Finite-dimensional Modules Research Articles

Reduced submodules of finite dimensional polynomial modules

Reduced submodules of finite dimensional polynomial modules

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.

An Elliptic Generalization of A1 Spherical DAHA at K = 2

Abstract We construct an algebra that is an elliptic generalization of $A_1$ spherical DAHA acting on its finite-dimensional module at $t=-q^{-K/2}$ with $K=2$. We prove that $PSL(2,\mathbb{Z})$ acts by automorphisms of the algebra we constructed, and provide an explicit representation of automorphisms and algebra operators alike by $3 \times 3$ matrices with matrix elements given by products of elliptic functions. A relation of this construction to the $K$-theory character of affine Laumon space is conjectured. We point out two potential applications, respectively to $SL(3,\mathbb{Z})$ symmetry of Felder–Varchenko functions and to new elliptic invariants of torus knots and Seifert manifolds.

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.

Modules with finitely generated cohomology

Let G be a finite group and k a field of characteristic p. It is conjectured in a paper of the first author and John Greenlees that the thick subcategory of the stable module category StMod(kG) consisting of modules whose cohomology is finitely generated over H⁎(G,k) is generated by finite dimensional modules and modules with no cohomology. If the centraliser of every element of order p in G is p-nilpotent, this statement follows from previous work. Our purpose here is to prove this conjecture in two cases with non p-nilpotent centralisers. The groups involved are Z/3r×Σ3 (r⩾1) in characteristic three and Z/2×A4 in characteristic two.As a consequence, in these cases the bounded derived category of C⁎BG (cochains on BG with coefficients in k) is generated by C⁎BS, where S is a Sylow p-subgroup of G.

The periplectic q-Brauer category

We introduce the periplectic q-Brauer category over an integral domain of characteristic not 2. This is a strict monoidal supercategory and can be considered as a q-analogue of the periplectic Brauer category in [18]. We prove that the periplectic q-Brauer category admits a split triangular decomposition in the sense of [6]. When the ground ring is an algebraically closed field, the category of locally finite-dimensional right modules for the periplectic q-Brauer category is an upper finite fully stratified category in the sense of [6]. We prove that periplectic q-Brauer algebras defined in [1] are isomorphic to endomorphism algebras in the periplectic q-Brauer category. Furthermore, a periplectic q-Brauer algebra is a standardly based algebra in the sense of [13]. We construct a Jucys-Murphy basis for any standard module of the periplectic q-Brauer algebra with respect to a family of commutative elements called Jucys-Murphy elements. Via them, we classify blocks for both the periplectic q-Brauer category and periplectic q-Brauer algebras in the generic case. Our result shows that both the periplectic q-Brauer category and periplectic q-Brauer algebras are always not semisimple over any algebraically closed field.

On Borel subalgebras of quantum groups

For a quantum group, we study those right coideal subalgebras, for which all irreducible representations are one-dimensional. If a right coideal subalgebra is maximal with this property, then we call it a Borel subalgebra. Besides the positive part of the quantum group and its reflections, we find new unfamiliar Borel subalgebras, for example, ones containing copies of the quantum Weyl algebra. Given a Borel subalgebra, we study its induced (Verma-)modules and prove among others that they have all irreducible finite-dimensional modules as quotients. We give two structural conjectures involving the associated graded right coideal subalgebra, which we prove in certain cases. In particular, they predict the shape of all triangular Borel subalgebras. As examples, we determine all Borel subalgebras of [Formula: see text] and [Formula: see text] and discuss the induced modules.

Strong Duality Data of Type A and Extended T-Systems

AbstractThe extended T-systems are a number of relations in the Grothendieck ring of the category of finite-dimensional modules over the quantum affine algebras of types $$A_n^{(1)}$$ A n ( 1 ) and $$B_n^{(1)}$$ B n ( 1 ) , introduced by Mukhin and Young as a generalization of the T-systems. In this paper we establish the extended T-systems for more general modules, which are constructed from an arbitrary strong duality datum of type A. Our approach does not use the theory of q-characters, and so also provides a new proof to the original Mukhin–Young’s extended T-systems.

