McKay Quivers and Lusztig Algebras of Some Finite Groups

We are interested in the McKay quiver Γ(G) and skew group rings A ∗G, where G is a finite subgroup of GL(V ), where V is a finite dimensional vector space over a field K, and A is a K −G-algebra. These skew group rings appear in Auslander’s version of the McKay correspondence. In the first part of this paper we consider complex reflection groups $\mathsf {G} \subseteq \text {GL}(V)$ G ⊆ GL ( V ) and find a combinatorial method, making use of Young diagrams, to construct the McKay quivers for the groups G(r,p,n). We first look at the case G(1,1,n), which is isomorphic to the symmetric group Sn, followed by G(r,1,n) for r > 1. Then, using Clifford theory, we can determine the McKay quiver for any G(r,p,n) and thus for all finite irreducible complex reflection groups up to finitely many exceptions. In the second part of the paper we consider a more conceptual approach to McKay quivers of arbitrary finite groups: we define the Lusztig algebra $\widetilde {A}(\mathsf {G})$ A ~ ( G ) of a finite group $\mathsf {G} \subseteq \text {GL}(V)$ G ⊆ GL ( V ) , which is Morita equivalent to the skew group ring A ∗G. This description gives us an embedding of the basic algebra Morita equivalent to A ∗ G into a matrix algebra over A.

Torsion Pairs and Ringel Duality for Schur Algebras

Let A be a finite-dimensional algebra over an algebraically closed field. We use a functorial approach involving torsion pairs to construct embeddings of endomorphism algebras of basic projective A–modules P into those of the torsion submodules of P. As an application, we show that blocks of both the classical and quantum Schur algebras S(2,r) and Sq(2,r) in characteristic p > 0 are Morita equivalent as quasi-hereditary algebras to their Ringel duals if they contain 2pk simple modules for some k.

Strong Morita equivalence for inclusions of $C^*$-algebras induced by twisted actions of a countable discrete group

We consider two twisted actions of a countable discrete group on $\sigma$-unital $C^*$-algebras. Then by taking the reduced crossed products, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^*$-algebras. Also, we suppose that one of the inclusions of $C^*$-algebras is irreducible, that is, the relative commutant of one of the $\sigma$-unital $C^*$-algebra in the multiplier $C^*$-algebra of the reduced twisted crossed product is trivial. We show that the two actions are then strongly Morita equivalent up to some automorphism of the group.

Partial C*-dynamics and Rokhlin dimension

We develop the notion of the Rokhlin dimension for partial actions of finite groups, extending the well-established theory for global systems. The partial setting exhibits phenomena that cannot be expected for global actions, usually stemming from the fact that virtually all averaging arguments for finite group actions completely break down for partial systems. For example, fixed point algebras and crossed products are not in general Morita equivalent, and there is in general no local approximation of the crossed product $A\rtimes G$ by matrices over A. Using decomposition arguments for partial actions of finite groups, we show that a number of structural properties are preserved by formation of crossed products, including finite stable rank, finite nuclear dimension, and absorption of a strongly self-absorbing $C^*$ -algebra. Some of our results are new even in the global case. We also study the Rokhlin dimension of globalizable actions: while in general it differs from the Rokhlin dimension of its globalization, we show that they agree if the coefficient algebra is unital. For topological partial actions on spaces of finite covering dimension, we show that finiteness of the Rokhlin dimension is equivalent to freeness, thus providing a large class of examples to which our theory applies.

AS-REGULARITY OF GEOMETRIC ALGEBRAS OF PLANE CUBIC CURVES

In noncommutative algebraic geometry an Artin–Schelter regular (AS-regular) algebra is one of the main interests, and every three-dimensional quadratic AS-regular algebra is a geometric algebra, introduced by Mori, whose point scheme is either $\mathbb {P}^{2}$ or a cubic curve in $\mathbb {P}^{2}$ by Artin et al. [‘Some algebras associated to automorphisms of elliptic curves’, in: The Grothendieck Festschrift, Vol. 1, Progress in Mathematics, 86 (Birkhäuser, Basel, 1990), 33–85]. In the preceding paper by the authors Itaba and Matsuno [‘Defining relations of 3-dimensional quadratic AS-regular algebras’, Math. J. Okayama Univ. 63 (2021), 61–86], we determined all possible defining relations for these geometric algebras. However, we did not check their AS-regularity. In this paper, by using twisted superpotentials and twists of superpotentials in the Mori–Smith sense, we check the AS-regularity of geometric algebras whose point schemes are not elliptic curves. For geometric algebras whose point schemes are elliptic curves, we give a simple condition for three-dimensional quadratic AS-regular algebras. As an application, we show that every three-dimensional quadratic AS-regular algebra is graded Morita equivalent to a Calabi–Yau AS-regular algebra.

Morita Equivalence of Formal Poisson Structures

We extend the notion of Morita equivalence of Poisson manifolds to the setting of formal Poisson structures, that is, formal power series of bivector fields $\pi =\pi _0 + \lambda \pi _1 +\cdots $ satisfying the Poisson integrability condition $[\pi ,\pi ]=0$. Our main result gives a complete description of Morita equivalent formal Poisson structures deforming the zero structure ($\pi _0=0$) in terms of $B$-field transformations, relying on a general study of formal deformations of Poisson morphisms and dual pairs. Combined with previous work on Morita equivalence of star products [ 5], our results link the notions of Morita equivalence in Poisson geometry and noncommutative algebra via deformation quantization.

Morita equivalence of semirings with local units

In this paper we study some necessary and sufficient conditions for two semirings with local units to be Morita equivalent. Then we obtain some properties which remain invariant under Morita equivalence.

Gabor duality theory for Morita equivalent C∗-algebras

The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalence bimodules with some extra properties. For certain twisted group [Formula: see text]-algebras, the reformulation of the duality principle to the setting of Morita equivalence bimodules reduces to the well-known Gabor duality principle by localizing with respect to a trace. We may lift all results at the module level to matrix algebras and matrix modules, and in doing so, it is natural to introduce [Formula: see text]-matrix Gabor frames, which generalize multi-window super Gabor frames. We are also able to establish density theorems for module frames on equivalence bimodules, and these localize to density theorems for [Formula: see text]-matrix Gabor frames.

ON THE MORITA REDUCED VERSIONS OF SKEW GROUP ALGEBRAS OF PATH ALGEBRAS

Let $R$ be the skew group algebra of a finite group acting on the path algebra of a quiver. This article develops both theoretical and practical methods to do computations in the Morita-reduced algebra associated to $R$. Reiten and Riedtmann proved that there exists an idempotent $e$ of $R$ such that the algebra $eRe$ is both Morita equivalent to $R$ and isomorphic to the path algebra of some quiver, which was described by Demonet. This article gives explicit formulas for the decomposition of any element of $eRe$ as a linear combination of paths in the quiver described by Demonet. This is done by expressing appropriate compositions and pairings in a suitable monoidal category, which takes into account the representation theory of the finite group.