Irreducible weight modules for the Virasoro-like algebra and its q-analog

In this paper, using Larsson’s functor with irreducible 𝔰𝔩₂-modules V, we construct a class of ?²-graded modules for the Virasoro-like algebra and its q-analogs. We determine the irreducibility of these modules for finite-dimensional or infinite-dimensional V using a unified method. In particular, these modules provide new irreducible weight modules with infinite-dimensional weight spaces for the corresponding algebras.     

Representations of Nonstandard Poincaré Hopf Algebras

Let 𝔭 _q (1 + 1) be a nonstandard Poincaré Hopf algebra, we characterize all finite dimensional completely E-semisimple modules of 𝔭 _q (1 + 1). We also classify all finite dimensional E-semisimple modules of 𝔭 _q for a special quotient algebra of 𝔭 _q (1 + 1). Moreover, the decomposition of tensor product of two finite dimensional E-semisimple indecomposable modules is obtained.     

On classification of quasifinite representations of two classes of Lie superalgebras of block type

Motivated by a well-known theorem of Mathieu’s on Harish–Chandra modules over the Virasoro algebra and its super version, we show that an irreducible quasifinite module over two classes of Lie superalgebras 𝒮(q) of Block type is either a highest or lowest weight module or else a module of the intermediate series if q≠?1. For such a module over 𝒮(?1), we give a rough classification.     

TORSION THEORIES FOR FINITE VON NEUMANN ALGEBRAS

ABSTRACT

The study of modules over a finite von Neumann algebra 𝒜 can be advanced by the use of torsion theories. In this work, some torsion theories for 𝒜 are presented, compared, and studied. In particular, we prove that the torsion theory (T, P) (in which a module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie torsion theories for 𝒜.

Using torsion theories, we describe the injective envelope of a finitely generated projective 𝒜-module and the inverse of the isomorphism K ₀(𝒜) → K ₀ (𝒰), where 𝒰 is the algebra of affiliated operators of 𝒜. Then the formula for computing the capacity of a finitely generated module is obtained. Lastly, we study the behavior of the torsion and torsion-free classes when passing from a subalgebra ? of a finite von Neumann algebra 𝒜 to 𝒜. With these results, we prove that the capacity is invariant under the induction of a ?-module.     

The Yoneda Algebra of a 𝒦2 Algebra Need not be Another 𝒦2 Algebra

The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. 𝒦₂ algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a 𝒦₂ algebra would be another 𝒦₂ algebra. We show that this is not necessarily the case by constructing a monomial 𝒦₂ algebra for which the corresponding Yoneda algebra is not 𝒦₂.     

Non-weight representations of Cartan type S Lie algebras

For the two Cartan type S subalgebras of the Witt algebra 𝒲_n, called Lie algebras of divergence-zero vector fields, we determine all module structures on the universal enveloping algebra of their Cartan subalgebra 𝔥_n. We also give all submodules of these modules.     

Hopf coactions on commutative algebras generated by a quadratically independent comodule

Let 𝒜 be a commutative unital algebra over an algebraically closed field k of characteristic ≠2, whose generators form a finite-dimensional subspace V, with no nontrivial homogeneous quadratic relations. Let 𝒬 be a Hopf algebra that coacts on 𝒜 inner-faithfully, while leaving V invariant. We prove that 𝒬 must be commutative when either: (i) the coaction preserves a non-degenerate bilinear form on V; or (ii) 𝒬 is co-semisimple, finite-dimensional, and char(k) = 0.     

The Terwilliger Algebra Associated with a Set of Vertices in a Distance-Regular Graph

Let Γ be a distance-regular graph of diameter D. Let X denote the vertex set of Γ and let Y be a nonempty subset of X. We define an algebra τ = τ(Y). This algebra is finite dimensional and semisimple. If Y consists of a single vertex then τ is the corresponding subconstituent algebra defined by P. Terwilliger. We investigate the irreducible τ-modules. We define endpoints and thin condition on irreducible τ-modules as a generalization of the case when Y consists of a single vertex. We determine when an irreducible module is thin. When the module is generated by the characteristic vector of Y, it is thin if and only if Y is a completely regular code of Γ. By considering a suitable subset Y, every irreducible τ(x)-module of endpoint i can be regarded as an irreducible τ(Y)-module of endpoint 0.This research was partially supported by the Grant-in-Aid for Scientific Research (No. 12640039), Japan Society of the Promotion of Science. A part of the research was done when the author was visiting the Ohio State University.     

