Finite-dimensional Modules Research Articles

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.

Grothendieck rings of Queer Lie superalgebras

We determine the Grothendieck rings of the category of finite-dimensional modules over Queer Lie superalgebras via their rings of characters. In particular, we show that the Q \mathbb {Q} -span of the ring of characters of the Queer Lie supergroup Q ( n ) Q(n) is isomorphic to the ring of Laurent polynomials in x 1 , … , x n x_{1},\ldots ,x_{n} for which the evaluation x n − 1 = − x n = t x_{n-1}=-x_{n}=t is independent of t t . We thus complete the description of Grothendieck rings for all classical Lie superalgebras.

Dimension vectors of elementary modules of generalized Kronecker quivers

Let k be an algebraically closed field. The generalized or n-Kronecker quiver K(n) is the quiver with two vertices, called a source and a sink, and n arrows from source to sink. Given a finite-dimensional module M of the path algebra , we consider its dimension vector . Let , and let . We construct a module X(x, y) of , and we prove it to be elementary. Suppose that . We show that: if M is an elementary module, then , and when , the module M is elementary if and only if M is of the form X(x, y).

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.

Galois G-covering of quotients of linear categories

In this paper, we introduce the notion of G -liftable ideals, which extends the liftable ideas defined by Assem and Le Meur. We characterize the G -liftable ideals and construct the Galois G -coverings of quotients of categories associated to the G -liftable ideals. In particular, we study the behavior of G -liftable admissible ideals under Galois G -coverings. Furthermore, we show that the ideals generated by finite dimensional projective modules over a locally bounded linear categories are admissible G -liftable ideals. As an application, we provide a reduction technique for dealing with the existence of Serre functors in the stable categories of Gorenstein projective objects.

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).

Ribbon tensor structure on the full representation categories of the singlet vertex algebras

We show that the category of finite-length generalized modules for the singlet vertex algebra M(p), p∈Z>1, is equal to the category OM(p) of C1-cofinite M(p)-modules, and that this category admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. Since OM(p) includes the uncountably many typical M(p)-modules, which are simple M(p)-module structures on Heisenberg Fock modules, our results substantially extend our previous work on tensor categories of atypical M(p)-modules. We also introduce a tensor subcategory OM(p)T, graded by an algebraic torus T, which has enough projectives and is conjecturally tensor equivalent to the category of finite-dimensional weight modules for the unrolled restricted quantum group of sl2 at a 2pth root of unity. We compute all tensor products involving simple and projective M(p)-modules, and we prove that both tensor categories OM(p) and OM(p)T are rigid and thus also ribbon. As an application, we use vertex operator algebra extension theory to show that the representation categories of all finite cyclic orbifolds of the triplet vertex algebras W(p) are non-semisimple modular tensor categories, and we confirm a conjecture of Adamović-Lin-Milas on the classification of simple modules for these finite cyclic orbifolds.

Simple modules over and at root of unity

In this article the smash product algebras as in the title due to V. V. Bavula and T. Lu have been studied at the root of unity. A full classification of simple modules over along with a class of finite dimensional indecomposable modules are presented. Also simple X, Y-torsionfree modules over are classified.

Decomposition of pointwise finite-dimensional 𝕊1 persistence modules

We prove that, over an arbitrary field, pointwise finite-dimensional persistence modules indexed by [Formula: see text] decompose uniquely, up to isomorphism, into the direct sum of a bar code and finitely-many Jordan cells. In the language of representation theory, this is a direct sum of string modules and band modules. Persistence modules indexed on [Formula: see text] have also been called angle-valued or circular persistence modules. We allow either a cyclic order or partial order on [Formula: see text] and do not have additional finiteness requirements on the modules. We also show that a pointwise finite-dimensional [Formula: see text] persistence module is indecomposable if and only if it is a bar or Jordan cell. Along the way we classify the isomorphism classes of such indecomposable modules.

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.

