Minimal varieties of algebras of exponential growth

The exponent of a variety of algebras over a field of characteristic zero has been recently proved to be an integer. Through this scale we can now classify all minimal varieties of given exponent and of finite basic rank. As a consequence, we describe the corresponding T-ideals of the free algebra and we compute the asymptotics of the related codimension sequences, verifying in this setting some known conjectures. We also show that the number of these minimal varieties is finite for any given exponent. We finally point out some relations between the exponent of a variety and the Gelfand-Kirillov dimension of the corresponding relatively free algebras of finite rank.     

An uncountable family of almost nilpotent varieties of polynomial growth

A non-nilpotent variety of algebras is almost nilpotent if any proper subvariety is nilpotent. Let the base field be of characteristic zero. It has been shown that for associative or Lie algebras only one such variety exists. Here we present infinite families of such varieties. More precisely we shall prove the existence of1) a countable family of almost nilpotent varieties of at most linear growth and2) an uncountable family of almost nilpotent varieties of at most quadratic growth.     

Graded polynomial identities and codimensions: Computing the exponential growth

Let G be a finite abelian group and A a G-graded algebra over a field of characteristic zero. This paper is devoted to a quantitative study of the graded polynomial identities satisfied by A. We study the asymptotic behavior of , n=1,2,…, the sequence of graded codimensions of A and we prove that if A satisfies an ordinary polynomial identity, exists and is an integer. We give an explicit way of computing such integer by proving that it equals the dimension of a suitable finite dimension semisimple G×Z₂-graded algebra related to A.     

Polynomial identities on superalgebras: Classifying linear growth

We classify, up to PI-equivalence, the superalgebras over a field of characteristic zero whose sequence of codimensions is linearly bounded. As a consequence we determine the linear functions describing the graded codimensions of a superalgebra.     

Gröbner-Shirshov bases for associative algebras with multiple operators and free Rota-Baxter algebras

In this paper, we establish the Composition-Diamond lemma for associative algebras with multiple linear operators. As applications, we obtain Gröbner-Shirshov bases of free Rota-Baxter algebra, free λ-differential algebra and free λ-differential Rota-Baxter algebra, respectively. In particular, linear bases of these three free algebras are respectively obtained, which are essentially the same or similar to the recent results obtained by K. Ebrahimi-Fard-L. Guo, and L. Guo-W. Keigher by using other methods.     

Topological and bornological characterisations of ideals in von Neumann algebras: I

Suppose is a von Neumann algebra on a Hilbert space and is any ideal in . We determine a topology on , for which the members of that are to norm continuous are exactly those in ; and a bornology on such that the elements of which map the unit ball to an element of , equivalently those members of that are norm to bounded, are exactly those in . This is achieved via analogues of the notions of injectivity and surjectivity in the theory of operator ideals on Banach spaces.     

Poisson identities of enveloping algebras

Let L be a restricted Lie algebra. The symmetric algebra S_p(L) of the restricted enveloping algebra u(L) has the structure of a Poisson algebra. We give necessary and sufficient conditions on L in order for the symmetric algebra S_p(L) to satisfy a multilinear Poisson identity. We also settle the same problem for the symmetric algebra S(L) of a Lie algebra L over an arbitrary field. The first author was partially supported by MIUR of Italy. The second author was partially supported by Grant RFBR-04-01- 00739. Received: 31 October 2005     

Crystals, quiver varieties, and coboundary categories for Kac-Moody algebras

Henriques and Kamnitzer have defined a commutor for the category of crystals of a finite-dimensional complex reductive Lie algebra that gives it the structure of a coboundary category (somewhat analogous to a braided monoidal category). Kamnitzer and Tingley then gave an alternative definition of the crystal commutor, using Kashiwara's involution on Verma crystals, that generalizes to the setting of symmetrizable Kac-Moody algebras. In the current paper, we give a geometric interpretation of the crystal commutor using quiver varieties. Equipped with this interpretation we show that the commutor endows the category of crystals of a symmetrizable Kac-Moody algebra with the structure of a coboundary category, answering in the affirmative a question of Kamnitzer and Tingley.     

