Finite-dimensional Quotients Research Articles

Cellularity for weighted KLRW algebras of types B$B$, A(2)$A^{(2)}$, D(2)$D^{(2)}$

This paper constructs homogeneous affine sandwich cellular bases of weighted KLRW algebras in types $B$, $A^{(2)}$, $D^{(2)}$. Our construction immediately gives homogeneous sandwich cellular bases for the finite dimensional quotients of these algebras. Since weighted KLRW algebras generalize KLR algebras, we also obtain bases and cellularity results for the (infinite and finite dimensional) KLR algebras.

Hopf actions of some quantum groups on path algebras

Our first collection of results parametrize (filtered) actions of a quantum Borel Uq(b)⊂Uq(sl2) on the path algebra of an arbitrary (finite) quiver. When q is a root of unity, we give necessary and sufficient conditions for these actions to factor through corresponding finite-dimensional quotients, generalized Taft algebras T(r,n) and small quantum groups uq(sl2).In the second part of the paper, we shift to the language of tensor categories. Here we consider a quiver path algebra equipped with an action of a Hopf algebra H to be a tensor algebra in the tensor category of representations H. Such a tensor algebra is generated by an algebra and bimodule in this tensor category. Our second collection of results describe the corresponding bimodule categories via an equivalence with categories of representations of certain explicitly described quivers with relations.

Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball

In this paper we study the relationships between a reproducing kernel Hilbert space, its multiplier algebra, and the geometry of the point set on which they live. We introduce a variant of the Banach-Mazur distance suited for measuring the distance between reproducing kernel Hilbert spaces, that quantifies how far two spaces are from being isometrically isomorphic as reproducing kernel Hilbert spaces. We introduce an analogous distance for multiplier algebras, that quantifies how far two algebras are from being completely isometrically isomorphic. We show that, in the setting of finite dimensional quotients of the Drury-Arveson space, two spaces are “close” to one another if and only if their multiplier algebras are “close”, and that this happens if and only if one of the underlying point sets is close to an image of the other under a biholomorphic automorphism of the unit ball. These equivalences are obtained as corollaries of quantitative estimates that we prove.

Representations of Finite-Dimensional Quotient Algebras of the 3-String Braid Group

We consider quotients of the group algebra of the $3$-string braid group $B_3$ by $p$-th order generic polynomial relations on the elementary braids. In cases $p=2,3,4,5$ these quotient algebras are finite dimensional. We give semisimplicity criteria for these algebras and present explicit formulas for all their irreducible representations.

Indecomposable tilting modules for the blob algebra

The blob algebra is a finite-dimensional quotient of the Hecke algebra of type B which is almost always quasi-hereditary. We construct the indecomposable tilting modules for the blob algebra over a field of characteristic 0 in the doubly critical case. Every indecomposable tilting module of maximal highest weight is either a projective module or an extension of a simple module by a projective module. Moreover, every indecomposable tilting module is a submodule of an indecomposable tilting module of maximal highest weight. We conclude that the graded Weyl filtration multiplicities of the indecomposable tilting modules in this case are given by inverse Kazhdan–Lusztig polynomials of type A˜1.

Representations of Leavitt path algebras

We study representations of a Leavitt path algebra L of a finitely separated digraph Γ over a field. We show that the category of L-modules is equivalent to a full subcategory of quiver representations. When Γ is a (non-separated) row-finite digraph we determine all possible finite dimensional quotients of L after giving a necessary and sufficient graph theoretic criterion for the existence of a nonzero finite dimensional quotient. This criterion is also equivalent to L having UGN (Unbounded Generating Number) as well as being algebraically amenable. We also realize the category of L-modules as a retract, hence a quotient by an explicit Serre subcategory of the category of quiver representations (that is, FΓ-modules) via a new colimit model for M⊗FΓL.

Poisson geometry of PI three‐dimensional Sklyanin algebras

We give the 3-dimensional Sklyanin algebras $S$ that are module-finite over their center $Z$ the structure of a Poisson $Z$-order (in the sense of Brown-Gordon). We show that the induced Poisson bracket on $Z$ is non-vanishing and is induced by an explicit potential. The ${\mathbb Z}_3 \times \Bbbk^\times$-orbits of symplectic cores of the Poisson structure are determined (where the group acts on $S$ by algebra automorphisms). In turn, this is used to analyze the finite-dimensional quotients of $S$ by central annihilators: there are 3 distinct isomorphism classes of such quotients in the case $(n,3) \neq 1$ and 2 in the case $(n,3)=1$, where $n$ is the order of the elliptic curve automorphism associated to $S$. The Azumaya locus of $S$ is determined, extending results of Walton for the case $(n,3)=1$.

