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

Venue: | J. Algebra |

Citations: | 4 - 4 self |

### BibTeX

@ARTICLE{Drensky_constantsof,

author = {Vesselin Drensky and C. K. Gupta},

title = {Constants of Weitzenböck derivations and invariants of unipotent transformations acting on relatively free algebras},

journal = {J. Algebra},

year = {},

pages = {393--428}

}

### OpenURL

### Abstract

Abstract. In commutative algebra, a Weitzenböck derivation is a nonzero triangular linear derivation of the polynomial algebra K[x1,..., xm] 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 GL2(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. 1.

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Citation Context ...algebra of invariants of a single unipotent transformation. (This contrasts to the counterexample of Nagata described above.) The original proof of Weitzenböck from 1932 was for K = C. Later Seshadri =-=[63]-=- found a proof for any field K of charactersitic 0. A simple proof for K = C using ideas from [63] has been recently given by Tyc [72]. To the best of our knowledge, no constructive proof, with effect... |

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Citation Context ...(n) )t n . Dicks and Formanek [15] proved also an analogue of the Molien formula for the Hilbert series of K〈X〉 G , |G| < ∞, which was generalized for compact groups G by Almkvist, Dicks and Formanek =-=[4]-=- (an analogue of the Molien-Weyl formula in classical invariant theory). In particular, Almkvist, Dicks and Formanek showed that the Hilbert series of the algebra of invariants K〈X〉 g is an algebraic ... |

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Citation Context ...e Algebras. By a theorem of Lane [53] and Kharchenko [45], the algebra of invariants K〈X〉 G is always a free algebra (independently of the properties of G ⊂ GLm). 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 Vm = ∑m j=1 Kxj as a group of scalar multiplications. This result was generalized for a m... |

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Citation Context ...ply directly [53] and [45] because the group is not linear.) Similar study of the algebra of constants in a very large class of (not only associative) algebras was performed by Gerritzen and Holtkamp =-=[41]-=- and Drensky and Holtkamp [27]. We shall finish the survey section with the following, probably folklore known lemma. Lemma 2.1. Let W be any variety of algebras and let F(W) be the relatively free al... |

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Citation Context ...rst to recognize the connection between the Hilbert 14-th problem and constants of derivations (but his derivations were not always locally nilpotent) and the counterexample of Daigle and Freudenburg =-=[12]-=- of a triangular (but not linear) derivation of K[x1, . . .,x5] with not finitely generated algebra of constants. For more counterexamples to the Hilbert 14-th problem we refer to the recent survey by... |

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Citation Context ...s isomorphic to the algebra generated by two generic 2 × 2 matrices x and y. So, the results are stated in the natural setup of the trace algebra. We start with the necessary background, see Formanek =-=[36]-=-, Alev and Le Bruyn [1], or Drensky and Gupta [26]. We consider the polynomial algebra in 8 variables Ω = K[xij, yij | i, j = 1, 2]. The algebra R of two generic 2 × 2 matrices ( ) ( ) x11 x12 y11 y12... |

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Citation Context ...G be a subgroup of GLm and let K[X] G = K[x1, . . .,xm] G be the algebra of G-invariants. The problem for finite generation of K[X] G was the main motivation for the famous Hilbert Fourteenth Problem =-=[44]-=-. The theorem of Emmy Noether [60] gives the finite generation of K[X] G for finite groups G. More generally, the Hilbert-Nagata theorem states the finite generation of K[X] G for reductive groups G, ... |

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Citation Context ...on of K[x1, . . .,xm] with the same algebra of constantsWEITZENBÖCK DERIVATIONS AND UNIPOTENT TRANSFORMATIONS 5 as δ and exp ∆ is an automorphism of K[x1, . . . , xm]. By the theorem of Martha Smith =-=[67]-=-, all such automorphisms are stably tame and become tame if extended to K[x1, . . . , xm, xm+1] by (exp ∆)(xm+1) = xm+1. The famous Nagata automorphism of K[x, y, z], see [59], also can be obtained in... |

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Citation Context ...s (K〈x, y〉/I) δ is spanned by the highest weight vectors of the GL2-irreducible components of K〈x, y〉/I. Remarks 4.7. 1. A direct proof of Theorem 4.6 can be obtained using the criterion of Koshlukov =-=[49]-=- which states: A multihomogeneous of degree λ = (λ1, . . . , λm) polynomial w(x1, . . .,xm) ∈ K〈x1, . . . , xm〉 is a highest weight vector of an irreducible GLm-submodule W(λ) of K〈x1, . . .,xm〉 if an... |