Hopf PBW-deformations of a new type quantum group Uq(𝔰𝔩2∗) and deformed preprojective algebras

We classify all the Hopf PBW-deformations of a new type quantum group [Formula: see text] from which the classical Drinfeld–Jimbo quantum group [Formula: see text] can arise as an almost unique nontrivial one. Different from the [Formula: see text] case, the category of finite-dimensional [Formula: see text]-modules is non-semisimple. We establish a block decomposition theorem for the category [Formula: see text] of finite-dimensional weight modules of [Formula: see text]. On the level of tensor category, we show that [Formula: see text] (respectively, the category [Formula: see text] of finite-dimensional [Formula: see text]-modules) can be realized via (respectively, deformed) preprojective algebras of Dynkin type [Formula: see text].

Dimension of uncountably generated submodules

In this article we introduce and study the concepts of uncountably generated Krull dimension and uncountably generated Noetherian dimension of an $R$-module, where $R$ is an arbitrary associative ring. These dimensions are ordinal numbers and extend the notion of Krull dimension and Noetherian dimension.
 They respectively rely on the behavior of descending and ascending chains of uncountably generated submodules.
 It is proved that a quotient finite dimensional module $M$ has uncountably generated Krull dimension if and only if it has Krull dimension, but
 the values of these dimensions might differ.
 Similarly, a quotient finite dimensional module $M$ has uncountably generated Noetherian dimension if and only if it has Noetherian dimension.
 We also show that the Noetherian dimension of a quotient finite dimensional module $M$ with uncountably generated Noetherian dimension $\beta$ is less than or equal to $\omega _{1}+\beta $, where $\omega_{1}$ is the first uncountable ordinal number.

Tensor ideals for quantum groups via minimal tilting complexes

We use minimal tilting complexes to construct an explicit bijection between the set of thick tensor ideals with the two-out-of-three property in the category of finite-dimensional modules over a quantum group at a root of unity and the set of thick tensor ideals in the subcategory of tilting modules. We also explain why the analogous construction for rational representations of a reductive algebraic group over a field of positive characteristic does not give rise to a bijection.

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.

Generalized Schur-Weyl dualities for quantum affine symmetric pairs and orientifold KLR algebras

Let g be a complex simple Lie algebra and UqLg the corresponding quantum affine algebra. We construct a functor Fθ between finite-dimensional modules over a quantum symmetric pair subalgebra of affine type Uqk⊂UqLg and an orientifold KLR algebra arising from a framed quiver with a contravariant involution, providing a boundary analogue of the Kang-Kashiwara-Kim-Oh generalized Schur-Weyl duality. With respect to their construction, our combinatorial model is further enriched with the poles of a trigonometric K-matrix intertwining the action of Uqk on finite-dimensional UqLg-modules. By construction, Fθ is naturally compatible with the Kang-Kashiwara-Kim-Oh functor in that, while the latter is a functor of monoidal categories, Fθ is a functor of module categories. Relying on a suitable isomorphism à la Brundan-Kleshchev-Rouquier, we prove that Fθ recovers the Schur-Weyl dualities due to Fan-Lai-Li-Luo-Wang-Watanabe in quasi-split type AIII.

On some Ringel-Hall polynomials associated to tame indecomposable modules

Let k be a finite field and Q an acyclic quiver of tame type (i.e. of type A˜m,D˜m,E˜6,E˜7,E˜8). Consider the path algebra kQ and the category of finite dimensional right modules mod-kQ. Using Schofield induction combined with some reduction tools, we determine all the Ringel-Hall polynomials associated to indecomposable modules of defect ranging from −2 to 2. In this way we obtain the full list of Ringel-Hall polynomials corresponding to indecomposables in the tame cases A˜m, D˜m and a partial list in the tame cases E˜6, E˜7, E˜8.

