Invariants of the action of a semisimple finite-dimensional Hopf algebra on special algebras

In this paper we extend classical results of the invariant theory of finite groups to the action of a finite-dimensional semisimple Hopf algebra H on a special algebra A, which is homomorphically mapped onto a commutative integral domain, and the kernel of this map contains no nonzero H-stable ideals. We prove that the algebra A is finitely generated as a module over a subalgebra of invariants, and the latter is finitely generated as a k-algebra. We give a counterexample to the finite generation of a non-semisimple Hopf algebra.     

Actions of the dipper-donkin quantization GL 2 on the clifford algebra C(l,3)

Following the method already developed for studying the actions of GL_q (2,C) on the Clifford algebra C(l,3) and its quantum invariants [1], we study the action on C(l, 3) of the quantum GL ₂ constructed by Dipper and Donkin [2]. We are able of proving that there exits only two non-equivalent cases of actions with nontrivial “perturbation” [1]. The spaces of invariants are trivial in both cases.

We also prove that each irreducible finite dimensional algebra representation of the quantum GL ₂ q^m ≠1, is one dimensional.

By studying the cases with zero “perturbation” we find that the cases with nonzero “perturbation” are the only ones with maximal possible dimension for the operator algebra ?.     

On locally trivial G a -actions

If the additive group of complex numbers acts algebraically on a normal affine variety, then the associated ring of invariants need not be finitely generated, but is an ideal transform of some normal affine algebra (Theorem 1). We investigate such normal affine algebras in the case of a locally trivial action on a factorial variety. If the variety is a complex affine space and the ring of invariants is isomorphic to a polynomial ring, then the action is conjugate to a translation (Theorem 3). Equivalently, ifC ⁿ , is the total space for a principalG _a -bundle over some open subset ofC ^n–1 then the bundle is trivial. On the other hand, there is a locally trivialG _a -action on a normal affine variety with nonfinitely generated ring of invariants (Theorem 2).Supported in part by NSA Grant No. MDA904-96-1-0069     

Multiplicative Invariants and Semigroup Algebras

Let G be a finite group acting by automorphism on a lattice A, and hence on the group algebra S=k[A]. The algebra of G-invariants in S is called an algebra of multiplicative invariants. We present an explicit version of a result of Farkas stating that multiplicative invariants of finite reflection groups are semigroup algebras.     

Nonlinear Approximation with Local Fourier Bases

It is shown that local Fourier bases are unconditional bases for the modulation spaces on R, including the Bessel potential spaces and the Segal algebra S ₀ . As a consequence, the abstract function spaces, that are defined by the approximation properties with respect to a local Fourier basis, are precisely the modulation space s. April 22, 1998. Date accepted: May 18, 1999.     

Complexity and rank of homogeneous spaces

We study an algebraic varieties with the action of a reductive group G. The relation is elucidated between the notions of complexity and rank of an arbitrary G-variety and the structure of stabilizers of general position of some actions of G itself and its Borel subgroup. The application of this theory to homogeneous spaces provides the explicit formulas for the rank and the complexity of quasiaffine G/H in terms of co-isotropy representation of H. The existence of Cartan subspace (and hence the freeness of algebra of invariants) for co-isotropy representation of a connected observable spherical subgroup H is proved.     

Analogue of the Cayley–Sylvester formula and the Poincaré series for the algebra of invariants of n-ary form

The number of linearly independent homogeneous invariants of degree k for the n-ary form of order d is calculated. Also, a formula for the Poincaré series for the algebra of invariants of the n-ary form is found.     

Quantum affine Gelfand�CTsetlin bases and quantum toroidal algebra via K-theory of affine Laumon spaces

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GL _n . We construct the action of the quantum loop algebra U_v(L\mathfraksl_n){U_v({\bf L}\mathfrak{sl}_n)} in the K-theory of Laumon spaces by certain natural correspondences. Also we construct the action of the quantum toroidal algebra ü _v ([^(\mathfraksl)]_n){(\widehat{\mathfrak{sl}}_n)} in the K-theory of the affine version of Laumon spaces.     

Exceptional Components

For a wild acyclic quiver Q, Kerner introduced the notion of exceptional components for the Auslander–Reiten quiver of Q over an algebraically closed field k. He then defined two invariants for these exceptional components and asked whether these invariants coincide for each exceptional component. He showed that for each exceptional component there is a related hereditary factor algebra B of the path algebra kQ. He then proved that B is tame or representation finite and asked whether the representation finite case does occur, at all. We will answer both of Kerner's questions.     

