Showing papers in "Algebras and Representation Theory in 2015"

Uniqueness Theorems for Steinberg Algebras

Abstract: We prove Cuntz-Krieger and graded uniqueness theorems for Steinberg algebras. We also show that a Steinberg algebra is basically simple if and only if its associated groupoid is both effective and minimal. Finally we use results of Steinberg to characterise the center of Steinberg algebras associated to minimal groupoids.

Continuous Cluster Categories I

Abstract: In Igusa and Todorov(2013) we constructed topological triangulated categories \(\mathcal {C}_{c}\) as stable categories of certain topological Frobenius categories \(\mathcal {F}_{c}\). In this paper we show that these categories have a cluster structure for certain values of c including c = π. The continuous cluster categories are those \(\mathcal {C}_{c}\) which have cluster structure. We study the basic structure of these cluster categories and we show that \(\mathcal {C}_{c}\) is isomorphic to an orbit category \(\mathcal {D}_{r}/\underline F_{s}\) of the continuous derived category\(\mathcal {D}_{r}\) if c = rπ/s. In \(\mathcal {C}_{\pi }\), a cluster is equivalent to a discrete lamination of the hyperbolic plane. We give the representation theoretic interpretation of these clusters and laminations.

Representations of Leibniz Algebras

Abstract: In this paper we prove that every irredicuble representation of a Leibniz algebra can be obtained from irreducible representations of the semisimple Lie algebra from the Levi decomposition. We also prove that - in general - for (semi)simple Leibniz algebras it is not true that a representation can be decomposed to a direct sum of irreducible ones.

Noncommutative (Crepant) Desingularizations and the Global Spectrum of Commutative Rings

Abstract: In this paper we study endomorphism rings of finite global dimension over not necessarily normal commutative rings. These objects have recently attracted attention as noncommutative (crepant) resolutions, or NC(C)Rs, of singularities. We propose a notion of a NCCR over any commutative ring that appears weaker but generalizes all previous notions. Our results yield strong necessary and sufficient conditions for the existence of such objects in many cases of interest. We also give new examples of NCRs of curve singularities, regular local rings and normal crossing singularities. Moreover, we introduce and study the global spectrum of a ring R, that is, the set of all possible finite global dimensions of endomorphism rings of MCM R-modules. Finally, we use a variety of methods to compute global dimension for many endomorphism rings.

Determination of the Representations and a Basis for the Yokonuma–Temperley–Lieb Algebra

Abstract: We determine the representations of the Yokonuma–Temperley–Lieb algebra, which is defined as a quotient of the Yokonuma–Hecke algebra by generalising the construction of the classical Temperley–Lieb algebra. We then deduce the dimension of this algebra, and produce an explicit basis for it.

Minuscule Schubert Varieties: Poset Polytopes, PBW-Degenerated Demazure Modules, and Kogan Faces

Abstract: We study a family of posets and the associated chain and order polytopes. We identify the order polytope as a maximal Kogan face in a Gelfand-Tsetlin polytope of a multiple of a fundamental weight. We show that the character of such a Kogan face equals to the character of a Demazure module which occurs in the irreducible representation of $\mathfrak {sl}_{n+1}$ having highest weight multiple of fundamental weight and for any such Demazure module there exists a corresponding poset and associated maximal Kogan face. We prove that the chain polytope parametrizes a monomial basis of the associated PBW-graded Demazure module and further, that the Demazure module is a favourable module, e.g. interesting geometric properties are governed by combinatorics of convex polytopes. Thus, we obtain for any minuscule Schubert variety a flat degeneration into a toric projective variety which is projectively normal and arithmetically Cohen-Macaulay. We provide a necessary and sufficient condition on the Weyl group element such that the toric variety associated to the chain polytope and the toric variety associated to the order polytope are isomorphic.