Triangular Decomposition of Composition Algebras of Domestic Canonical Algebras

Let A be a domestic canonical algebra over a finite field. In this article, we prove that the composition algebra C(A) of A has a triangular decomposition 𝒫·𝒯·? corresponding to the division of the indecomposable modules into the preprojectives, the regulars, and the preinjectives.     

The Coradical Filtration for Drinfeld Double Ringel-Hall Algebras

Abstract

Let 𝒟(Λ) be the Drinfeld double Ringel-Hall algebra with Λ being any finite dimensional hereditary algebra over a finite field k. We determine the coradical filtration for 𝒟(Λ). As an application, we describe the group of Hopf algebra automorphisms of the Drinfeld double Ringel composition algebra of Λ.     

Algebras of twisted chiral differential operators and affine localization of {\mathfrak {g}} -modules

We propose a notion of algebra of twisted chiral differential operators over algebraic manifolds with vanishing 1st Pontrjagin class. We show that such algebras possess families of modules depending on infinitely many complex parameters, which we classify in terms of the corresponding algebra of twisted differential operators. If the underlying manifold is a flag manifold, our construction recovers modules over an affine Lie algebra parameterized by opers over the Langlands dual Lie algebra. The spaces of global sections of “smallest” such modules are irreducible [^(\mathfrakg)]{{\hat{{\mathfrak{g}}}}} -modules, and all irreducible \mathfrakg{{\mathfrak{g}}} -integrable [^(\mathfrakg)]{{\hat{{\mathfrak{g}}}}} -modules at the critical level arise in this way.     

Representations of the virasoro-like algebra and its q-Analog

In this paper, some irreducible graded modules with 1-dimensional homogeneous spaces over the Virasoro-like algebra and its q-analogs are constructed. The unitarizability of these modules, and the conditions under which two of such irreducible graded modules are ismorphic are determined. Some other kinds of irreducible graded modules with 1-dimensional homogeneous spaces over the Virasorolike algebra and its q-analogs are also given.     

Extensions of Schrödinger–Virasoro conformal modules

In this paper, we study extensions between two finite irreducible conformal modules over the Schrödinger–Virasoro conformal algebra and the extended Schrödinger–Virasoro conformal algebra. Also, we classify all finite nontrivial irreducible conformal modules over the extended Schrödinger–Virasoro conformal algebra. As a byproduct, we obtain a classification of extensions of Heisenberg–Virasoro conformal modules.     

W-algebras related to parafermion algebras

We study a W-algebra of central charge 2(k−1)/(k+2), k=2,3,…, contained in the commutant of a Heisenberg algebra in a simple affine vertex operator algebra L(k,0) of type with level k. We calculate the operator product expansions of the W-algebra. We also calculate some singular vectors in the case k6 and determine the irreducible modules and Zhu's algebra. Furthermore, the rationality and the C₂-cofiniteness are verified for such k.     

On the composition subalgebras of Bridgeland’s Hall algebras

Let A be a finite dimensional hereditary algebra over a finite field k and 𝒫 the category consisting of finite dimensional projective (left) A-modules. In this paper, we consider the composition subalgebra of Bridgeland’s Hall algebra of the category 𝒞_m(𝒫) of m-cyclic complexes for any positive integer m≥2 and determine its generating relations.     

Universal Poisson Envelope for Binary-Lie Algebras

In this article the universal Poisson enveloping algebra for a binary-Lie algebra is constructed. Taking a basis 𝔹 of a binary-Lie algebra B, we consider the symmetric algebra S(B) of polynomials in the elements of 𝔹. We consider two products in S(B), the usual product of polynomials fg and the braces {f, g}, defined by the product in B and the Leibniz rule. This algebra is a general Poisson algebra. We find an ideal I of S(B) such that the factor algebra S(B)/I is the universal Poisson envelope of B. We provide some examples of this construction for known binary-Lie algebras.     

Simple modules of the quantum double of the Nichols algebra of unidentified diagonal type 𝔲𝔣𝔬(7)

The finite-dimensional simple modules over the Drinfeld double of the bosonization of the Nichols algebra 𝔲𝔣𝔬(7) are classified.     

Polynomial identity rings as rings of functions

We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where “varieties” carry a PGL_n-action, regular and rational “functions” on them are matrix-valued, “coordinate rings” are prime polynomial identity algebras, and “function fields” are central simple algebras of degree n. In particular, a prime polynomial identity algebra of degree n is finitely generated if and only if it arises as the “coordinate ring” of a “variety” in this setting. For n=1 our definitions and results reduce to those of classical affine algebraic geometry.