Subregular J-rings of Coxeter systems via quiver path algebras

We use quivers and their representations to bring new perspectives on the subregular J-ring JC of a Coxeter system (W,S), a subring of Lusztig's J-ring. We prove that JC is isomorphic to a suitable quotient of the path algebra of the double quiver of (W,S). Up to Morita equivalence, such quotients include the group algebras of all free products of finite cyclic groups. We then use quiver representations to study the category mod-AK of finite dimensional right modules of the algebra AK=K⊗ZJC over an algebraically closed field K of characteristic zero. Our results include classifications of Coxeter systems for which mod-AK is semisimple, has finitely many simple modules up to isomorphism, or has a bound on the dimensions of simple modules.

Taylor Spectrum for Modules over Lie Algebras

In this paper we generalize the notion of the Taylor spectrum to modules over an arbitrary Lie algebra and study it for finite-dimensional modules. We show that the spectrum can be described as the set of simple submodules in the case of nilpotent and semisimple Lie algebras. We also show that this result does not hold for solvable Lie algebras and obtain a precise description of the spectrum in the case of Borel subalgebras of semisimple Lie algebras.

The q-characters of minimal affinizations of type G 2 arising from cluster algebras

Hernandez and Leclerc proved that the Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional modules of any untwisted quantum affine algebra has a cluster algebra structure. This makes it possible to compute -characters of real prime simple modules associated to cluster variables as a result of a sequence of cluster mutations. In this paper, for minimal affinizations of type G 2, we describe a cluster algebra algorithm to compute their -characters and prove the geometric -character formula conjectured by Hernandez and Leclerc.

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.

Compact Schur-Weyl duality and the Type B/C VW-algebra

In this paper, we study a compact analogue of Schur-Weyl duality for real orthogonal and symplectic groups. We analyse the functors Fμ,k defined in [12], and introduce a new algebra Bkθ. This algebra's finite dimensional modules are naturally a co-domain for the functors Fμ,k. With this new insight, we prove that principle series modules map to principles series modules of graded Hecke algebras and these functors preserve Langlands quotients and unitarity of Harish-Chandra modules.

Whittaker category for the Lie algebra of polynomial vector fields

For any positive integer n, let An=C[t1,…,tn], Wn=Der(An) and Δn=Span{∂∂t1,…,∂∂tn}. Then (Wn,Δn) is a Whittaker pair. A Wn-module M is called a Whittaker module if the action of Δn on M is locally finite. We show that each block ΩaW˜ of the category of (An,Wn)-Whittaker modules with finite dimensional Whittaker vector spaces is equivalent to the category of finite dimensional modules over Ln, where Ln is the Lie subalgebra of Wn consisting of vector fields vanishing at the origin. As a corollary, we classify all simple non-singular Whittaker Wn-modules with finite dimensional Whittaker vector spaces using gln-modules. We also obtain an analogue of Skryabin's equivalence for the non-singular block ΩaW.

Modules with chain condition on uncountably generated submoduled

In this paper, we study modules with chain condition on uncountably generated submodules. We show that if an [Formula: see text]-module [Formula: see text] satisfies the ascending chain condition on uncountably generated submodules, then its Goldie dimension is less than or equal to [Formula: see text], where [Formula: see text] is the first uncountable cardinal number. We also show that if a quotient finite dimensional module [Formula: see text] satisfies the ascending chain condition on uncountably generated submodules, then it has Noetherian dimension and its Noetherian dimension is less than or equal to [Formula: see text], where [Formula: see text] is the first uncountable ordinal number. We also investigate that if a quotient finite dimensional module [Formula: see text] satisfies the descending chain condition on uncountably generated submodules, then [Formula: see text] has Krull dimension and its Krull dimension may be any ordinal number [Formula: see text].

Affine highest weight categories and quantum affine Schur-Weyl duality of Dynkin quiver types

For a Dynkin quiver Q Q (of type A D E \mathrm {ADE} ), we consider a central completion of the convolution algebra of the equivariant K K -group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclerc’s monoidal category C Q \mathcal {C}_{Q} of modules over the quantum loop algebra U q ( L g ) U_{q}(L\mathfrak {g}) via Nakajima’s homomorphism. As an application, we show that Kang-Kashiwara-Kim’s generalized quantum affine Schur-Weyl duality functor gives an equivalence between the category of finite-dimensional modules over the quiver Hecke algebra associated with Q Q and Hernandez-Leclerc’s category C Q \mathcal {C}_{Q} , assuming the simpleness of some poles of normalized R R -matrices for type E \mathrm {E} .