Central invariants of H-module algebras

Let H be a Hopf algebra over a field k:, and A an H-module algebra, with subalgebra of H-invariants denoted by A^H . When (H, R) is quasitriangular and A is quantum commutative with respect to (H,R), (e.g. quantum planes, graded commutative superalgebras), then A^H ? center of A = Z(A). In this paper we are mainly concerned with actions of H for which A^H ? Z(A). We show that under this hypothesis there exists strong relations between the ideal structures of A^H A and A#H.

We demonstrate the theorems by constructing an example of a quantum commutative A, so that A/A^H is H ^?-Galois. This is done by giving (C G)^? G = Z_n × Z_n , a nontrivial quasitriangular structure and defining an action of it on a localization of the quantum plane.     

Martindale quotients of Jordan algebras

In this paper we introduce Martindale quotients of Jordan algebras over arbitrary rings of scalars with respect to denominator filters of ideals. For any denominatored algebra, we show the existence of maximal Martindale quotients naturally containing all Martindale quotients of the algebra with respect to the given denominator filter.     

Homogeneous polynomials with isomorphic milnor algebras

We recall first Mather’s Lemma providing effective necessary and sufficient conditions for a connected submanifold to be contained in an orbit. We show that two homogeneous polynomials having isomorphic Milnor algebras are right-equivalent.     

Trace polynomials and invariant theory

LetA be a finite-dimensional simple (non-associative) algebra over an algebraically closed fieldF of characteristic 0. LetG be the group of its automorphisms which acts onkA, the direct sum ofk copies ofA. SupposeA is generated byk elements. In this paper, generators of the field of rational invariantF(kA) ^G are described in terms of operations of the algebraA.     

Singularities of generic characteristic polynomials and smooth finite splittings of Azumaya algebras over surfaces

Let k be an algebraically closed field. Let P(X ₁₁, . . . , X _nn , T) be the characteristic polynomial of the generic matrix (X _ij ) over k. We determine its singular locus as well as the singular locus of its Galois splitting. If X is a smooth quasi-projective surface over k and A an Azumaya algebra on X of degree n, using a method suggested by M. Artin, we construct finite smooth splittings for A of degree n over X whose Galois closures are smooth.     

Gorenstein rings and inverse polynomials

We characterize left perfect rings using locally, finitely projective and flat covers, thus answering a question of Xue in the positive.     

Hilbert polynomials and Arveson's curvature invariant

We define and study Hilbert polynomials for certain holomorphic Hilbert spaces. We obtain several estimates for these polynomials and their coefficients. Our estimates inspire us to investigate the connection between the leading coefficients of Hilbert polynomials for invariant subspaces of the symmetric Fock space and Arveson's curvature invariant for coinvariant subspaces. We are able to obtain some formulas relating the curvature invariant with other invariants. In particular, we prove that Arveson's version of the Gauss-Bonnet-Chern formula is true when the invariant subspaces are generated by any polynomials.     

Quasi-triangular Hopf algebras and invariant Jacobians

We show that two module homomorphisms for groups and Lie algebras established by Xi can be generalized to the setting of quasi-triangular Hopf algebras. These module homomorphisms played a key role in his proof of a conjecture of Yau (1998). They will also be useful in the problem of decomposition of tensor products of modules. Additionally, we give another generalization of result of Xi in terms of Chevalley-Eilenberg complex.     

DJKM algebras and non-classical orthogonal polynomials

We describe families of polynomials arising in the study of the universal central extensions of Lie algebras introduced by Date, Jimbo, Kashiwara, and Miwa (1983) [6] in their work on the Landau–Lifshitz equations. We show these two families of polynomials satisfy certain fourth order linear differential equations by direct computation and one of the families is a particular collection of associated ultraspherical polynomials.     

The Terwilliger algebras of certain association schemes over the Galois rings of characteristic 4

In a cyclotomic scheme over a finite field, there are some relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the field. These relations were investigated in [3]. In this paper, we replace the finite field by a commutative local ring which is called a Galois ring of characteristic 4. Hence we want to find similar relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the local ring. Specifically, if we let be a Galois ring of characteristic 4,X a cyclotomic scheme over with classD and the Terwilliger algebra ofX, then we show that most of the irreducible -modules have standard forms; otherwise, certain relations of the Jacobi sums hold. When the classD is three, we can completely determine the irreducible -modules using Jacobi sums.     

Graded monads and rings of polynomials

Models for free graded monads over the category of sets are constructed. Certain rings of generalized noncommutative polynomials, generated by an operation of arbitrary arity, are implemented as subrings of classical rings of noncommutative polynomials. It is shown that natural homomorphisms from rings of generalized polynomials to rings of the usual commutative polynomials are not inclusions as a rule. For instance, the natural homomorphism , where t is a binary variable, is not an inclusion even if t is subject to the alternating condition. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 349, 2007, pp. 174–210.