Finite-dimensional Representations for a Class of Generalized Intersection Matrix Lie Algebras

In this paper, the authors study a class of generalized intersection matrix Lie algebras gim(Mn), and prove that its every finite-dimensional semisimple quotient is of type M(n, a, c, d). Particularly, any finite dimensional irreducible gim(Mn) module must be an irreducible module of Lie algebra of type M(n, a, c, d) and any finite dimensional irreducible module of Lie algebra of type M(n, a, c, d) must be an irreducible module of gim(Mn).

Finite-dimensional quotients of Hecke algebras

Let W be a complex reflection group. We prove that there is the maximal finite dimensional quotient of the Hecke algebra H_q(W) of W and that the dimension of this quotient coincides with |W|. This is a weak version of a Brou\'e-Malle-Rouquier conjecture from 1998. The proof is based on categories O for Rational Cherednik algebras.

Presenting Queer Schur Superalgebras

Associated to the two types of finite dimensional simple superalgebras, there are the general linear Lie superalgebra and the queer Lie superalgebra. The universal enveloping algebras of these Lie superalgebras act on the tensor spaces of the natural representations and, thus, define certain finite dimensional quotients, the Schur superalgebras and the queer Schur superalgebra. In this paper, we introduce the quantum analogue of the queer Schur superalgebra and investigate the presentation problem for both the queer Schur superalgebra and its quantum analogue.

Invariant connections and invariant holomorphic bundles on homogeneous manifolds

AbstractLet X be a differentiable manifold endowed with a transitive action α: A×X→X of a Lie group A. Let K be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects:equivalence classes of α-invariant K-connections on X α-invariant gauge classes of K-connections on X, andα-invariant isomorphism classes of pairs (Q,P) consisting of a holomorphic Kℂ-bundle Q → X and a K-reduction P of Q (when X has an α-invariant complex structure).

ON THE SMOOTHNESS OF CENTRES OF RATIONAL CHEREDNIK ALGEBRAS IN POSITIVE CHARACTERISTIC

AbstractIn this paper we study rational Cherednik algebras att= 1 in positive characteristic. We study a finite-dimensional quotient of the rational Cherednik algebra called the restricted rational Cherednik algebra. When the corresponding pseudo-reflection group belongs to the infinite seriesG(m, d, n), we describe explicitly the block decomposition of the restricted algebra. We also classify all pseudo-reflection groups for which the centre of the corresponding rational Cherednik algebra is regular for generic values of the deformation parameter.

A cubic defining algebra for the Links–Gould polynomial

We define a finite-dimensional cubic quotient of the group algebra of the braid group, endowed with a (essentially unique) Markov trace which affords the Links–Gould invariant of knots and links. We investigate several of its properties, and state several conjectures about its structure.

Morse theory, Higgs fields, and Yang–Mills–Higgs functionals

In this mostly expository paper we describe applications of Morse theory to moduli spaces of Higgs bundles. The moduli spaces are finite-dimensional analytic varieties but they arise as quotients of infinite-dimensional spaces. There are natural functions for Morse theory on both the infinite-dimensional spaces and the finite-dimensional quotients. The first comes from the Yang–Mills–Higgs energy, while the second is provided by the Hitchin function. After describing what Higgs bundles are, we explore these functions and how they may be used to extract topological information about the moduli spaces.