Canonical Traces and Directly Finite Leavitt Path Algebras

Abstract: Motivated by the study of traces on graph C∗-algebras, we consider traces (additive, central maps) on Leavitt path algebras, the algebraic counterparts of graph C∗-algebras. In particular, we consider traces which vanish on nonzero graded components of a Leavitt path algebra and refer to them as canonical since they are uniquely determined by their values on the vertices. A desirable property of a \(\mathbb {C}\)-valued trace on a C∗-algebra is that the trace of an element of the positive cone is nonnegative. We adapt this property to traces on a Leavitt path algebra LK(E) with values in any involutive ring. We refer to traces with this property as positive. If a positive trace is injective on positive elements, we say that it is faithful. We characterize when a canonical, K-linear trace is positive and when it is faithful in terms of its values on the vertices. As a consequence, we obtain a bijective correspondence between the set of faithful, gauge invariant, \(\mathbb {C}\)-valued (algebra) traces on \(L_{\mathbb {C}}(E)\) of a countable graph E and the set of faithful, semifinite, lower semicontinuous, gauge invariant (operator theory) traces on the corresponding graph C∗-algebra C∗(E). With the direct finite condition (i.e xy=1 implies yx=1) for unital rings adapted to rings with local units, we characterize directly finite Leavitt path algebras as exactly those having the underlying graphs in which no cycle has an exit. Our proof involves consideration of “local” Cohn-Leavitt subalgebras of finite subgraphs. Lastly, we show that, while related, the class of locally noetherian, the class of directly finite, and the class of Leavitt path algebras which admit a faithful trace are different in general.

Representations of Hopf-Ore Extensions of Group Algebras and Pointed Hopf Algebras of Rank One

Abstract: In this paper, we study the representation theory of Hopf-Ore extensions of group algebras and pointed Hopf algebras of rank one over an arbitrary field k. Let H=kG(χ,a,δ) be a Hopf-Ore extension of kG and H′ a rank one quotient Hopf algebra of H, where k is a field, G is a group, a is a central element of G and χ is a k-valued character for G with χ(a)≠1. We first show that the simple weight modules over H and H′ are finite dimensional. Then we describe the structures of all simple weight modules over H and H′, and classify them. We also consider the decomposition of the tensor product of two simple weight modules over H′ into the direct sum of indecomposable modules. Furthermore, we describe the structures of finite dimensional indecomposable weight modules over H and H′, and classify them. Finally, when χ(a) is a primitive n-th root of unity for some n≥2, we determine all finite dimensional indecomposable projective objects in the category of weight modules over H′.

PBW Deformations of Skew Group Algebras in Positive Characteristic

Abstract: We investigate deformations of a skew group algebra that arise from a finite group acting on a polynomial ring. When the characteristic of the underlying field divides the order of the group, a new type of deformation emerges that does not occur in characteristic zero. This analogue of Lusztig’s graded affine Hecke algebra for positive characteristic can not be forged from the template of symplectic reflection and related algebras as originally crafted by Drinfeld. By contrast, we show that in characteristic zero, for arbitrary finite groups, a Lusztig-type deformation is always isomorphic to a Drinfeld-type deformation. We fit all these deformations into a general theory, connecting Poincare-Birkhoff-Witt deformations and Hochschild cohomology when working over fields of arbitrary characteristic. We make this connection by way of a double complex adapted from Guccione, Guccione, and Valqui, formed from the Koszul resolution of a polynomial ring and the bar resolution of a group algebra.

Automorphisms of the Doubles of Purely Non-Abelian Finite Groups

Abstract: Using a recent classification of End $(\mathcal {D}(G))$ , we determine a number of properties for Aut $(\mathcal {D}(G))$ , where $\mathcal {D}(G)$ is the Drinfel’d double of a finite group G. Furthermore, we completely describe Aut $(\mathcal {D}(G))$ for all purely non-abelian finite groups G. A description of the action of Aut $(\mathcal {D}(G))$ on Rep $(\mathcal {D}(G))$ is also given. We are also able to produce a simple proof that $\mathcal {D}(G)\cong \mathcal {D}(H)$ if and only if $\mathcal {G}\cong H$ , for G and H finite groups.

Noncommutative blowups of elliptic algebras.

