Certain categories of modules for twisted affine Lie algebras

In this paper, using generating functions, we study two categories ? and ? of modules for twisted affine Lie algebras g^[σ], which were firstly introduced and studied for untwisted affine Lie algebras by H. -S. Li [Math Z, 2004, 248: 635-664]. We classify integrable irreducible g^[σ]-modules in categories ? and ?, where ? is proved to contain the well-known evaluation modules and ? to unify highest weight modules, evaluation modules and their tensor product modules. We determine also the isomorphism classes of those irreducible modules.     

Classification of irreducible non-zero level quasifinite modules over twisted affine Nappi-Witten algebra

We obtain that every irreducible quasifinite module with non-zero level over the twisted affine Nappi-Witten algebra is either a highest weight module or a lowest one.     

On twisted representations of vertex algebras

In this paper, we develop a formalism for working with representations of vertex and conformal algebras by generalized fields—formal power series involving non-integer powers of the variable. The main application of our technique is the construction of a large family of representations for the vertex superalgebra corresponding to an integer lattice Λ. For an automorphism coming from a finite-order automorphism we find the conditions for existence of twisted modules of . We show that the category of twisted representations of is semisimple with finitely many isomorphism classes of simple objects.     

A twisted quantum toroidal algebra

As an analog of the quantum TKK algebra, a twisted quantum toroidal algebra of type A ₁ is introduced. Explicit realization of the new quantum TKK algebra is constructed with the help of twisted quantum vertex operators over a Fock space.     

Degenerate Sklyanin algebras and generalized twisted homogeneous coordinate rings

In this work, we introduce the point parameter ring B, a generalized twisted homogeneous coordinate ring associated to a degenerate version of the three-dimensional Sklyanin algebra. The surprising geometry of these algebras yields an analogue to a result of Artin–Tate–van den Bergh, namely that B is generated in degree one and thus is a factor of the corresponding degenerate Sklyanin algebra.     

Filtrations and completions of certain positive level modules of affine algebras

We define a filtration indexed by the integers on the tensor product of a simple highest weight module and a loop module for a quantum affine algebra. We prove that such a filtration is either trivial or strictly decreasing and give sufficient conditions for this to happen. In the first case we prove that the tensor product is simple and in the second case we prove that the intersection of all the modules in the filtration is zero, thus allowing us to define the completed tensor product. In certain special cases, we identify the subsequent quotients of filtration.     

Trees, set compositions and the twisted descent algebra

We first show that increasing trees are in bijection with set compositions, extending simultaneously a recent result on trees due to Tonks and a classical result on increasing binary trees. We then consider algebraic structures on the linear span of set compositions (the twisted descent algebra). Among others, a number of enveloping algebra structures are introduced and studied in detail. For example, it is shown that the linear span of trees carries an enveloping algebra structure and embeds as such in an enveloping algebra of increasing trees. All our constructions arise naturally from the general theory of twisted Hopf algebras.     

A certain clifford-like algebra and quantum vertex algebras

We show a new game characterizing various types of σ-porosity for Souslin sets in terms of winning strategies. We use the game to prove and reprove some new and older inscribing theorems for σ-ideals of σ-porous type in locally compact metric spaces.     

P-twisted affine Lie algebra and its realizations by twisted vertex operators

In this paper, we define a P-twisted affine Lie algebra, and construct its realizations by twisted vertex operators.     

Lattice construction of logarithmic modules for certain vertex algebras

A general method for constructing logarithmic modules in vertex operator algebra theory is presented. By utilizing this approach, we give explicit vertex operator construction of certain indecomposable and logarithmic modules for the triplet vertex algebra W(p){\mathcal{W}(p)} and for other subalgebras of lattice vertex algebras and their N = 1 super extensions. We analyze in detail indecomposable modules obtained in this way, giving further evidence for the conjectural equivalence between the category of W(p){\mathcal{W}(p)}-modules and the category of modules for the restricted quantum group [`(U)]_q(sl₂){\overline{\mathcal{U}}_q(sl_2)} , q = e ^{π i/p} . We also construct logarithmic representations for a certain affine vertex operator algebra at admissible level realized in Adamović (J. Pure Appl. Algebra 196:119–134, 2005). In this way we prove the existence of the logarithmic representations predicted in Gaberdiel (Int. J. Modern Phys. A 18, 4593–4638, 2003). Our approach enlightens related logarithmic intertwining operators among indecomposable modules, which we also construct in the paper.     

Interpolation in nest algebra modules

Let be a nest algebra and its invariant projection (or subspace) lattice. In this paper, using order homomorphisms of , we give necessary and sufficient conditions on bounded linear operators and on a Hilbert space to guarantee the existence of an operator in a certain -module such that .

     

On certain vertex algebras and their modules associated with vertex algebroids

We study the family of vertex algebras associated with vertex algebroids, constructed by Gorbounov, Malikov, and Schechtman. As the main result, we classify all the (graded) simple modules for such vertex algebras and we show that the equivalence classes of graded simple modules one-to-one correspond to the equivalence classes of simple modules for the Lie algebroids associated with the vertex algebroids. To achieve our goal, we construct and exploit a Lie algebra from a given vertex algebroid.     

Characteristic modules and tensor products over quasi-hereditary algebras

Let A be a monomial quasi-hereditary algebra with a pure strong exact Borel subalgebra B.It is proved that the category of induced good modules over B is contained in the category of good modules over A;that the characteristic module of A is an induced module of that of B via the exact functor-(?)_B A if and only if the induced A-module of an injective B-module remains injective as a B-module.Moreover,it is shown that an exact Borel subalgebra of a basic quasi-hereditary serial algebra is right serial and that the characteristic module of a basic quasi-hereditary serial algebra is exactly the induced module of that of its exact Borel subalgebra.     

Tilting modules over path algebras of Dynkin type and complete slice modules

Let T be a tilting A-module over a path algebra of Dynkin type.We prove thatif the indecomposable direct summands of T are in the different (?)-orbits of the AR-quiverof A,then T is a complete slice module.