Noncommutative Invariant Theory Research Articles

非可換不変式論としての Capelli 型恒等式

非可換不変式論としての Capelli 型恒等式

Noncommutative Knörrer periodicity and noncommutative Kleinian singularities

We establish a version of Knörrer's Periodicity Theorem in the context of noncommutative invariant theory. Namely, let A be a left noetherian AS-regular algebra, let f be a normal and regular element of A of positive degree, and take B=A/(f). Then there exists a bijection between the set of isomorphism classes of indecomposable non-free maximal Cohen-Macaulay modules over B and those over (a noncommutative analog of) its second double branched cover (B#)#. Our results use and extend the study of twisted matrix factorizations, which was introduced by the first three authors with Cassidy. These results are applied to the noncommutative Kleinian singularities studied by the second and fourth authors with Chan and Zhang.

Noncommutative invariant theory of symplectic and orthogonal groups

We present a method for computing the Hilbert series of the algebra of invariants of the complex symplectic and orthogonal groups acting on graded noncommutative algebras with homogeneous components which are polynomial modules of the general linear group. We apply our method to compute the Hilbert series for different actions of the symplectic and orthogonal groups on the relatively free algebras of the varieties of associative algebras generated, respectively, by the Grassmann algebra and the algebra of 2×2 upper triangular matrices. These two varieties are remarkable with the property that they are the only minimal varieties of exponent 2.

NAKAYAMA AUTOMORPHISMS OF ORE EXTENSIONS OVER POLYNOMIAL ALGEBRAS

AbstractNakayama automorphisms play an important role in the fields of noncommutative algebraic geometry and noncommutative invariant theory. However, their computations are not easy in general. We compute the Nakayama automorphism ν of an Ore extension R[x; σ, δ] over a polynomial algebra R in n variables for an arbitrary n. The formula of ν is obtained explicitly. When σ is not the identity map, the invariant EG is also investigated in terms of Zhang’s twist, where G is a cyclic group sharing the same order with σ.

Rationality of Hilbert series in noncommutative invariant theory

It is a fundamental result in commutative algebra and invariant theory that a finitely generated graded module over a commutative finitely generated graded algebra has a rational Hilbert series, and consequently the Hilbert series of the algebra of polynomial invariants of a group of linear transformations is rational, whenever this algebra is finitely generated. This basic principle is applied here to prove rationality of Hilbert series of algebras of invariants that are neither commutative nor finitely generated. Our main focus is on linear groups acting on certain factor algebras of the tensor algebra that arise naturally in the theory of polynomial identities.

Some aspects of noncommutative invariant theory and the Noether’s problem

This survey discusses the classical Noether’s problem and presents some important cases where it has a positive solution. With the Weyl algebras taking place of the polynomial algebras, some aspects of noncommutative invariant theory are introduced: in particular, the noncommutative version of the Noether’s problem. Some known cases are discussed and some new results are presented. As an application one obtains a new proof of the Gelfand–Kirillov conjecture for $$gl_n$$ . There is a striking similarity between the cases with positive solution of the Noether’s problem for both commutative and noncommutative versions, and this leads to many conjectures.

Noncommutative complete intersections

Several generalizations of a commutative ring that is a graded complete intersection are proposed for a noncommutative graded k-algebra; these notions are justified by examples from noncommutative invariant theory.

Quantum binary polyhedral groups and their actions on quantum planes

Abstract We classify quantum analogues of actions of finite subgroups G of SL 2 ⁢ ( k ) ${\mathrm{SL}_{2}(k)}$ on commutative polynomial rings k ⁢ [ u , v ] ${k[u,v]}$ . More precisely, we produce a classification of pairs ( H , R ) ${(H,R)}$ where H is a finite-dimensional Hopf algebra that acts inner faithfully and preserves the grading of an Artin–Schelter regular algebra R of global dimension 2. Remarkably, the corresponding invariant rings R H ${R^{H}}$ share similar regularity and Gorenstein properties as the invariant rings k ⁢ [ u , v ] G ${k[u,v]^{G}}$ in the classical setting. We also present several questions and directions for expanding this work in noncommutative invariant theory.

A noncrossing basis for noncommutative invariants of [formula omitted

Noncommutative invariant theory is a generalization of the classical invariant theory of the action of SL ( 2 , C ) on binary forms. The dimensions of the spaces of invariant noncommutative polynomials coincide with the numbers of certain noncrossing partitions. We give an elementary combinatorial explanation of this fact by constructing a noncrossing basis of the homogeneous components. Using the theory of free stochastic measures this provides a combinatorial proof of the Molien–Weyl formula in this setting.

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.

Young diagrammatic methods in non-commutative invariant theory

In this paper we will study some aspects of non-commutative invariant theory. Let V be a finite-dimensional vector space over a field K of characteristic zero and letK[V] = K⊕V⊕S2(V)⊕…, andK′V› = K⊕V⊕⊕2(V)⊕⊕3V⊕&be respectively the symmetric algebra and the tensor algebra over V. Let G be a subgroup of GL(V). Then G acts on K[V] and K′V›. Much of this paper is devoted to the study of the (non-commutative) invariant ring K′V›G of G acting on K′V›.In the first part of this paper, we shall study the invariant ring in the following situation.

The noncommutative invariants of the finite subgroups of SL(2, [formula omitted]))

The commutative invariant theory of the finite subgroups of SL(2, C ) (the binary polyhedral groups) is classical; it was developed in the 19th century. In this article, we will be connected with the noncommutative invariant theory of these groups.