Invariant theory, old and new

“…Now we give a series of statements which are straightforward generalizations of statements used in Nagata's proof of the above theorem. Their commutative version can be found for example in [4,Chapter 3]. We assume that R is a finitely generated (not necessarily commutative or unitary) K-algebra and G acts on R in such a way that the conditions (i) and (ii) hold.…”

Section: Preliminariesmentioning

confidence: 99%

“…Nagata (see [4,Chapter 3]) found a fundamental condition assuring the finite generation of K[V ] G . This condition applies to a large class of group actions, including all rational representations of reductive linear algebraic groups.…”

Section: Introductionmentioning

confidence: 99%

“…Again, we give a complete description of the T-ideals of K + V such that finite generation of the invariant algebra in the corresponding non-unitary relatively free algebra holds for any group action satisfying the condition of Nagata. These Tideals are characterized by the property that they are contained neither in the square of the commutator ideal nor in the T-ideal generated by the polynomial 4 . In particular, we obtain examples of non-Noetherian varieties, where the Hilbert-Nagata theorem holds.…”

Section: Introductionmentioning

confidence: 99%

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A Hilbert-Nagata theorem in noncommutative invariant theory

Domokos¹,

Drensky²

1998

Trans. Amer. Math. Soc.

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Abstract. Nagata gave a fundamental sufficient condition on group actions on finitely generated commutative algebras for finite generation of the subalgebra 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 corresponding 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.

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“…for all x, y E V, and where / denotes the identity transformation of V. p is a %(«)-concomitant [3]. Further we define a second %(n)-concomitant p* which, together with p and J, suffices to write down all %(n)-concomitants of 61(F) in A2(F) following the methods of [15].…”

mentioning

confidence: 99%

Curvature Tensors on Almost Hermitian Manifolds

Tricerri¹,

Vanhecke²

1981

Transactions of the American Mathematical Society

143

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Abstract.A complete decomposition of the space of curvature tensors over a Hermitian vector space into irreducible factors under the action of the unitary group is given. The dimensions of the factors, the projections, their norms and the quadratic invariants of a curvature tensor are determined. Several applications for almost Hermitian manifolds are given. Conformal invariants are considered and a general Bochner curvature tensor is introduced and shown to be a conformal invariant. Finally curvature tensors on four-dimensional manifolds are studied in detail.1. Introduction. Let (V, g) be an n-dimensional real vector space with positive definite inner product g and denote by 61 (V) the subspace of V* <8> V* <8> V* <8> V* consisting of all tensors having the same symmetries as the curvature tensor of a Riemannian manifold, including the first Bianchi identity. In a well-known paper [21] Singer and Thorpe considered 61(F) (in particular for n = 4) and gave a geometrical useful description of the splitting of 61(F) under the action of &(n) into three components. This was also studied by Nomizu [18] for generalized curvature tensor fields.A similar decomposition was given in [16], [17] and [22] when V is a 2«-dimensional real vector space endowed with a complex structure J compatible with a positive definite inner product g and for the subspace %(V) of 61(F) consisting of

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Invariant theory, old and new

Cited by 118 publications

References 4 publications

Representation of spectra of algebras of block-symmetric analytic functions of bounded type

Representation of spectra of algebras of block-symmetric analytic functions of bounded type

A Hilbert-Nagata theorem in noncommutative invariant theory

Curvature Tensors on Almost Hermitian Manifolds

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