On the Hochschild cohomology ring of tensor products of algebras

We prove that, as Gerstenhaber algebras, the Hochschild cohomology ring of the tensor product of two algebras is isomorphic to the tensor product of the respective Hochschild cohomology rings of these two algebras, when at least one of them is finite dimensional. In case of finite dimensional symmetric algebras, this isomorphism is an isomorphism of Batalin–Vilkovisky algebras. As an application, we explain by examples how to compute the Batalin–Vilkovisky structure, in particular, the Gerstenhaber Lie bracket, over the Hochschild cohomology ring of the group algebra of a finite abelian group.     

一类特征单的n-李代数

利用结合代数与n-李代数的张量积构造了一类无限维特征单的n-李代数,且证明了除n=3的情形以外,这类特征单n-李代数的内导子代数是特征单李代数.     

Locally finite dimensional lie algebras and their derivation algebras

The derivation algebras of all locally finite dimensional locally simple Lie algebras over a field of characteristic 0 are determined. Every locally finite dimensional Lie algebra of countable dimension is a subalgebra of the outer derivation algebra outder (ℒ) for every Lie algebra ℒ, which is the direct limit of diagonally embedded classical Lie algebras. These outer derivation algebras have dimension ℒ and are never locally finite dimensional. Dedicated to Prof. H. Petersson on the occasion of his 60th birthday     

Completions of cellular algebras

We introduce procellular algebras, so called because they are inverse limits of finite dimensional cellular algebras as defined by Graham and Lehrer. A procellular algebra is defined as a certain completion of an infinite dimensional cellular algebra whose cell datum is of “proflnite type”. We show how these notions overcome some known obstructions to the theory of cellular agebras in infinite dimensions.     

Affine cellular algebras

Graham and Lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras. Many classes of finite dimensional algebras, including various Hecke algebras and diagram algebras, have been shown to be cellular, and the theory due to Graham and Lehrer successfully has been applied to these algebras.We will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field. Affine Hecke algebras of type A and infinite dimensional diagram algebras like the affine Temperley–Lieb algebras are shown to be examples of our definition. The isomorphism classes of simple representations of affine cellular algebras are shown to be parameterised by the complement of finitely many subvarieties in a finite disjoint union of affine varieties. In this way, representation theory of non-commutative algebras is linked with commutative algebra. Moreover, conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification; these conditions are shown to be satisfied for the affine Hecke algebra of type A if the quantum parameter is not a root of the Poincaré polynomial.     

Motivic zeta functions of infinite-dimensional Lie algebras

The paper shows how to associate a motivic zeta function with a large class of infinite dimensional Lie algebras. These include loop algebras, affine Kac-Moody algebras, the Virasoro algebra and Lie algebras of Cartan type. The concept of a motivic zeta functions provides a good language to talk about the uniformity in p of local p-adic zeta functions of finite dimensional Lie algebras. The theory of motivic integration is employed to prove the rationality of motivic zeta functions associated to certain classes of infinite dimensional Lie algebras.     

Tensor Product Weight Representations of the Neveu–Schwarz Algebra

We study the tensor product of a highest weight module with an intermediate series module over the Neveu–Schwarz algebra. If the highest weight module is nontrivial, the weight spaces of such a tensor product are infinite dimensional. We show that such a tensor product is indecomposable. Using a “shifting technique” developed by H. Chen, X. Guo, and K. Zhao for the Virasoro algebra case, we give necessary and sufficient conditions for such a tensor product to be irreducible. Furthermore, we give necessary and sufficient conditions for two such tensor products to be isomorphic.     

扭的Heisenberg-Virasoro顶点算子代数的一个特征化（英文）

The twisted Heisenberg-Virasoro algebra is the universal central extension of the Lie algebra of differential operators on a circle of order at most one. In this paper, we first study the variety of semi-conformal vectors of the twisted Heisenberg-Virasoro vertex operator algebra, which is a finite set consisting of two nontrivial elements. Based on this property,we also show that the twisted Heisenberg-Virasoro vertex operator algebra is a tensor product of two vertex operator algebras. Moreover, associating to properties of semi-conformal vectors of the twisted Heisenberg-Virasoro vertex operator algebra, we charaterized twisted Heisenberg-Virasoro vertex operator algebras. This will be used to understand the classification problems of vertex operator algebras whose varieties of semi-conformal vectors are finite sets.     

Representation type of Jordan algebras

The problem of classification of Jordan bimodules over (non-semisimple) finite dimensional Jordan algebras with respect to their representation type is considered. The notions of diagram of a Jordan algebra and of Jordan tensor algebra of a bimodule are introduced and a mapping Qui is constructed which associates to the diagram of a Jordan algebra J the quiver of its universal associative enveloping algebra S(J). The main results are concerned with Jordan algebras of semi-matrix type, that is, algebras whose semi-simple component is a direct sum of Jordan matrix algebras. In this case, criterion of finiteness and tameness for one-sided representations are obtained, in terms of diagram and mapping Qui, for Jordan tensor algebras and for algebras with radical square equals to 0.     