Contraction groups, ergodicity, and distal properties of automorphisms of compact groups

Given an automorphism $\tau$ of a compact group $G$, we study the factorization of $C(\tau,K)$, the contraction group of $\tau$ modulo a closed $\tau$-invariant subgroup $K$, into the product $C(\tau)K$, of the contraction group $C(\tau)$ of $\tau$, and $K$. We prove that the factorization $C(\tau,K)=C(\tau)K$ holds for every closed $\tau$-invariant subgroup $K$ if and only if $G$ contains arbitrarily small closed normal $\tau$-invariant subgroups $N$ with finite-dimensional quotients $G/N$. For metrizable groups, we obtain that $C(\tau)K$ is a dense subgroup of $C(\tau,K)$, for every closed $\tau$-invariant subgroup $K$. These results are used to link the contraction group to the properties of the dynamical system $(G,\tau)$. It follows that $\tau$ is distal if and only if $C(\tau)$ is trivial, while ergodicity of $\tau$ implies that $C(\tau)$ is nontrivial. When $G$ is metrizable, the closure of $C(\tau)$ is the largest closed $\tau$-invariant subgroup on which $\tau$ acts ergodically and, at the same time, it is the smallest among closed normal $\tau$-invariant subgroups $N$ such that $\tau$ acts distally on $G/N$. If $\tau$ is ergodic, then its restriction to any closed connected normal $\tau$-invariant subgroup $N$ with finite-dimensional quotient $G/N$ is also ergodic. Moreover, when $G$ is connected, the largest closed $\tau$-invariant subgroup on which $\tau$ acts ergodically is necessarily connected.

Codimension and pseudometric in co-Heyting algebras

In this paper, we introduce a notion of dimension and codimension for every element of a bounded distributive lattice L. These notions prove to have a good behavior when L is a co-Heyting algebra. In this case the codimension gives rise to a pseudometric on L which satisfies the ultrametric triangle inequality. We prove that the Hausdorff completion of L with respect to this pseudometric is precisely the projective limit of all its finite dimensional quotients. This completion has some familiar metric properties, such as the convergence of every monotonic sequence in a compact subset. It coincides with the profinite completion of L if and only if it is compact or equivalently if every finite dimensional quotient of L is finite. In this case we say that L is precompact. If L is precompact and Hausdorff, it inherits many of the remarkable properties of its completion, specially those regarding the join/meet irreducible elements. Since every finitely presented co-Heyting algebra is precompact Hausdorff, all the results we prove on the algebraic structure of the latter apply in particular to the former. As an application, we obtain the existence for every positive integers n, d of a term t n, d such that in every co-Heyting algebra generated by an n-tuple a, t n, d(a) is precisely the maximal element of codimension d.

Quantum subgroups of GLα,β(n)

Let α , β ∈ C ∖ { 0 } and ℓ ∈ N be odd with ℓ ⩾ 3 . We determine all Hopf algebra quotients of the quantized coordinate algebra O α , β ( GL n ) when α − 1 β is a primitive ℓ -th root of unity and α , β satisfy certain mild conditions, and we characterize all finite-dimensional quotients when α − 1 β is not a root of unity. As a byproduct we find a new family of non-semisimple and non-pointed Hopf algebras with non-pointed duals which are quotients of O α , β ( GL n ) .

Cuspidal representations of rational Cherednik algebras at t = 0

We study those finite dimensional quotients of the rational Cherednik algebra at t = 0 that are supported at a point of the centre. It is shown that each such quotient is Morita equivalent to a certain “cuspidal” quotient of a rational Cherednik algebra associated to a parabolic subgroup of W.

Irreducible finite-dimensional modules of prime characteristic down–up algebras

A down–up algebra A= A( α, β, γ), as defined in a 1998 paper by Benkart and Roby [J. Algebra 209 (1998) 305–344; 213 (1999) 378 (Addendum)], is a unital associative algebra over a field K with two generators d and u and defining relations d 2u=αdud+βud 2+γd, du 2=αudu+βu 2d+γu, where α, β, γ are fixed scalars in K . This paper investigates the modules of down–up algebras over fields of characteristic p>0. We start with the Verma modules and consider their weight spaces relative to h= K du⊕ K ud . We calculate exactly when a Verma module will break up into a finite number of infinite-dimensional weight spaces and when it splits into an infinite number of one-dimensional spaces. Using that result we then find all the finite-dimensional irreducible quotients of the Verma modules. Under the additional assumption that K is algebraically closed we determine all finite-dimensional irreducible modules for A, describing the actions of A on those modules and computing their dimension.

On idempotents in reduced enveloping algebras

Explicit constructions are given for idempotents that generate all projective indecomposable modules for certain finite-dimensional quotients of the universal enveloping algebra of the Lie algebra s ℓ ( 2 ) s\ell (2) in odd prime characteristic. The program is put in a general context, although constructions are only carried through in the case of s ℓ ( 2 ) s\ell (2) .