Subring Of Invariants Research Articles

The construction of modular invariants

We use the k [ V ] k[V] -module generator of the dual module of the polynomial ring k [ V ] k[V] over its subring of invariants of a finite group to construct modular invariants and show that it behaves better than the transfer homomorphism.

Galois bimodules and integrality of PI comodule algebras over invariants

Let A be a comodule algebra for a finite dimensional Hopf algebra K over an algebraically closed field k , and let A^K be the subalgebra of invariants. Let Z be a central subalgebra in A , which is a domain with quotient field Q . Assume that Q\otimes_Z A is a central simple algebra over Q , and either A is a finitely generated torsion-free Z -module and Z is integrally closed in Q , or A is a finite projective Z -module. Then we show that A and Z are integral over the subring of central invariants Z\cap A^K . More generally, we show that this statement is valid under the same assumptions if Z is a reduced algebra with quotient ring Q ,and Q\otimes_Z A is a semisimple algebra with center Q . In particular, the statement holds for a coaction of K on a prime PI algebra A whose center Z is an integrally closed finitely generated domain over k . For the proof, we develop a theory of Galois bimodules over semisimple algebras finite over the center.

Affine Actions of Uq(sl(2)) on Polynomial Rings

AbstractWe classify the affine actions of Uq(sl(2)) on commutative polynomial rings in m ≥ 1 variables. We show that, up to scalar multiplication, there are two possible actions. In addition, for each action, the subring of invariants is a polynomial ring in either m or m−1 variables, depending upon whether q is or is not a root of 1.

GK DIMENSION AND LOCALLY NILPOTENT SKEW DERIVATIONS

AbstractLet A be a domain over an algebraically closed field with Gelfand–Kirillov dimension in the interval [2,3). We prove that if A has two locally nilpotent skew derivations satisfying some natural conditions, then A must be one of five algebras. All five algebras are Noetherian, finitely generated, and have Gelfand–Kirillov dimension equal to 2. We also obtain some results comparing the Gelfand–Kirillov dimension of an algebra to its subring of invariants under a locally nilpotent skew derivation.

The action of the special orthogonal group on planar vectors: integrity bases via a generalization of the symbolic interpretation of Molien functions

The present article completes the mathematical description initiated in the paper by Dhont and Zhilinskií (2013 The action of the orthogonal group on planar vectors: invariants, covariants and syzygies J. Phys. A: Math. Theor. 46 455202) of the algebraic structures that emerge from the symmetry-adapted polynomials in the coordinates of n planar vectors under the action of the SO(2) group. The set of -covariant polynomials contains all the polynomials that transform according to the weight of SO(2) and is a free module for but a non-free module for . The sum of the rational functions of the Molien function for -covariants describes the decomposition of the ring of invariants or the module of -covariants as a direct sum of submodules. A method for extracting the generating function for -covariants from the comprehensive generating function for all polynomials is introduced. The approach allows the direct construction of the integrity basis for the module of -covariants decomposed as a direct sum of submodules and gives insight into the expressions for the Molien functions found in our earlier paper. In particular, a generalized symbolic interpretation in terms of the integrity basis of a rational function is discussed, where the requirement of associating the different terms in the numerator of one rational function with the same subring of invariants is relaxed.

Invariants of Algebraic Automorphisms

Let R be a prime ring with extended centroid C and let σ be a C-algebraic automorphism of R. We let \(R^{(\sigma)}\mathop{=}\limits^{\rm def.}\{x\in R\mid \sigma(x)=x\}\), the subring of invariants of σ in R, and let Out-deg(σ) and Inn-deg(σ) denote the outer and inner degrees of σ, respectively. In the paper we first prove the nilpotence of the prime radical of R(σ) with a bound and characterize the semiprimeness and primeness of R(σ). Moreover, we show that if R(σ) is a prime PI-ring, then PI-deg(R) = PI-deg(R(σ)) × Inn-deg(σ).

Gorenstein subrings of invariants under Hopf algebra actions

This paper concerns conditions on the action of a finite dimensional semisimple Hopf algebra on an Artin–Schelter regular algebra that force the subring of invariants to satisfy the Artin–Schelter Gorenstein condition. Classical results such as Watanabe's Theorem and Stanley's Theorem are extended from the case of a group action to the context of a Hopf algebra action. A Hopf algebra version of the homological determinant is introduced, and it becomes an important tool in the generalization from group actions to Hopf algebra actions.

