Realizable polynomial algebras

In this paper we investigate whether a polynomial algebra can be realized as a cohomology ring of a topological space. Our main results are that we can split the realizable polynomial algebra into a tensor product of certain simple factors and that these factors are given explicitly whenp>7. What is worth mentioning is that most of these factors are known to be realizable.     

Representations of skew polynomial algebras

C. De Concini and C. Procesi have proved that in many cases the degree of a skew polynomial algebra is the same as the degree of the corresponding quasi polynomial algebra. We prove a slightly more general result. In fact we show that in case the skew polynomial algebra is a P.I. algebra, then its degree is the degree of the quasi polynomial algebra.

Our argument is then applied to determine the degree of some algebras given by generators and relations.

     

DG polynomial algebras and their homological properties

In this paper, we introduce and study differential graded(DG for short) polynomial algebras. In brief, a DG polynomial algebra A is a connected cochain DG algebra such that its underlying graded algebra A~# is a polynomial algebra K[x_1, x_2,..., x_n] with |xi| = 1 for any i ∈ {1, 2,..., n}. We describe all possible differential structures on DG polynomial algebras, compute their DG automorphism groups, study their isomorphism problems, and show that they are all homologically smooth and Gorenstein DG algebras. Furthermore, it is proved that the DG polynomial algebra A is a Calabi-Yau DG algebra when its differential ?_A≠ 0 and the trivial DG polynomial algebra(A, 0) is Calabi-Yau if and only if n is an odd integer.     

Group algebras of polynomial growth

Let A be a finite dimensional, basic and connected algebra (associative, with 1) over an algebraically closed field, G a finite group and AG the group algebra. The main result gives necessary and sufficient conditions for AG to be of polynomial growth, that is, there is a natural number m such that the indecomposable finite dimensional AG-modules occur, in each dimension d2, in a finite number of discrete and at most d one-parameter families.     

Co-Poisson structures on polynomial Hopf algebras

The Hopf dual H° of any Poisson Hopf algebra H is proved to be a co-Poisson Hopf algebra provided H is noetherian. Without noetherian assumption, unlike it is claimed in literature, the statement does not hold. It is proved that there is no nontrivial Poisson Hopf structure on the universal enveloping algebra of a non-abelian Lie algebra. So the polynomial Hopf algebra, viewed as the universal enveloping algebra of a finite-dimensional abelian Lie algebra, is considered. The Poisson Hopf structures on polynomial Hopf algebras are exactly linear Poisson structures. The co-Poisson structures on polynomial Hopf algebras are characterized. Some correspondences between co-Poisson and Poisson structures are also established.     

New classes of test polynomials of polynomial algebras

A polynomialp in a polynomial algebra over a field is called a test polynomial if any endomorphism of the polynomial algebra that fixesp is an automorphism. some classes of new test polynomials recognizing nonlinear automorphisms of polynomial algebras are given. In the odd prime characteristic case, test polynomials recognizing non-semisimple automorphisms are also constructed. Project supported by the National Natural Science Foundation of China (Grant No. 19631100) and University of Hong Kong RGC Fundable Grant 344/024/0004.     

Weighted group algebras on groups of polynomial growth

Let G be a compactly generated group of polynomial growth and a weight function on G. For a large class of weights we characterize symmetry of the weighted group algebra L ¹ (G,). In particular, if the weight is sub-exponential, then the algebra L ¹ (G,) is symmetric. For these weights we develop a functional calculus on a total part of L ¹ (G,) and use it to prove the Wiener property. Mathematics Subject Classification (2000):43A20, 22D15, 22D12.Supported by the Austrian Science Foundation project FWF P-14485.Supported by the research grants MEN/CUL/98/007 and CUL/01/014.     

Automorphic orbit problem for polynomial algebras

It is proved that every endomorphism preserving the automorphic orbit of a non-trivial element of the rank two polynomial algebra over the complex number field is an automorphism.     

Derivations of Lie algebras

In this paper we prove a theorem on the expansion into a sum of a derivation of a splittable finite-dimensional Lie algebra over a field of characteristic 0. We investigate the structure of the derivation algebra D(L) of a free nilpotent Lie algebra L and we show that the algebra D(D(L)) is perfect.Translated from Matematicheskie Zametki, Vol. 20, No. 1, pp. 3–10, July, 1976.The author thanks L. A. Bokut' and G. P. Kukin for a great deal of assistance in laying out and writing the article.     