Abstract: We develop a ring-theoretic approach for blowing up many noncommutative projective surfaces. Let T be an elliptic algebra (meaning that, for some central element ∈T 1, T/g T is a twisted homogeneous coordinate ring of an elliptic curve E at an infinite order automorphism). Given an effective divisor d on E whose degree is not too big, we construct a blowup T(d) of T at d and show that it is also an elliptic algebra. Consequently it has many good properties: for example, it is strongly noetherian, Auslander-Gorenstein, and has a balanced dualizing complex. We also show that the ideal structure of T(d) is quite rigid. Our results generalise those of the first author in Rogalski (Advances Math. 226, 1433–1473, 2011). In the companion paper Rogalski et al. (2013), we apply our results to classify orders in (a Veronese subalgebra of) a generic cubic or quadratic Sklyanin algebra.

The Singularity Categories of the Cluster-Tilted Algebras of Dynkin Type

Abstract: We use the stable categories of some selfinjective algebras to describe the singularity categories of the cluster-tilted algebras of Dynkin type. Furthermore, in this way, we settle the problem of singularity equivalence classification of the cluster-tilted algebra of type A, D and E respectively.

Classification of simple weight modules over the 1-spatial ageing algebra.

Abstract: 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 Schrodinger algebra \(\mathcal {S}\) that have a simple \(\mathfrak {age(1)}\)-submodule thus constructing many new simple weight \(\mathcal {S}\)-modules.

The ϕ-Dimension: A New Homological Measure

Abstract: In Igusa and Todorov (2005) introduced two functions ϕ and ψ, which are natural and important homological measures generalising the notion of the projective dimension. These Igusa-Todorov functions have become a powerful tool to understand better the finitistic dimension conjecture. In this paper, for an artin R-algebra A and the Igusa-Todorov function ϕ, we characterise the ϕ-dimension of A in terms of the bi-functors $\text{Ext}^{i}_{A}(-, -)$ and in terms of Tor’s bi-functors $\text{Tor}^{A}_{i}(-,-).$ Furthermore, by using the first characterisation of the ϕ-dimension, we show that the finiteness of the ϕ-dimension of an artin algebra is invariant under derived equivalences. As an application of this result, we generalise the classical Bongartz’s result (Bongartz, Lect. Notes Math. 903, 26–38, (1981), Corollary 1) as follows: For an artin algebra A, a tilting A-module T and the endomorphism algebra B = End A (T) o p , we have that ϕ dim (A) − pd T ≤ ϕ dim (B) ≤ ϕ dim (A) + pd T.

Cofiniteness with Respect to Ideals of Small Dimensions

Abstract: Let R be a Noetherian ring, I an ideal of R and M an R-module. It is shown that if \({\operatorname {Ext}^{i}_{R}}(R/I,M)\) is finitely generated, for all i≤ dimM, then \({\operatorname {Ext}^{i}_{R}}(N, M)\) is finitely generated for all i≥0 and all finitely generated R-modules N with Supp N⊆V(I) and dim N≤1. In addition, we show that if R is local, then \({\operatorname {Ext}^{i}_{R}}(N, M)\) is finitely generated for each i≥0 and for each finitely generated R-module N with Supp N⊆V(I) and dim N≤2. As a consequence we deduce that if dim R/I=2 and Supp M⊆V(I), then M is I-cofinite if (and only if) HomR(R/I,M), \({\operatorname {Ext}^{1}_{R}}(R/I,M)\) and \({\operatorname {Ext}^{2}_{R}}(R/I,M)\) are finitely generated. These generalize the main results of Melkersson [(J. Algebra. 372, 459–462, 2012) Theorem 2.3] and Bahmanpour et al. [(Proc. Amer. Math. Soc. 142, 1101–1107, 2014) Proposition 2.6].