Obtaining the one-holed torus from pants: Duality in anSL(3,ℂ)-character variety

The SL(3,C)-representation variety R of a free group F arises naturally by considering surface group representations for a surface with boundary. There is a SL(3,C)-action on the coordinate ring of R. The geometric points of the subring of invariants of this action is an affine variety X. The points of X parametrize isomorphism classes of completely reducible representations. The coordinate ring C[X] is a complex Poisson algebra with respect to a presentation of F imposed by the surface. In previous work, we have worked out the bracket on all generators when the surface is a three-holed sphere and when the surface is a one-holed torus. In this paper, we show how the symplectic leaves corresponding to these two different Poisson structures on X relate to each other. In particular, they are symplectically dual at a generic point. Moreover, the topological gluing map which turns the three-holed sphere into the one-holed torus induces a rank preserving Poisson map on C[X].

Poisson geometry of 𝑆𝐿(3,ℂ)-character varieties relative to a surface with boundary

TheSL(3,C)\mathrm {SL}(3,\mathbb {C})-representation varietyR\mathfrak {R}of a free groupFr\mathtt {F}_rarises naturally by considering surface group representations for a surface with boundary. There is anSL(3,C)\mathrm {SL}(3,\mathbb {C})-action on the coordinate ring ofR\mathfrak {R}by conjugation. The geometric points of the subring of invariants of this action is an affine varietyX\mathfrak {X}. The points ofX\mathfrak {X}parametrize isomorphism classes of completely reducible representations. We show the coordinate ringC[X]\mathbb {C}[\mathfrak {X}]is a complex Poisson algebra with respect to a presentation ofFr\mathtt {F}_rimposed by the surface. Lastly, we work out the bracket on all generators when the surface is a three-holed sphere or a one-holed torus.

Maximal Crossed Product Orders Over Discrete Valuation Rings

The problem of determining when a (classical) crossed product T = S f *G of a finite group G over a discrete valuation ring S is a maximal order, was answered in the 1960s for the case where S is tamely ramified over the subring of invariants S G . The answer was given in terms of the conductor subgroup (with respect to f) of the inertia. In this article we solve this problem in general when S/S G is residually separable. We show that the maximal order property entails a restrictive structure on the subcrossed product graded by the inertia subgroup. In particular, the inertia is abelian. Using this structure, one is able to extend the notion of the conductor. As in the tame case, the order of the conductor is equal to the number of maximal two-sided ideals of T and hence to the number of maximal orders containing T in its quotient ring. Consequently, T is a maximal order if and only if the conductor subgroup is trivial.

Actions of Lie Superalgebras on Reduced Rings

In this paper, we look at the question of whether the subring of invariants is always nontrivial when a finite dimensional Hopf algebra acts on a reduced ring. Affirmative answers where given by Kharchenko for group algebras and by Beidar and Grzeszczuk for finite dimensional restricted Lie algebras. Our main result is

Generators, relations and symmetries in pairs of [formula omitted] unimodular matrices

Denote the free group on two letters by F 2 and the SL ( 3 , C ) -representation variety of F 2 by R = Hom ( F 2 , SL ( 3 , C ) ) . There is a SL ( 3 , C ) -action on the coordinate ring of R , and the geometric points of the subring of invariants is an affine variety X . We determine explicit minimal generators and defining relations for the subring of invariants and show X is a degree 6 hyper-surface in C 9 mapping onto C 8 . Our choice of generators exhibit Out ( F 2 ) symmetries which allow for a succinct expression of the defining relations.

Differents in modular invariant theory

Let $A=k[x_1,\ldots,x_n]$ be a graded polynomial algebra over a field k, such that each variable is homogeneous of positive degree. No restrictions are made with respect to the field. Let the finite group G act on A by graded algebra automorphisms and denote the subalgebra of invariants by B. In this paper the various "different ideals" of the extension $B\subset A$ are studied that define the ramification locus. We prove, for example, that the subring of invariants is itself a polynomial ring if and only if the ramification locus is pure of height one. Here the ramification locus is defined by either the Kahler different, the Noether different or the Galois different. As a consequence we prove that the invariant ring is itself a polynomial ring if and only if there are invariants $f_1,\ldots, f_n$ whose Jacobian determinant does not vanish and is of degree δ, where δ is the degree of the Dedekind different. Using this criterion we give a quick proof of Serre's result that if the invariant ring is a polynomial algebra, then the group is generated by generalized reflections.

Intermediate moduli spaces of stable maps

We describe the Chow ring with rational coefficients of $\overline{M}_{0,1}(\mathbb{P}^n,d)$ as the subring of invariants of a ring $B^*(\overline{M}_{0,1}(\mathbb{P}^n,d);\mathbb{Q})$ , relative to the action of the group of symmetries Sd. We compute $B^*(\overline{M}_{0,1}(\mathbb{P}^n,d);\mathbb{Q})$ by following a sequence of intermediate spaces for $\overline{M}_{0,1}(\mathbb{P}^n,d)$ .

