Modules For Algebras Research Articles

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.

Change of base for operator space modules

We prove a change of base theorem for operator space modules over C*-algebras, analogous to the change of rings for algebraic modules. We demonstrate how this can be used to show that the category of (right) matrix normed modules and completely bounded module maps has enough injectives.

Trace decategorification of tensor product algebras

We show that in ADE type the trace of Webster's categorification of a tensor product of irreducibles for the quantum group is isomorphic to a tensor product of Weyl modules for the current algebra U ˙ ( g [ t ] ) . This extends a result of Beliakova, Habiro, Lauda, and Webster who showed that the trace of the categorified quantum group U ˙ ⁎ ( g ) is isomorphic to U ˙ ( g [ t ] ) , and the trace of a cyclotomic quotient of U ˙ ⁎ ( g ) , which categorifies a single irreducible for the quantum group, is isomorphic to a Weyl module for U ˙ ( g [ t ] ) . We use a deformation argument based on Webster's technique of unfurling 2-representations.

A Constructive Approach to Freyd Categories

We discuss Peter Freyd’s universal way of equipping an additive category mathbf {P} with cokernels from a constructive point of view. The so-called Freyd category mathcal {A}(mathbf {P}) is abelian if and only if mathbf {P} has weak kernels. Moreover, mathcal {A}(mathbf {P}) has decidable equality for morphisms if and only if we have an algorithm for solving linear systems X cdot alpha = beta for morphisms alpha and beta in mathbf {P}. We give an example of an additive category with weak kernels and decidable equality for morphisms in which the question whether such a linear system admits a solution is computationally undecidable. Furthermore, we discuss an additional computational structure for mathbf {P} that helps solving linear systems in mathbf {P} and even in the iterated Freyd category construction mathcal {A}( mathcal {A}(mathbf {P})^{mathrm {op}} ), which can be identified with the category of finitely presented covariant functors on mathcal {A}(mathbf {P}). The upshot of this paper is a constructive approach to finitely presented functors that subsumes and enhances the standard approach to finitely presented modules in computer algebra.

Twisted Modules and G-equivariantization in Logarithmic Conformal Field Theory

A two-dimensional chiral conformal field theory can be viewed mathematically as the representation theory of its chiral algebra, a vertex operator algebra. Vertex operator algebras are especially well suited for studying logarithmic conformal field theory (in which correlation functions have logarithmic singularities arising from non-semisimple modules for the chiral algebra) because of the logarithmic tensor category theory of Huang, Lepowsky, and Zhang. In this paper, we study not-necessarily-semisimple or rigid braided tensor categories $\mathcal{C}$ of modules for the fixed-point vertex operator subalgebra $V^G$ of a vertex operator (super)algebra $V$ with finite automorphism group $G$. The main results are that every $V^G$-module in $\mathcal{C}$ with a unital and associative $V$-action is a direct sum of $g$-twisted $V$-modules for possibly several $g\in G$, that the category of all such twisted $V$-modules is a braided $G$-crossed (super)category, and that the $G$-equivariantization of this braided $G$-crossed (super)category is braided tensor equivalent to the original category $\mathcal{C}$ of $V^G$-modules. This generalizes results of Kirillov and M\"{u}ger proved using rigidity and semisimplicity. We also apply the main results to the orbifold rationality problem: whether $V^G$ is strongly rational if $V$ is strongly rational. We show that $V^G$ is indeed strongly rational if $V$ is strongly rational, $G$ is any finite automorphism group, and $V^G$ is $C_2$-cofinite.

The last formula of Jean-Louis Koszul

Since the first international conference in France on the Information Geometry, GSI2013, Jean-Louis Koszul’s interest in Information Geometry went increasing. Motivated by the impact of the cohomology of Koszul-Vinberg algebras on the Information Geometry and moved by some issues raised by Albert Nijenhuis, Jean-Louis Koszul undertook another rewriting of the Brut formula of the coboundary operator of the KV complex. The source of other motivations of Jean-Louis Koszul was the relationships between the theory of KV cohomology and the theory of deformation of locally flat manifolds.In 2015 Jean-Louis Koszul sent me his Last formula of the KV boundary operator. In that Last formula Jean-Louis Koszul dealt with the case where spaces of coefficients are trivial modules of KV algebras. A part of the present work is devoted to extending the Last formula of Jean-Louis Koszul to KV cochain complexes whose spaces of coefficients are non-trivial two-sided modules of KV algebras. At another side, I also aim to highlight other significant impacts of the theory of KV cohomology of Koszul-Vinberg algebras. In particular I will use the KV cohomology to widely revisit the theory of statistical models of measurable sets. The reader will see why the source of the theory of statistical models is of homological nature.I also intend to highlight several impacts of the KV cohomology on the quantitative differential topology. I am particularly concerned with problems regarding the existence of Riemannian foliations, the existence of symplectic foliations as well as the existence of multi-dimensional webs. The homological theory of statistical models is presented as branches of rooted trees whose roots are weakly Jensen random cohomology classes.