Bounded derived categories and repetitive algebras

By a theorem due to the first author, the bounded derived category of a finite dimensional algebra over a field embeds fully faithfully into the stable category over its repetitive algebra. This embedding is an equivalence if the algebra is of finite global dimension. The purpose of this paper is to investigate the relationship between the derived category and the stable category over the repetitive algebra from various points of view for algebras of infinite global dimension. The most satisfactory results are obtained for Gorenstein algebras, especially for selfinjective algebras.     

Smash products of quasi-hereditary graded algebras

The Smash product of a finite dimensional quasi-hereditary algebra graded by a finite group with the group is proved to be a quasi-hereditary algebra. Some elementary relations between the good modules of the two quasi-hereditary algebras are given.     

The socle of a nondegenerate Lie algebra

We define the socle of a nondegenerate Lie algebra as the sum of all its minimal inner ideals. The socle turns out to be an ideal which is a direct sum of simple ideals, and satisfies the descending chain condition on principal inner ideals. Every classical finite dimensional Lie algebra coincides with its socle, while relevant examples of infinite dimensional Lie algebras with nonzero socle are the simple finitary Lie algebras and the classical Banach Lie algebras of compact operators on an infinite dimensional Hilbert space. This notion of socle for Lie algebras is compatible with the previous ones for associative algebras and Jordan systems. We conclude with a structure theorem for simple nondegenerate Lie algebras containing abelian minimal inner ideals, and as a consequence we obtain that a simple Lie algebra over an algebraically closed field of characteristic 0 is finitary if and only if it is nondegenerate and contains a rank-one element.     

Tensor Products and Support Varieties for Some Noncocommutative Hopf Algebras

We explore questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. We construct many classes of examples in which tensor powers of nonprojective modules are projective and tensor products of modules in one order are projective but in the other order are not. Our examples are smash coproducts with duals of group algebras, some having algebra and coalgebra structures twisted by cocycles. We apply support variety theory for these Hopf algebras as a tool in our investigations.     

On the codimension growth of almost nilpotent Lie algebras

We study codimension growth of infinite dimensional Lie algebras over a field of characteristic zero. We prove that if a Lie algebra L is an extension of a nilpotent algebra by a finite dimensional semisimple algebra then the PI-exponent of L exists and is a positive integer.     

Decomposability of free Tarski algebras

In this paper we study the direct decomposability of free Tarski algebras. We show that infinite freely generated Tarski algebras are directly indecomposable, whereas finite freely generated Tarski algebras can only be decomposed into a direct product of two factors, one of which is the two-element Tarski algebra.     

On the existence of coproducts in the category of noncommutative algebras

The weakly multiplicative bilinear maps among R-algebras are introduced and the corresponding to them category of pointed algebras is defined. By the resolution of the respective universal problem, a new tensor product of two R-algebras is realized and its connection with the usual one is pointed out. Furthermore, after establishing the existence of an infinite (new) tensor product algebra, as a special colimit, it is shown that coproducts there always exist in the category Al of noncommutative (associative, unitary) R-algebras. Certain canonical decompositions of a given algebra in some particular cases are also investigated.     

Towers of derivation for Lie rings and some results on complete Lie rings

The well-known tower theorem of groups (resp. Lie algebras) shows that the tower of automorphism groups (resp. derivation algebras) of a finite group (resp. a finite dimensional Lie algebra) with trivial center terminates after finitely many steps. We generalize these results for Lie rings, and present some necessary and sufficient conditions for Lie rings to be complete.     

TOWERS OF DERIVATION FOR LIE RINGS AND SOME RESULTS ON COMPLETE LIE RINGS

The well-known tower theorem of groups （resp. Lie algebras） shows that the tower of automorphism groups （resp. derivation algebras） of a finite group （resp. a finite dimensional Lie algebra） with trivial center terminates after finitely many steps. We generalize these results for Lie rings, and present some necessary and sufficient conditions for Lie rings to be complete.     

Effect Algebras as Presheaves on Finite Boolean Algebras

For an effect algebra A, we examine the category of all morphisms from finite Boolean algebras into A. This category can be described as a category of elements of a presheaf R(A) on the category of finite Boolean algebras. We prove that some properties (being an orthoalgebra, the Riesz decomposition property, being a Boolean algebra) of an effect algebra A can be characterized in terms of some properties of the category of elements of the presheaf R(A). We prove that the tensor product of effect algebras arises as a left Kan extension of the free product of finite Boolean algebras along the inclusion functor. The tensor product of effect algebras can be expressed by means of the Day convolution of presheaves on finite Boolean algebras.