On Cohen–Macaulay Rings of Invariants

We investigate the transfer of the Cohen–Macaulay property from a commutative ring to a subring of invariants under the action of a finite group. Our point of view is ring theoretic and not a priori tailored to a particular type of group action. As an illustration, we discuss the special case of multiplicative actions, that is, actions on group algebras k[Zn] via an action on Zn.

Skew Derivations Whose Invariants Satisfy a Polynomial Identity

If σ is an automorphism and δ is a q-skew σ-derivation of a ring R, then the subring of invariants is the set R(δ)={r∈R|δ(r)=0}. The main result of this paper isTheorem.Let R be a prime algebra with a q-skew σ-derivation δ, where δ and σ are algebraic. If R(δ)satisfies a P.I., then R satisfies a P.I.If δ is separable, then we also obtain the following result:Theorem.Let δ be a separable q-skew σ-derivation of an algebra R, where δ and σ are algebraic.•(i)•(ii)∩ σWhen R is a domain, it is necessary to assume neither that σ is algebraic nor that δ is q-skew as we proveTheorem.If R is a domain with an algebraic σ-derivation δ such that R(δ) satisfies a P.I., then R also satisfies a P.I.

Skew Derivations with Central Invariants

If r is an automorphism and d is a r-derivation of a ring R, then the subring of invariants is the set R(d) fl† r ‘ R r d(r) fl 0·. The main result of this paper is ‘let R be a semiprime ring with an algebraic r-derivation d such that R(d) is central; then R is commutative’. This theorem generalizes results on the invariants of automorphisms and derivations and is proved by reducing down to the special cases of automorphisms and derivations.

Invariants of Uq( sℓ(2)) and q-skew derivations

If δ is a q- rmskew derivation of a ring R, then the subring of invariants is R (δ) − r ϵ R ¦ δ(r) = 0 . We prove Theorem. Let δ be a q-skew derivation which is algebraic in its action on the K-algebra R. If R is (σ, δ)-semiprime and I ≠ 0 is a (σ, δ)-stable ideal of R, then I ( δ) is a nonnilpotent ideal of R ( δ) . This result is used to examine the actions of the Hopf algebra H = U q ( sℓ(2)). We show, under certain natural hypotheses, that for any H-stable ideal I ≠ 0 of a semiprime ring, the invariants of I under the action of U q ( sℓ(2)) are nonnilpotent.

The Cohomological Dimension of a Resultant Variety in Prime Characteristicp

Ž . 2 2 Ž . Let F s, t s X s q 2 X st q X t 1 F i F 3 be a collection of bii i0 i1 i2 nary quadratic forms with indeterminate coefficients over a field k and let w x R s k X . The aim of this paper is to compute the cohomoi, j 1F iF 3, 0 F jF 2 logical dimension of the ideal I ; R generated by the following Sylvester Ž . Ž . Ž . Ž . resultants: Res F , F , Res F , F , Res F , F , and Res F q F , F . 1 2 1 3 2 3 1 2 3 The closed points of the affine variety defined by I correspond to coefficients of the three forms having a common zero. Thus, in the w x Ž terminology of L1 , I is a resultant system for F , F , and F compare 1 2 3 w x. with Theorem 1.b in L1 . In general, if F , . . . , F are binary forms of 1 s degrees d G d G ??? G d , one can show that a resultant system is 1 2 s s Ž . Ž w x. generated by at least 1 q Ý 1 q d elements Theorem 1 in L1 and ts3 t one would like to know if this lower bound is sharp. In our case, where s s 3 and d s d s d s 2, the question is whether 1 2 3 4Ž . I can be generated by three elements. This would imply that H R s 0 I and this is the motivation behind the main result of this paper. 4Ž . It turns out that it is possible to show that H R s 0 working in the I Ž . subring of joint invariants of G s GL 2, k acting on R via change of Ž . variables in s, t . This calculation is based on some recent results by Lyubeznik on F-modules, which include an algorithm for deciding whether local cohomology modules over regular rings of prime characteristic p vanish. To the best of the author’s knowledge, Theorem 9 in this paper is the first application of this algorithm. Yan has recently produced a 4Ž . w x topological argument showing that H R vanishes in characteristic 0 Y . I

Invariants of Domains under the Actions of Restricted Lie Algebras

The goal of this paper is to examine the relationship between a domain R and its subring of invariants R L , under the action of a finite-dimensional restricted Lie algebra L. We first show that there exists a nondegenerate ( R L , R L )-bimodule "trace-like" map g: R → R L and therefore I ∩ R L ≠ 0, for every ideal I ≠ 0 of R. Next, we show that there exists a right R L -module embedding φ of R into a finite direct sum of copies of R L . Using the maps g and φ, we obtain the following, which is a combination of several of the main results of this paper.