New R-matrices for small quantum groups

Abstract: Quasitriangular Hopf algebras, i.e. Hopf algebras with an (universal) R-matrix, have braided categories of their modules. In particular, quasitriangular quantum groups yield ribbon categories, which have many interesting applications, especially in low- dimensional topology and topological quantum field theories. In this thesis we construct new universal R-matrices of the small quantum groups u_q(g) and extensions thereof. Here, g is always a finite-dimensional complex, simple Lie algebra and q is an l-th root of unity for an arbitrary integer l > 2. The first chapter of this thesis provides the necessary algebraic notions and gives the definition of the quantum groups in interest. Since in general the small quantum group is not quasitriangular, i.e. there exist no R-matrix, we consider certain extensions of the usual quantum group u_q(g), parametrized by a Lie theoretic input datum, namely a lattice Λ, containing the root lattice Λ_R and contained in the weight lattice Λ_W of g. This leads to the notion of u_q(g,Λ,Λ′) for the extension by Λ of the small quantum group uq(g,Λ′). Here, the lattice Λ′ ⊂ Λ_R determines a certain quotient in the construction of the quantum group. In the second chapter we review the ansatz R = R_0 Θ for R-matrices by Lusztig. This ansatz introduces a fixed element, the so-called quasi-R-matrix Θ, which is an intertwiner between the comultiplication ∆ and a new comultiplication ∆ ,obtained by conjugating with a certain antilinear involution. The free element R_0 is an intertwiner for ∆ and the opposed comultiplication ∆^{opp}. Eric Muller gives in his dissertation an ansatz and equations for the coefficients of R_0 in this ansatz and determines R-matrices of the form R = R_0 Θ for quadratic extensions of u_q(sl_n). In this thesis, we develop this ansatz further and get a new set of equations for the coefficients of R0, which we split in two different types. The first type of equations depends only on the fundamental group π_1 = Λ_W /Λ_R of the Lie algebra g and will be called group-equations. The second type depends on some sublattices of Λ and is very sensitive to different choices of l. This type of equations will be called diamond-equations. In Chapter 3 we determine the solutions to the group-equations. For the case of cyclic fundamental group π_1 = Z_N , N ∈ |N, (which is the case for all simple root systems but for D_{2m}, m ≥ 2), a theorem about certain idempotents of the group algebra C[Z_N ×Z_N ] is required. In order to prove this, we prove a main theorem about roots of unity first, which is of independent interest. The proof of this theorem is carried out along the lines of a new combinatorial principle, which to the best of our knowledge is introduced in this thesis. The solutions of the group-equations for the case of fundamental group π_1 = Z_2 × Z_2 are determined by a Maple-calculation which is documented in Appendix A. In Chapter 4 we determine all solutions of the diamond-equations from the set of the solutions of the group-equations. Again, we consider the case of cyclic fundamental group first and find that the existence of solutions then depends only on the funda- mental group π_1 of g and on the order l of the root of unity q. Depending on the data g, q, Λ (and fixed Λ′) we then determine in the main theorem all R-matrices obtained through Lusztig’s ansatz for different variants u_q(g,Λ,Λ′) of u_q(g).

Twisted Semigroup Algebras

Abstract: We study 2-cocycle twists, or equivalently Zhang twists, of semigroup algebras over a field ${\mathbb K}$ . If the underlying semigroup is affine, that is abelian, cancellative and finitely generated, then $\mathsf {Spec}~{\mathbb K}[S]$ is an affine toric variety over ${\mathbb K}$ , and we refer to the twists of ${\mathbb K}[S]$ as quantum affine toric varieties. We show that every quantum affine toric variety has a “dense quantum torus”, in the sense that it has a localization isomorphic to a quantum torus. We study quantum affine toric varieties and show that many geometric regularity properties of the original toric variety survive the deformation process.

Bounding Cohomology for Finite Groups and Frobenius Kernels

Abstract: Let G be a simple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p. Let σ :G → G be a strict endomorphism (i.e., the subgroup G(σ) of σ-fixed points is finite). Also, let Gσ be the scheme-theoretic kernel of σ, an infinitesimal subgroup of G. This paper shows that the dimension of the degree m cohomology group Hm(G(σ),L) for any irreducible kG(σ)-module L is bounded by a constant depending on the root system Φ of G and the integer m. These bounds are actually established for the degree m extension groups \( Ext^{m}_{G(\sigma )}(L,L^{\prime })\) between irreducible kG(σ)-modules \(L,L^{\prime }\), with a similar result holding for Gσ. In these Extm results, the bounds also depend on the highest weight associated to L, but are, nevertheless, independent of the characteristic p.