Hochschild cohomology for periodic algebras of polynomial growth

We describe the dimensions of low Hochschild cohomology spaces of exceptional periodic representation-infinite algebras of polynomial growth. As an application we obtain that an indecomposable non-standard periodic representation-infinite algebra of polynomial growth is not derived equivalent to a standard self-injective algebra.     

Weighted infinitesimal unitary bialgebras,pre-Lie,matrix algebras,and polynomial algebras

Motivated by comatrix coalgebras, we introduce the concept of a Newtonian comatrix coalgebra. We construct an infinitesimal unitary bialgebra on matrix algebras, via the construction of a suitable coproduct. As a consequence, a Newtonian comatrix coalgebra is established. Furthermore, an infinitesimal unitary Hopf algebra, under the view of Aguiar, is constructed on matrix algebras. By the close relationship between pre-Lie algebras and infinitesimal unitary bialgebras, we erect a pre-Lie algebra and a new Lie algebra on matrix algebras. Finally, a weighted infinitesimal unitary bialgebra on non-commutative polynomial algebras is also given.     

Lie algebras with finite-dimensional polynomial centralizer

We give criteria for finite or infinite dimensionality of the polynomial centralizer of the Lie algebra of a linear Lie group, in terms of invariants and relative invariants of the group. In the finite-dimensional scenario some applications to normal forms and to certain equations with fundamental solutions are presented.     

Flag algebras and the stable coefficients of the Jones polynomial

We study the structure of the stable coefficients of the Jones polynomial of an alternating link. We start by identifying the first four stable coefficients with polynomial invariants of a (reduced) Tait graph of the link projection. This leads us to introduce a free polynomial algebra of invariants of graphs whose elements give invariants of alternating links which strictly refine the first four stable coefficients. We conjecture that all stable coefficients are elements of this algebra, and give experimental evidence for the fifth and the sixth stable coefficient. We illustrate our results in tables of all alternating links with at most 10 crossings and all irreducible planar graphs with at most 6 vertices.     

Group-graded algebras with polynomial identity

LetG be a finite group and letR=Σ _g∈G R _g be any associative algebra over a field such that the subspacesR _g satisfyR _g R _h ⊆R _gh . We prove that ifR ₁ satisfies a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the order ofG. This result implies the following: ifH is a finite-dimensional semisimple commutative Hopfalgebra andR is anyH-module algebra withR ^H satisfying a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the dimension ofH.     

C~*-index of observable algebras in G-spin model

In two-dimensional lattice spin systems in which the spins take values in afinite group G,one can define a field algebra F which carries an action of a Hopf algebraD(G),the double algebra of G and moreover,an action of D(G;H),which is a subalgebraof D(G)determined by a subgroup H of G,so that F becomes a modular algebra.Theconcrete construction of D(G;H)-invariant subspace A_H in F is given.By constructingthe quasi-basis of conditional expectation γ_G of A_H onto A_G,the C~*-index of γ_G is exactlythe index of H in G.     

Poisson (co)homology of truncated polynomial algebras in two variables

We study the Poisson (co)homology of the algebra of truncated polynomials in two variables viewed as the semi-classical limit of a quantum complete intersection studied by Bergh and Erdmann. We show in particular that the Poisson cohomology ring of such a Poisson algebra is isomorphic to the Hochschild cohomology ring of the corresponding quantum complete intersection. To cite this article: S. Launois, L. Richard, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Lie superalgebras whose enveloping algebras satisfy a non-matrix polynomial identity

Let L be a Lie superalgebra with its enveloping algebra U(L) over a field F. A polynomial identity is called non-matrix if it is not satisfied by the algebra of 2×2 matrices over F. We characterize L when U(L) satisfies a non-matrix polynomial identity. We also characterize L when U(L) is Lie solvable, Lie nilpotent, or Lie super-nilpotent.     

Linear gradings of polynomial algebras

Let k be a field, let be a finite group. We describe linear -gradings of the polynomial algebra k[x ₁, ..., x _m ] such that the unit component is a polynomial k-algebra.