Robust sensorless low‐speed trajectory tracking for a permanent magnet synchronous motor: An extended state observer based backstepping control approach

This article deals with the low‐speed sensorless trajectory tracking control of a permanent magnet synchronous motor (PMSM). The rotor position and angular speed are obtained through back electromotive forces (back‐EMF), using extended state observers (ESOs) in the alpha‐beta coordinates. Additionally, the estimation of the back‐EMF is used by an algebraic module to reconstruct online the position and speed using an off‐line estimation of the back‐EMF parameter . The control law is derived using a robust recursive controller design methodology, namely; the backstepping design approach in the d‐q coordinates. Estimation schemes allow the adaptation of the angular position, angular speed, and the load torque parameters in the control law. With this adaptation, the controller achieves the necessary robustness to reduce the effects of endogenous and exogenous perturbations present in the PMSM system. The trajectory tracking task is achieved at low angular speed, with the presence of a load torque applied to the motor shaft. Experimental results at low‐speed and rated load/no‐load conditions are presented to demonstrate the effectiveness and robustness of the proposed scheme.

Logarithmic modules for chiral differential operators of nilmanifolds

We describe explicitly the vertex algebra of (twisted) chiral differential operators on certain nilmanifolds and construct their logarithmic modules. This is achieved by generalizing the construction of vertex operators in terms of exponentiated scalar fields to Jacobi theta functions naturally appearing in these nilmanifolds. This construction furnishes non-trivial examples of logarithmic vertex algebra modules, a notion recently introduced by Bakalov.

A combinatorial classification of 2-regular simple modules for Nakayama algebras

Enomoto showed for finite dimensional algebras that the classification of exact structures on the category of finitely generated projective modules can be reduced to the classification of 2-regular simple modules. In this article, we give a combinatorial classification of 2-regular simple modules for Nakayama algebras and we use this classification to answer several natural questions such as when there is a unique exact structure on the category of finitely generated projective modules for Nakayama algebras. We also classify 1-regular simple modules, quasi-hereditary Nakayama algebras and Nakayama algebras of global dimension at most two. It turns out that most classes are enumerated by well-known combinatorial sequences, such as Fibonacci, Riordan and Narayana numbers. We first obtain interpretations in terms of the Auslander-Reiten quiver of the algebra using homological algebra, and then apply suitable bijections to relate these to combinatorial statistics on Dyck paths.

Level one Weyl modules for toroidal Lie algebras

We identify level one global Weyl modules for toroidal Lie algebras with certain twists of modules constructed by Moody–Eswara Rao–Yokonuma via vertex operators for type ADE and by Iohara–Saito–Wakimoto and Eswara Rao for general type. The twist is given by an action of \(\mathrm {SL}_{2}(\mathbb {Z})\) on the toroidal Lie algebra. As a by-product, we obtain a formula for the character of the level one local Weyl module over the toroidal Lie algebra and that for the graded character of the level one graded local Weyl module over an affine analog of the current Lie algebra.

Vertex algebraic intertwining operators among generalized Verma modules for affine Lie algebras

We find sufficient conditions for the construction of vertex algebraic intertwining operators, among generalized Verma modules for an affine Lie algebra gˆ, from g-module homomorphisms. When g=sl2, these results extend previous joint work with J. Yang, but the method used here is different. Here, we construct intertwining operators by solving Knizhnik-Zamolodchikov equations for three-point correlation functions associated to gˆ, and we identify obstructions to the construction arising from the possible non-existence of series solutions having a prescribed form.

Finite irreducible conformal modules of rank two Lie conformal algebras

In the present paper, we prove that any finite nontrivial irreducible module over a rank two Lie conformal algebra [Formula: see text] is of rank one. We also describe the actions of [Formula: see text] on its finite irreducible modules explicitly. Moreover, we show that all finite nontrivial irreducible modules of finite Lie conformal algebras whose semisimple quotient is the Virasoro Lie conformal algebra are of rank one.

Vertex representations of quantum N-toroidal algebras for type C

Quantum [Formula: see text]-toroidal algebras are generalizations of quantum affine algebras and quantum toroidal algebras. In this paper, we construct a level-one vertex representation of the quantum [Formula: see text]-toroidal algebra for type [Formula: see text]. In particular, we also obtain a level-one module of the quantum toroidal algebra for type [Formula: see text] as a special case.

