra $\mathfrak {age(1)}$ . Most of these modules have infinite dimensional weight spaces and so far the algebra $\mathfrak {age(1)}$ is the only Lie algebra having simple weight modules with infinite dimensional weight spaces for which such a classification exists. As an application we classify all simple weight modules over the (1+1)-dimensional space-time Schrödinger algebra $\mathcal {S}$ that have a simple $\mathfrak {age(1)}$ -submodule thus constructing many new simple weight $\mathcal {S}$ -modules.

Non-restricted representations of simple Lie superalgebras of special type and Hamiltonian type

Let \(\mathbb{F}\) be an algebraically closed field of prime characteristic p > 2, and g be a simple Lie superalgebra of special type or Hamiltonian type over \(\mathbb{F}\). We construct the simple g-modules with non-singular characters of height more than one, and some simple modules with singular characters of height more than five. Furthermore, for the case of special type Lie superalgebras, we also construct a class of simple modules with regular semisimple characters of height one. All those simple modules mentioned above are proved to be reduced Kac modules.

The algebra of integro-differential operators on an affine line and its modules

For the algebra I1=K〈x,ddx,∫〉 of polynomial integro-differential operators over a field K of characteristic zero, a classification of simple modules is given. It is proved that I1 is a left and right coherent algebra. The Strong Compact-Fredholm Alternative is proved for I1. The endomorphism algebra of each simple I1-module is a finite dimensional skew field. In contrast to the first Weyl algebra, the centralizer of a nonscalar integro-differential operator can be a noncommutative, non-Noetherian, non-finitely generated algebra which is not a domain. It is proved that neither left nor right quotient ring of I1 exists but there exists the largest left quotient ring and the largest right quotient ring of I1, they are not I1-isomorphic but I1-anti-isomorphic. Moreover, the factor ring of the largest right quotient ring modulo its only proper ideal is isomorphic to the quotient ring of the first Weyl algebra. An analogue of the Theorem of Stafford (for the Weyl algebras) is proved for I1: each finitely generated one-sided ideal of I1 is 2-generated.

Schur algebras of Brauer algebras I

A classical problem of invariant theory and of Lie theory is to determine endomorphism rings of representations of classical groups, for instance of tensor powers of the natural module (Schur–Weyl duality) or of full direct sums of tensor products of exterior powers (Ringel duality). In this article, the endomorphism rings of full direct sums of tensor products of symmetric powers over symplectic and orthogonal groups are determined. These are shown to be isomorphic to Schur algebras of Brauer algebras as defined in Henke and Koenig (Math Z 272(3–4):729–759, 2012). This implies structural properties of the endomorphism rings, such as double centraliser properties, quasi-hereditary, and a universal property, as well as a classification of simple modules.

Quiver varieties and cluster algebras

Motivated by a recent conjecture by Hernandez and Leclerc, we embed a Fomin-Zelevinsky cluster algebra into the Grothendieck ring R of the category of representations of quantum loop algebras Uq(Lg) of a symmetric Kac-Moody Lie algebra, studied earlier by the author via perverse sheaves on graded quiver varieties. Graded quiver varieties controlling the image can be identified with varieties which Lusztig used to define the canonical base. The cluster monomials form a subset of the base given by the classes of simple modules in R, or Lusztig’s dual canonical base. The conjectures that cluster monomials are positive and linearly independent (and probably many other conjectures) of Fomin and Zelevinsky follow as consequences when there is a seed with a bipartite quiver. Simple modules corresponding to cluster monomials factorize into tensor products of “prime” simple ones according to the cluster expansion.

Extremal Projectors of q -Boson Algebras

We define the extremal projector of the q-boson Kashiwara algebra and study their basic properties. Applying their properties to the representation theory of the category , whose objects are ‘‘upper bounded’’ -modules, we obtain its semi-simplicity and the classification of simple modules.

