Noncommutative invariants of reductive groups

Koryukin, A. N.

doi:10.1007/bf02071789

Cited by 10 publications

(9 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By the theorem of Dicks and Formanek [15] and Kharchenko [45], if G is finite, then K X G is finitely generated if and only if G is cyclic and acts on V m = m j =1 Kx j as a group of scalar multiplications. This result was generalized for a much larger class of groups by Koryukin [48] who also established a finite generation of K X G if we equip it with a proper action of the symmetric group.…”

Section: Free Associative Algebrasmentioning

confidence: 82%

Constants of Weitzenböck derivations and invariants of unipotent transformations acting on relatively free algebras

Drensky

Gupta

2005

Journal of Algebra

View full text Add to dashboard Cite

In commutative algebra, a Weitzenböck derivation is a nonzero triangular linear derivation of the polynomial algebra K[x 1 , . . . , x m ] in several variables over a field K of characteristic 0. The classical theorem of Weitzenböck states that the algebra of constants is finitely generated. (This algebra coincides with the algebra of invariants of a single unipotent transformation.) In this paper we study the problem of finite generation of the algebras of constants of triangular linear derivations of finitely generated (not necessarily commutative or associative) algebras over K assuming that the algebras are free in some sense (in most of the cases relatively free algebras in varieties of associative or Lie algebras). In this case the algebra of constants also coincides with the algebra of invariants of some unipotent transformation.The main results are the following: (1) We show that the subalgebra of constants of a factor algebra can be lifted to the subalgebra of constants.(2) For all varieties of associative algebras which are not nilpotent in Lie sense the subalgebras of constants of the relatively free algebras of rank 2 are not finitely generated. (3) We describe the generators of the subalgebra of constants for all factor algebras K x, y /I modulo a GL 2 (K)-invariant ideal I . (4) Applying known results from commutative algebra, we construct classes of automorphisms of the algebra generated by two generic 2 × 2 matrices. We obtain also some partial results on relatively free Lie algebras.

show abstract

Section: Free Associative Algebrasmentioning

confidence: 82%

Constants of Weitzenböck derivations and invariants of unipotent transformations acting on relatively free algebras

Drensky

Gupta

2005

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…But we have something that is almost as good: the algebra of noncommutative invariants is finitely generated if we allow permutations of the factors in homogeneous polynomials. This has been proved by Koryukin [14]. Hence, it should be interesting to study the invariant algebras taking into account this new structure (which is degenerate in the commutative case).…”

Section: Forewordmentioning

confidence: 83%

“…If B is finitely generated as S-algebra, it does not have to be finitely generated as an algebra. For more information on S-algebras, see Koryukin'spaper [14]. We now concentrate onthe tensor algebra T(R'~)~-k (a0 ..... a4).…”

Section: S-algebrasmentioning

confidence: 99%

Noncommutative classical invariant theory

Tambour

1991

Ark. Mat.

View full text Add to dashboard Cite

In this thesis, we consider some aspects of noncommutative classical invariant theory, i.e., noncommutative invariants of the classical group SL(2, k). We develop a symbolic method for invariants and covariants, and we use the method to compule some invariant algebras. The subspace ar~ of the noncommutative invariant algebra J'~ consisting of homogeneous elements of degree m has the structure of a module over the symmetric group S,,. We find the explicit decomposition into irreducible modules. As a consequence, we obtain the Hilbert series of the commutative classical invariant algebras. The Cayley---Sylvester theorem and the Hermite reciprocity law are studied in some detail. We consider a new power series //(I'd, t) whose coefficients are the number of irreducible S,,-modules in the decomposition of J'~, and show that it is rational. Finally, we develop some analogues of all this for covariants.

show abstract

“…The result of Dicks and Formanek [5] and Kharchenko [12] gives that the algebra of invariants K X n G is finitely generated if and only if G is a cyclic group acting by scalar multiplication. On the other hand, Koryukin [13] studied K X n G with the additional action of the symmetric group S d permuting the positions of the variables in the homogeneous component of degree d of K X n :…”

Section: Theorem 1 the Algebra K[x N ]mentioning

confidence: 99%

Symmetric Polynomials in the Free Metabelian Lie Algebras

2020

View full text Add to dashboard Cite

Let K[Xn] be the commutative polynomial algebra in the variables Xn = {x1,. .. , xn} over a field K of characteristic zero. A theorem from undergraduate course of algebra states that the algebra K[Xn] Sn of symmetric polynomials is generated by the elementary symmetric polynomials which are algebraically independent over K. In the present paper, we study a noncommutative and nonassociative analogue of the algebra K[Xn] Sn replacing K[Xn] with the free metabelian Lie algebra Fn of rank n ≥ 2 over K. It is known that the algebra F Sn n is not finitely generated, but its ideal (F n) Sn consisting of the elements of F Sn n in the commutator ideal F n of Fn is a finitely generated K[Xn] Sn-module. In our main result, we describe the generators of the K[Xn] Sn-module (F n) Sn which gives the complete description of the algebra F Sn n .

show abstract

Noncommutative invariants of reductive groups

Cited by 10 publications

References 3 publications

Constants of Weitzenböck derivations and invariants of unipotent transformations acting on relatively free algebras

Constants of Weitzenböck derivations and invariants of unipotent transformations acting on relatively free algebras

Noncommutative classical invariant theory

Symmetric Polynomials in the Free Metabelian Lie Algebras

Contact Info

Product

Resources

About