Involutions and representations for reduced quantum algebras

In the context of deformation quantization, there exist various procedures to deal with the quantization of a reduced space Mred. We shall be concerned here mainly with the classical Marsden-Weinstein reduction, assuming that we have a proper action of a Lie group G on a Poisson manifold M, with a moment map J for which zero is a regular value. For the quantization, we follow Bordemann et al. (2000) [6] (with a simplified approach) and build a star product red_? on Mred from a strongly invariant star product ? on M. The new questions which are addressed in this paper concern the existence of natural ^∗-involutions on the reduced quantum algebra and the representation theory for such a reduced ^∗-algebra.We assume that ? is Hermitian and we show that the choice of a formal series of smooth densities on the embedded coisotropic submanifold C=J−1(0), with some equivariance property, defines a ^∗-involution for red_? on the reduced space. Looking into the question whether the corresponding ^∗-involution is the complex conjugation (which is a ^∗-involution in the Marsden-Weinstein context) yields a new notion of quantized modular class.We introduce a left (C^∞(M)?λ?,?)-submodule and a right (C^∞(Mred)?λ?,red_?)-submodule of C^∞(C)?λ?; we define on it a C^∞(Mred)?λ?-valued inner product and we establish that this gives a strong Morita equivalence bimodule between C^∞(Mred)?λ? and the finite rank operators on . The crucial point is here to show the complete positivity of the inner product. We obtain a Rieffel induction functor from the strongly non-degenerate ^∗-representations of (C^∞(Mred)?λ?,red_?) on pre-Hilbert right D-modules to those of (C^∞(M)?λ?,?), for any auxiliary coefficient ^∗-algebra D over C?λ?.     

Finitistic dimension of properly stratified algebras

We prove that the finitistic dimension of a properly stratified algebra having a simple preserving duality and for which every tilting module is cotilting, equals twice the projective dimension of the characteristic tilting module. As a corollary, we get that the global dimension of a quasi-hereditary algebra with duality equals twice the projective dimension of the characteristic tilting module. As another corollary, we obtain an affirmative answer to the conjecture of Erdmann and Parker. Finally, we calculate the finitistic dimension of the blocks of certain parabolic generalizations of the category .     

Skew differential operator algebras of twisted Hopf algebras

By introducing a twisted Hopf algebra we unify several important objects of study. Skew derivations of such an algebra are defined and the corresponding skew differential operator algebras are studied. This generalizes results in the Weyl algebra. Applying this investigation to the twisted Ringel-Hall algebra we get, in particular, a natural realization of the non-positive part of a quantized generalized Kac-Moody algebra, by identifying the canonical generators with some linear, skew differential operators. This also induces some algebras which are quantum-group like.     

The GK dimension of relatively free algebras of PI-algebras

We prove a strict relation between the Gelfand–Kirillov (GK) dimension of the relatively free (graded) algebra of a PI-algebra and its (graded) exponent. As a consequence we show a Bahturin–Zaicev type result relating the GK dimension of the relatively free algebra of a graded PI-algebra and the one of its neutral part. We also get that the growth of the relatively free graded algebra of a matrix algebra is maximal when the grading is fine. Finally we compute the graded GK dimension of the matrix algebra with any grading and the graded GK dimension of any verbally prime algebra endowed with an elementary grading.     

Iyama’s finiteness theorem via strongly quasi-hereditary algebras

Let Λ be an artin algebra and X a finitely generated Λ-module. Iyama has shown that there exists a module Y such that the endomorphism ring Γ of X⊕Y is quasi-hereditary, with a heredity chain of length n, and that the global dimension of Γ is bounded by this n. In general, one only knows that a quasi-hereditary algebra with a heredity chain of length n must have global dimension at most 2n−2. We want to show that Iyama’s better bound is related to the fact that the ring Γ he constructs is not only quasi-hereditary, but even left strongly quasi-hereditary. By definition, the left strongly quasi-hereditary algebras are the quasi-hereditary algebras with all standard left modules of projective dimension at most 1.     

General presentations of algebras

For any finite dimensional basic associative algebra, we study the presentation spaces and their relation with the representation spaces. We prove two theorems about a general presentation, one on its subrepresentations and the other on its canonical decomposition. As a special case, we consider rigid presentations. We show how to complete a rigid presentation and study the number of nonisomorphic direct summands and different complements. Based on that, we construct a simplicial complex governing the canonical decompositions of rigid presentations and provide some examples.     

Decomposable compositions, symmetric quasisymmetric functions and equality of ribbon Schur functions

We define an equivalence relation on integer compositions and show that two ribbon Schur functions are identical if and only if their defining compositions are equivalent in this sense. This equivalence is completely determined by means of a factorization for compositions: equivalent compositions have factorizations that differ only by reversing some of the terms. As an application, we can derive identities on certain Littlewood-Richardson coefficients.Finally, we consider the cone of symmetric functions having a nonnnegative representation in terms of the fundamental quasisymmetric basis. We show the Schur functions are among the extremes of this cone and conjecture its facets are in bijection with the equivalence classes of compositions.