On mod <Emphasis Type="Italic">p</Emphasis> properties of Siegel modular forms

We prove two results on mod p properties of Siegel modular forms. First, we use theta series in order to construct of a Siegel modular form of weight p−1 which is congruent to 1 mod p. Second, we define a theta operator on q-expansions and show that the algebra of Siegel modular forms mod p is stable under , by exploiting the relation between and generalized Rankin-Cohen brackets.     

Analog of the Cayley–Sylvester formula and the Poincaré series for an algebra of invariants of ternary form

An explicit formula is obtained for the number ν _d (n) of linearly independent homogeneous invariants of degree n of a ternary form of order d. A formula for the Poincaré series of the algebra of invariants of the ternary form is also deduced.     

On the ring of invariants ofF_{2^n }^*

In [1] the first and last authors studied a decomposition ofH ^*(R P ^∞×…×R P ^∞;F ₂) into modules over the Steenrod algebra obtained from an action of the cyclic group . Here a minimal set of generators for the ring of invariants is characterized and counted by analyzing the associated ring of Laurent polynomials. A structure theorem for the ring of invariant Laurent polynomials is given and a ‘destabilisation cancels localisation’ theorem is obtained. The authors gratefully acknowledge the support of NSERC. 1980 Mathematics Subject classification, 13F20, 55. Keywords: Invariant theory, Steenrod algebra.     

Imbedding a Filial Ring with Identity

We obtain that a uniformly bounded simple module over a high rank Virasoro algebra is a module of the intermediate series, and that a simple module with finite dimensional weight spaces is either a module of the intermediate series or a so-called GHW module.

     

Moduli for pairs of elliptic curves¶with isomorphic N-torsion

We study the moduli surface for pairs of elliptic curves together with an isomorphism between their N-torsion groups. The Weil pairing gives a “determinant” map from this moduli surface to (Z/N Z)^*; its fibers are the components of the surface. We define spaces of modular forms on these components and Hecke correspondences between them, and study how those spaces of modular forms behave as modules for the Hecke algebra. We discover that the component with determinant −1 is somehow the “dominant” one; we characterize the difference between its spaces of modular forms and the spaces of modular forms on the other components using forms with complex multiplication. In addition, we prove Atkin–Lehner-style results about these spaces of modular forms. Finally, we show some simplifications that arise when N is prime, including a complete determination of such CM-forms, and give numerical examples. Received: 20 September 2000 / Revised version: 7 February 2001     

The Cohen–Macaulay property of separating invariants of finite groups

In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply the ring of invariants is non-Cohen–Macaulay actually imply that no graded separating algebra is Cohen–Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen–Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen–Macaulay graded separating algebra implies the group is generated by bireections. Additionally, we give an example which shows that Cohen–Macaulay separating algebras can occur when the ring of invariants is not Cohen–Macaulay.     

Yang–Baxter Operators Arising from Algebra Structures and the Alexander Polynomial of Knots

ABSTRACT

In this paper, we consider the problem of constructing knot invariants from Yang–Baxter operators associated to algebra structures. We first compute the enhancements of these operators. Then, we conclude that Turaev's procedure to derive knot invariants from these enhanced operators, as modified by Murakami, invariably produces the Alexander polynomial of knots.     

Orthogonal invariants of skew-symmetric matrices

The algebra of invariants of d-tuples of n?×?n skew-symmetric matrices under the action of the orthogonal group by simultaneous conjugation is considered over an infinite field of characteristic different from two. For n?=?3 and d?>?0 a minimal set of generators is established. A homogeneous system of parameters (i.e. an algebraically independent set such that the algebra of invariants is a finitely generated free module over subalgebra generated by this set) is described for n?=?3 and d?>?0, for n?=?4 and d?=?2,?3, for n?=?5 and d?=?2.     

Representation-theoretic rank and double hopf algebras

We compute the intrinsic category-theoretic rank: for quasitriangular Hopf algebras in the case of the quantum double Hopf algebra of Drinfeld. The result is closely related ti recent Hopf algebra invariants of Larson and Radford.     

Relative invariants for representations of finite dimensional algebras

 Let M be a finite dimensional module over a finite dimensional basic K-algebra Λ, where K is an algebraically closed field. We associate with M a weight θ ^M (i.e. an element of the dual of the Grothendieck group of mod-Λ) in module theoretic terms. Let β be a dimension vector with θ ^M (β)=0. We generalize a construction of relative invariants of quivers due to Schofield [S] and define a relative invariant polynomial function d ^M _β on the variety of modules of dimension vector β, such that d ^M _β (N) = 0 for some module N if and only if there is a nonzero morphism from M to N. Assuming char (K) = 0, we conclude from the main result of Schofield-Van den Bergh [SV] that relative invariants of this form span all the spaces of relative invariants. To get algebra generators of the algebra of semi-invariants it is sufficient to take the d ^M _β with M indecomposable. Received: 31 July 2001