On the First Hochschild Cohomology Group of a Cluster-Tilted Algebra

Abstract: Given a cluster-tilted algebra B, we study its first Hochschild cohomology group HH1(B) with coefficients in the B-B-bimodule B. If C is a tilted algebra such that B is the relation-extension of C, then we show that if B is tame, then HH1(B) is isomorphic, as a k-vector space, to the direct sum of \({\text {HH}}^{1}(C)\) with \(k^{n_{B,C}}\), where nB,C is an invariant linking the bound quivers of B and C. In the representation-finite case, HH1(B) can be read off simply by looking at the quiver of B.

Finite Groups with K 4-Free Prime Graphs

Abstract: Let G be a finite group and let cd(G) be the set of all complex irreducible character degrees of G. Let ρ(G) be the set of all primes dividing some degrees in cd(G). The prime graph Δ(G) has vertex set ρ(G) and there is an edge between two distinct vertices p and q if pq divides some degree a∈cd(G). In this paper we show that if Δ(G) is K 4-free, then |ρ(G)|≤7; and moreover, if G is solvable, then |ρ(G)|≤6. These bounds are best possible.

Tubular Jacobian Algebras

Abstract: We show that the endomorphism ring of each cluster tilting object in a tubular cluster category is a finite dimensional Jacobian algebra which is tame of polynomial growth. Moreover, these Jacobian algebras are given by a quiver with a non-degenerate potential and mutation of cluster tilting objects is compatible with mutation of QPs.

Pivotal Tricategories and a Categorification of Inner-Product Modules

Abstract: This article investigates duals for bimodule categories over finite tensor categories. We show that finite bimodule categories form a tricategory and discuss the dualities in this tricategory using inner homs. We consider (bi)module categories over pivotal tensor categories with additional structure on the inner homs. These inner-product module categories are related to Frobenius algebras and lead to the notion of *-Morita equivalence for pivotal tensor categories. In particular they allow to transport the pivotal structure to the categories of endofunctors. We show that inner-product bimodule categories form a tricategory with two duality operations and an additional pivotal structure. This is work is motivated by defects in topological field theories.

On Filtered Multiplicative Bases of Some Associative Algebras

Abstract: We deal with the existing problem of filtered multiplicative bases of finite- dimensional associative algebras. For an associative algebra A over a field, we investigate when the property of having a filtered multiplicative basis is hereditated by homomorphic images or by the associated graded algebra of A. These results are then applied to some classes of group algebras and restricted enveloping algebras.

Fusion Product Structure of Demazure Modules

Abstract: Let 𝔤 be a finite–dimensional complex simple Lie algebra. Given a non–negative integer l, we define \(\mathcal {P}^{+}_{\ell }\) to be the set of dominant weights λ of 𝔤 such that lΛ0+λ is a dominant weight for the corresponding untwisted affine Kac–Moody algebra \(\widehat {{\mathfrak {g}}}\). For the current algebra 𝔤[t] associated to 𝔤, we show that the fusion product of an irreducible 𝔤–module V(λ) such that \(\lambda \in \mathcal {P}^{+}_{\ell }\) and a finite number of special family of 𝔤–stable Demazure modules of level l (considered in Fourier and Littelmann, Nagoya Math. J. 182, 171–198 (2006), Adv. Math. 211(2), 566–593 2007) again turns out to be a Demazure module. This fact is closely related with several important conjectures. We use this result to construct the 𝔤[t]–module structure of the irreducible \({\widehat {\mathfrak {g}}}\)–module V(l Λ0 + λ) as a semi–infinite fusion product of finite dimensional 𝔤[t]–modules as conjectured in Fourier and Littelmann, Adv. Math. 211(2), 566–593 (2007). As a second application we give further evidence to the conjecture on the generalization of Schur positivity (see Chari, Fourier and Sagaki 2013).

