Mátyás Domokos

On the generalized Davenport constant and the Noether number

Known results on the generalized Davenport constant relating zero-sum sequences over a finite abelian group are extended for the generalized Noether number relating rings of polynomial invariants of an arbitrary finite group. An improved general upper degree bound for polynomial invariants of a non-cyclic finite group that cut out the zero vector is given.

CONJUGATION COINVARIANTS OF QUANTUM MATRICES

A quantum deformation of the classical conjugation action of GL(N,C) on the space of N × N matrices M(N,C) is defined via a coaction of the quantum general linear group O(GLq(N,C)) on the algebra of quantum matrices O(Mq(N,C)). The coinvariants of this coaction are calculated. In particular, interesting commutative subalgebras of O(Mq(N,C)) generated by (weighted) sums of principal quantum minors are obtained. For general Hopf algebras, co-commutative elements are characterized as coinvariants with respect to a version of the adjoint coaction. 2000 Mathematics Subject Classification. 16P40, 16W30, 16W35, 16S15, 13A50, 20G42.

Cayley-Hamilton theorem for 2 × 2 matrices over the Grassmann algebra

It is shown that the characteristic polynomial of matrices over a Lie nilpotent ring introduced recently by Szigeti is invariant with respect to the conjugation action of the general linear group. Explicit generators of the corresponding algebra of invariants in the case of 2 × 2 matrices over an algebra over a field of characteristic zero satisfying the identity [[x, y], z] = 0 are described. In this case the coefficients of the characteristic polynomial are expressed by traces of powers of the matrix, yielding a compact form of the Cayley-Hamilton equation of 2 × 2 matrices over the Grassmann algebra.

ON SINGULARITIES OF QUIVER MODULI

Any moduli space of representations of a quiver (possibly with oriented cycles) has an embedding as a dense open subvariety into a moduli space of representations of a bipartite quiver having the same type of singularities. A connected quiver is Dynkin or extended Dynkin if and only if all moduli spaces of its representations are smooth.

Invariants of quivers and wreath products

Let be a partition of the positive integer α. The block diagonal subgroup of the general linear group Glα(K) (K is a field of characteristic 0) acts on the n-tuples of α × α matrices over K by simultaneous conjugation. We give the first and second fundamental theorems for the polynomial invariants of this action. We translate the second fundamental theorem into the language of block trace identities, and show that all of them are consequences of the fundamental trace identities. The results give generators and relations for the polynomial invariants of representations of quivers

Weakly multiplicative coactions of quantized function algebras

Abstract A condition is identified which guarantees that the coinvariants of a coaction of a Hopf algebra on an algebra form a subalgebra, even though the coaction may fail to be an algebra homomorphism. A Hilbert Theorem (finite generation of the subalgebra of coinvariants) is obtained for such coactions of a cosemisimple Hopf algebra. This is applied for two coactions α,β : A → A ⊗ O , where A is the coordinate algebra of the quantum matrix space associated with the quantized coordinate algebra O of a classical group, and α, β are quantum analogues of the conjugation action on matrices. Provided that O is cosemisimple and coquasitriangular, the α-coinvariants and the β-coinvariants form two finitely generated, commutative, graded subalgebras of A , having the same Hilbert series. Consequently, the cocommutative elements and the S2-cocommutative elements in O form finitely generated subalgebras. A Hopf algebra monomorphism from the quantum general linear group to Laurent polynomials over the quantum special linear group is found and used to explain the strong relationship between the corepresentation (and coinvariant) theories of these quantum groups.

A Hilbert-Nagata theorem in noncommutative invariant theory

Nagata gave a fundamental sufficient condition on group actions on finitely generated commutative algebras for finite generation of the subal[gebra of invariants. In this paper we consider groups acting on noncommutative algebras over a field of characteristic zero. We characterize all the T-ideals of the free associative algebra such that the algebra of invariants in the co:rresponding relatively free algebra is finitely generated for any group action from the class of Nagata. In particular, in the case of unitary algebras this condition is equivalent to the nilpotency of the algebra in Lie sense. As a consequence we extend the Hilbert-Nagata theorem on finite generation of the algebra of invariants to any finitely generated associative algebra which is Lie nilpotent. We also prove that the Hilbert series of the algebra of invariants of a group acting on a relatively free algebra with a non-matrix polynomial identity is rational, if the action satisfies the condition of Nagata.

A Generalization of a theorem of chang

Szigeti, Tuza and Revesz have developed a method in [6] to obtain polynomial identities for the n×n matrix ring over a commutative ring starting from directed Eulerian graphs. These polynomials are called Euler-ian. In the first part of this paper we show some polynomials that are in the T-ideal generated by a certain set of Eulerian polynomials, hence we get some identities of the n×n matrices. This result is a generalization of a theorem of Chang [l]. After that, using this theorem, we show that any Eulerian identity arising from a graph which lias d-fold multiple edges follows from the standard identity of degree d

Goldie's theorems for involution rings

In this paper we shall formulate necessary and sufficient conditions for a semiprime ring to be both left and right Goldie in terms of the symmetric notion of biideal instead of one-sided ideals. We use this result to characterize semiprime Goldie rings with involution by conditions consistent with the notion of involution rings, that is, in terms of ascending chain condition for annihilator ∗-biideals and of maximum condition for ∗-biideal direct sums.

The Noether numbers and the Davenport constants of the groups of order less than 32

Abstract The computation of the Noether numbers of all groups of order less than thirty-two is completed. It turns out that for these groups in non-modular characteristic the Noether number is attained on a multiplicity free representation, it is strictly monotone on subgroups and factor groups, and it does not depend on the characteristic. Algorithms are developed and used to determine the small and large Davenport constants of these groups. For each of these groups the Noether number is greater than the small Davenport constant, whereas the first example of a group whose Noether number exceeds the large Davenport constant is found, answering partially a question posed by Geroldinger and Grynkiewicz.