AGT basis in SCFT for c = 3/2 and Uglov polynomials

AGT allows one to compute conformal blocks of d = 2 CFT for a large class of chiral CFT algebras. This is related to the existence of a certain orthogonal basis in the module of the (extended) chiral algebra. The elements of the basis are eigenvectors of a certain integrable model, labeled in general by N-tuples of Young diagrams. In particular, it was found that in the Virasoro case these vectors are expressed in terms of Jack polynomials, labeled by 2-tuples of ordinary Young diagrams, and for the super-Virasoro case they are related to Uglov polynomials, labeled by two colored Young diagrams. In the case of a generic central charge this statement was checked in the case when one of the Young diagrams is empty. In this note we study the N = 1 SCFT and construct 4 point correlation function using the basis. To this end we need to clarify the connection between basis elements and Uglov polynomials, we also need to use two bosonizations and their connection to the reflection operator. For the central charge c=3/2 we checked that there is a connection with the Uglov polynomials for the whole set of diagrams.

A criterion for irreducibility of parabolic baby Verma modules of reductive Lie algebras

Let G be a connected, reductive algebraic group over an algebraically closed field k of prime characteristic p and g=Lie(G). In this paper, we study representations of g with a p-character χ of standard Levi form. When g is of type An,Bn,Cn or Dn, a sufficient condition for the irreducibility of standard parabolic baby Verma g-modules is obtained. This partially answers a question raised by Friedlander and Parshall (1990) in [2]. Moreover, as an application, in the special case that g is of type An or Bn, and χ lies in the sub-regular nilpotent orbit, we recover a result of Jantzen (1999) in [6].

BRAIDED COMMUTATIVE ALGEBRAS OVER QUANTIZED ENVELOPING ALGEBRAS

We produce braided commutative algebras in braided monoidal categories by generalizing Davydov’s full center construction of commutative algebras in centers of monoidal categories. Namely, we build braided commutative algebras in relative monoidal centers {mathcal{Z}}_{mathrm{mathcal{B}}}left(mathcal{C}right) from algebras in ℬ-central monoidal categories mathcal{C} , where ℬ is an arbitrary braided monoidal category; Davydov’s (and previous works of others) take place in the special case when ℬ is the category of vector spaces {mathbf{Vect}}_{mathbbm{K}} over a field mathbbm{K} . Since key examples of relative monoidal centers are suitable representation categories of quantized enveloping algebras, we supply braided commutative module algebras over such quantum groups. One application of our work is that we produce Morita invariants for algebras in ℬ-central monoidal categories. Moreover, for a large class of ℬ-central monoidal categories, our braided commutative algebras arise as a braided version of centralizer algebras. This generalizes the fact that centers of algebras in {mathbf{Vect}}_{mathbbm{K}} serve as Morita invariants. Many examples are provided throughout.

$$C ^*$$-Module Algebras

In this paper, we define $$C ^*$$ -module algebras as a generalization of Hilbert algebras and study their properties. We describe some structures of a $$C ^*$$ -module algebra including bounded elements and left (right) module adjointable maps. Furthermore we establish some properties of Hilbert algebras for $$C ^*$$ -module algebras.

Brauer–Clifford group of -Azumaya comodule algebras

In this paper, we extend the notion of the Brauer–Clifford group to the case of an -comodule algebra, where H is a commutative Hopf algebra, is a Lie algebra in the symmetric monoidal category of right H-comodules, and S is a commutative algebra which is an H-comodule algebra, a -module algebra and the H-coaction is compatible with the -action. This Brauer–Clifford group turns out to be an example of the Brauer group of a symmetric monoidal category.

On simple-minded systems and τ-periodic modules of self-injective algebras

Let A be a finite-dimensional self-injective algebra over an algebraically closed field, C a stably quasi-serial component (i.e. its stable part is a tube) of rank n of the Auslander-Reiten quiver of A, and S be a simple-minded system of the stable module category mod_A. We show that the intersection S∩C is of size strictly less than n, and consists only of modules with quasi-length strictly less than n. In particular, all modules in the homogeneous tubes of the Auslander-Reiten quiver of A cannot be in any simple-minded system.

Brauer–Clifford Group of ($$\varvec{S,{{\mathcal {G}}},H}$$)-Azumaya Algebras

In this paper we extend the notion of the Brauer–Clifford group to the case of an $$(S,{{\mathcal {G}}},H)$$ -algebra, when H is a cocommutative Hopf algebra, $${{\mathcal {G}}}$$ is a Lie algebra in the symmetric monoidal category of left H-modules, and S is a commutative algebra which is an H-module algebra, a $${\mathcal G}$$ -module algebra and the H-action is compatible with the $${\mathcal {G}}$$ -action. This Brauer–Clifford group turns out to be an example of the Brauer group of a symmetric monoidal category.