Endotrivial Modules for the Schur Covers of the Symmetric and Alternating Groups

Abstract: We investigate the endotrivial modules for the Schur covers of the symmetric and alternating groups and determine the structure of their group of endotrivial modules in all characteristics. We provide a full description of this group by generators and relations in all cases.

The Geometry of Representations of 3-Dimensional Sklyanin Algebras

Abstract: The representation scheme rep n A of the 3-dimensional Sklyanin algebra A associated to a plane elliptic curve and n-torsion point contains singularities over the augmentation ideal 𝔪. We investigate the semi-stable representations of the noncommutative blow-up algebra B = A ⊕ 𝔪t ⊕ 𝔪2 t 2 ⊕ … to obtain a partial resolution of the central singularity such that the remaining singularities in the exceptional fiber determine an elliptic curve and are all of type ℂ × ℂ2/ℤ n .

Crystal Isomorphisms in Fock Spaces and Schensted Correspondence in Affine Type A

Abstract: We are interested in the structure of the crystal graph of level l Fock spaces representations of \(\mathcal{U}_{q}^{\prime} (\widehat{\mathfrak{s}\mathfrak{l}_{e}}) \). Since the work of Shan (Ann. Sci. Ec Norm. Super. 44:147–182, 2011), we know that this graph encodes the branching rule for a corresponding cyclotomic rational Cherednik algebra. Besides, it appears to be closely related to the Harish-Chandra branching graph for the appropriate finite unitary group, according to [8]. In this paper, we make explicit a particular isomorphism between connected components of the crystal graphs of Fock spaces. This so-called “canonical” crystal isomorphism turns out to be expressible only in terms of: Schensted’s classic bumping procedure, the cyclage isomorphism defined in Jacon and Lecouvey (Algebras and Representation Theory 13:467–489, 2010), a new crystal isomorphism, easy to describe, acting on cylindric multipartitions.

Moduli Spaces of Modules of Schur-Tame Algebras

Abstract: In this paper, we first show that for an acyclic gentle algebra A, the irreducible components of any moduli space of A-modules are products of projective spaces. Next, we show that the nice geometry of the moduli spaces of modules of an algebra does not imply the tameness of the representation type of the algebra in question. Finally, we place these results in the general context of moduli spaces of modules of Schur-tame algebras. More specifically, we show that for an arbitrary Schur-tame algebra A and 𝜃-stable irreducible component C of a module variety of A-modules, the moduli space $\mathcal {M}(C)^{ss}_{\theta }$ is either a point or a rational projective curve.

Constructing Nearly Frobenius Algebras

Abstract: In the first part we study nearly Frobenius algebras. The concept of nearly Frobenius algebras is a generalization of the concept of Frobenius algebras. Nearly Frobenius algebras do not have traces, nor they are self-dual. We prove that the known constructions: direct sums, tensor, quotient of nearly Frobenius algebras admit natural nearly Frobenius structures. In the second part we study algebras associated to some families of quivers and the nearly Frobenius structures that they admit. As a main theorem, we prove that an indecomposable algebra associated to a bound quiver (Q, I) with no monomial relations admits a non trivial nearly Frobenius structure if and only if the quiver Q is linearly oriented of type \(\overrightarrow {\mathbb {A}_{n}}\) and I = 0. We also present an algorithm that determines the number of independent nearly Frobenius structures for gentle algebras without oriented cycles.

Erratum to: On the Homology of Completion and Torsion

Abstract: (1) There is an error in the proof of Theorem 7.12 (GM Duality) of [2]. The statement itself is correct, and it is repeated as Theorem 9 below, with a correct proof. (2) Just before formula (3.12) in [2], we said that “Moda-tor A is a thick abelian subcategory of ModA”. This is true if the ideal a is finitely generated, but might be false otherwise. There is no implication of this error on the rest of the paper, since WPR ideals are by definition finitely generated. We thank R. Vyas for mentioning this error to us.