Free products of cyclic groups in groups of infinite unitriangular matrices

Groups of infinite unitriangular matrices over associative unitary rings are considered. These groups naturally act on infinite dimensional free modules over underlying rings. They are profinite in case underlying rings are finite. Inspired by their connection with groups defined by finite automata the problem to construct faithful representations of free products of groups by banded infinite unitriangular matrices is considered.For arbitrary prime p a sufficient conditions on a finite set of banded infinite unitriangular matrices over unitary associative rings of characteristic p under which they generate the free product of cyclic p-groups is given. The conditions are based on certain properties of the actions on finite dimensional free modules over underlying rings.It is shown that these conditions are satisfied. For arbitrary free product of finite number of cyclic p-groups constructive examples of the sets of infinite unitriangular matrices over unitar associative rings of characteristic p that generate given free product are presented. These infinite matrices are constructed from finite dimensional ones that are nilpotent Jordan blocks.A few open questions concerning properties of presented examples and other types of faithful representations are formulated.

Composition series of arbitrary cardinality in modular lattices and abelian categories

For a certain family of complete modular lattices, we prove a “Jordan–Hölder–Schreier-like” theorem with no assumptions on cardinality or well-orderedness. This family includes both lattices with are both join- and meet-continuous, as well as the lattices of subobjects of any object in an abelian category satisfying properties related to Grothendieck's axioms (AB5) and (AB5⁎). We then give several examples of objects in abelian categories which satisfy these axioms, including pointwise finite-dimensional persistence modules, presheaves, and certain Prüfer modules. Moreover, we show that, over an arbitrary ring, the infinite product of isomorphic simple modules both fails to satisfy our axioms and admits at least two composition series with distinct cardinalities. We conclude by giving a lattice-theoretic proof that any object which is locally finitely generated and satisfies our axioms can be expressed as a direct sum of indecomposable subobjects. We conjecture that this decomposition is unique.

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)).

Geometric sieve over number fields for higher moments

The geometric sieve for densities is a very convenient tool proposed by Poonen and Stoll (and independently by Ekedahl) to compute the density of a given subset of the integers. In this paper we provide an effective criterion to find all higher moments of the density (e.g. the mean, the variance) of a subset of a finite dimensional free module over the ring of algebraic integers of a number field. More precisely, we provide a geometric sieve that allows the computation of all higher moments corresponding to the density, over a general number field K. This work advances the understanding of geometric sieve for density computations in two ways: on one hand, it extends a result of Bright, Browning and Loughran, where they provide the geometric sieve for densities over number fields; on the other hand, it extends the recent result on a geometric sieve for expected values over the integers to both the ring of algebraic integers and to moments higher than the expected value. To show how effective and applicable our method is, we compute the density, mean and variance of Eisenstein polynomials and shifted Eisenstein polynomials over number fields. This extends (and fully covers) results in the literature that were obtained with ad-hoc methods.

On linked modules over the super-Yangian of the superalgebra Q(1)

Let Q(n) be the queer Lie superalgebra. We determine conditions under which two one-dimensional modules over the super-Yangian of Q(1) can be extended nontrivially, and thus belong to the same block of the subcategory of finite-dimensional YQ(1)-modules admitting generalized central character χ = 0. We use these results to determine conditions under which two one-dimensional modules over the finite W-algebra for Q(n) can be extended nontrivially. We describe blocks in the category of finite-dimensional modules over the finite W-algebra for Q(2). In certain cases, we determine conditions under which two simple finite-dimensional YQ(1)-modules admitting central character χ ≠ 0 can be extended nontrivially and propose a conjecture in the general case.

Relative Serre functor for comodule algebras

Let C be a finite tensor category, and let M be an exact left C-module category. The relative Serre functor of M is an endofunctor S on M together with a natural isomorphism Hom_(M,N)⁎≅Hom_(N,S(M)) for M,N∈M, where Hom_ is the internal Hom functor of M. In this paper, we discuss the case where C and M are the category of finite-dimensional left modules over a finite-dimensional Hopf algebra H and a finite-dimensional left H-comodule algebra L, respectively. We give an explicit description of the relative Serre functor of M and its twisted module structure in terms of the Frobenius structure of L. We also study pivotal structures on M and give some concrete examples.