Rogawski’s conjecture on the Jantzen filtration for the degenerate affine Hecke algebra of type A

The functors constructed by Arakawa and the author relate the representation theory of g l n \mathfrak {gl}_n and that of the degenerate affine Hecke algebra H ℓ H_\ell of G L ℓ \mathrm {GL}_\ell . They transform the Verma modules over g l n \mathfrak {gl}_n to the standard modules over H ℓ H_\ell . In this paper we prove that they transform the simple modules to the simple modules (in more general situations than in the previous paper). We also prove that they transform the Jantzen filtration on the Verma modules to that on the standard modules. We obtain the following results for the representations of H ℓ H_\ell by translating the corresponding results for g l n \mathfrak {gl}_n through the functors: (i) the (generalized) Bernstein-Gelfand-Gelfand resolution for a certain class of simple modules, (ii) the multiplicity formula for the composition series of the standard modules, and (iii) its refinement concerning the Jantzen filtration on the standard modules, which was conjectured by Rogawski.

On the Block Structure of Supersolvable Restricted Lie Algebras

Block theory is an important tool in the modular representation theory Ž w x. Ž w x. of finite groups cf. 18 . Apart from a few papers e.g. 17, 13, 16 dealing with restricted simple Lie algebras there apparently has been no effort to do the same for other classes of restricted Lie algebras despite a good Ž w x . knowledge of the simple modules cf. e.g. 21 for the solvable case . The aim of this paper is to develop the block theory for reduced universal enveloping algebras of a finite dimensional supersolvable restricted Lie algebra as far as possible in close analogy to modular group algebras. In order to organize the paper in a concise way, we include some open questions on tensor products of simple modules. Under certain conditions either on the Lie algebra or on the character they have affirmative answers which are decisively used in the proofs of our main results. But unfortunately this is not true in general as we show by an example. In the following, we are going to describe the contents of the paper in more detail. The second section provides the necessary background from the block theory of associative algebras which perhaps is not as well-known to Lie theorists. In particular, we stress that the block decomposition induces an Ž . Ž . equivalence relation ‘‘linkage relation’’ on the finite set of isomorphism classes of simple modules and indicate the proof of a cohomological characterization of the linkage relation which is fundamental for some of the following results. The latter can be used to give a combinatorial description of the linkage relation via the Gabriel quiver. We include some results on the number of blocks, resp. the principal block, that hold for any restricted Lie algebra.

Extreme modules over the Weyl algebraA n

Certain classes of simple modules over generalized Weyl algebras (extreme modules, polynomial modules, and modules with strongD-torsion) are classified. For these algebras, analogs of the Bernshtein and Stafford theorems are proved.

Stable torsion radicals over FBN rings

Gabriel [9] called a localizing subcategory stable if it is closed under injective envelopes. (In this case, the associated torsion radical or torsion theory is also said to be stable.) He showed that over a commutative Noetherian ring R every localizing subcategory of R-Mod is stable. He noted the connections between stability and the Artin-Rees property for ideals, and considered the stability of the localizing subcategory generated by the class of simple modules. Papp [18] studied Noetherian rings for which every localizing subcategory is stable, and Louden [16] utilized stable torsion radicals to define a sheaf over the spectrum of an FBN ring. This paper gives a characterization of stable torsion radicals over left FBN rings. The equivalent conditions stated in Theorem 1.2 exhibit the connections with the Artin-Rees property and with the notion of ideal invariance introduced by Robson [19]. Over an FBN ring, there is a natural correspondence between torsion radicals in R-Mod and those in Mod-R. In Theorem 1.6 it is shown that corresponding torsion radicals are both stable if and only if they are both ideal invariant, and that such torsion radicals correspond to the biradicals defined by Jategaonkar [12], and thus to the link-closed hereditary sets of prime ideals in the sense of Miiller [17]. Finally, it is shown that every torsion radical over an FBN ring is stable if and only if every prime ideal is localizable. In the second section of the paper, applications are given to the study of certain specific torsion radicals. For a left FBN ring, necessary and sufficient conditions are given under which any finitely generated essential extension of a module of Krull dimension at most a again has Krull dimension at most a. These conditions can be checked easily for an FBN ring or for a left Noetherian ring integral over its center, yielding a unified approach to certain results of Jategaonkar [13] and Chamarie and Hudry [5]. Finally, the results in the first section are applied to determine several conditions under which finitely generated essential extensions of Artinian modules are again Artinian, extending a result of Ginn and Moss [lo].