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Citation Context ..., the free Lie algebra Lm = L(X) and relatively free algebras Lm(V) in varieties of Lie algebras V. For more detailed exposition we refer to the surveys on noncommutative invariant theory by Formanek =-=[35]-=-, Drensky [22] and the survey on algorithmic methods for relatively free semigroups, groups and algebras by Kharlampovich and Sapir [47]. 2.2.1. Free Associative Algebras. By a theorem of Lane [53] an... |

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Citation Context ...le as K〈X〉/I ∼ = ∑ m(λ)W(λ). λ In the case of two variables the Schur functions have the following simple expression tλ1−λ2+1 λ2 S (λ1,λ2)(t1, t2) = (t1t2) λ 1 − t λ1−λ2+1 2 t1 − t2 Drensky and Genov =-=[25]-=- defined the multiplicity series of f(t1, t2) = ∑ m(λ)Sλ(t1, t2) as the formal power series λ M(f)(t, u) = ∑ λ m(λ)t λ1 u λ2 , or, if one introduces a new variable v = tu, as M ′ (f)(t, v) = ∑ m(λ)t λ... |

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Citation Context ...ors of K[x1, . . .,xm] δ is known for small m only. Tan [71] presented an algorithm for computing the generators of the algebra of constants of a basic derivation. It was generalized by van den Essen =-=[32]-=- for any locally nilpotent derivation assuming that the finite generation of the algebra of constants is known. The algorithm involves Gröbner bases techniques. Examples 3.4. We have selected few exam... |

7 |
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Citation Context ...e bimodule action of K[Y ] on M). Obviously W satisfies the metabelian identity and hence belongs to M. The following proposition is a partial case of the main result of Lewin [56], see also Umirbaev =-=[73]-=- for further applications of this construction to automorphisms of relatively free associative algebras. Proposition 6.1. The mapping ι : xj → yj + aj, j = 1, . . .,m, defines an embedding ι of Fm(M) ... |

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Citation Context ...nerated by m generic 2 × 2 traceless matrices y1, . . . , ym are [v 2 1, v2] = 0, where v1, v2 run on the set of all Lie elements in K〈y1, . . .,ym〉 which is a restatement of the theorem of Razmyslov =-=[62]-=- for the weak polynomial identities of M2(K). An explicitly written system of defining relations consists of [y 2 i , yj] = 0, i, j = 1, . . .,m, and the standard polynomials s4(yi1, yi2, yi3, yi4) = ... |

6 |
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Citation Context ...for ∆ = wδ the Nagata automorphism is ν = exp∆: ν(x) = x + (−2y) w + (−2z)w2 1! 2! = x − 2(xz + y2 )y − (xz + y 2 ) 2 z, ν(y) = y + z w 1! = y + (xz + y2 )z, ν(z) = z. Recently Shestakov and Umirbaev =-=[64]-=- proved that the Nagata automorphism is wild. It is interesting to mention that their approach is based on Poisson algebras and methods of noncommutative, and even nonassociative, algebras. There are ... |

6 |
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Citation Context ...by δ(x3) = x2, δ(x2) = x1, δ(x5) = x4, δ(x1) = δ(x4) = 0 (see [61], Example 6.8.5): K[x1, x2, x3, x4, x5] δ = K[x1, x4, x1x5 − x2x4, x 2 2 − 2x1x3, 2x3x 2 4 − 2x2x4x5 + x1x 2 5]. Remark 3.5. Springer =-=[70]-=- found a formula for the Hilbert series of the algebra of invariants of SL2(K) acting on the forms of degree d. This is equivalent to the description of the Hilbert series of the algebra of constants ... |

6 |
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Citation Context ...ar linear derivations of the polynomial algebra K[X] = K[x1, . . .,xm] are called Weitzenböck derivations. The classicalWEITZENBÖCK DERIVATIONS AND UNIPOTENT TRANSFORMATIONS 3 theorem of Weitzenböck =-=[75]-=- states that the algebra of constants of such a derivation is finitely generated. This algebra coincides with the algebra of invariants of a single unipotent transformation. In this paper we study the... |

5 |
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Citation Context ...s of SL2(K) acting on the forms of degree d. This is equivalent to the description of the Hilbert series of the algebra of constants of the basic Weitzenböck derivation of K[x1, . . .,xd+1]. Almkvist =-=[2, 3]-=- related these invariants with invariants of the modular action of a cyclic group of order p. 4. Lifting and Description of the Constants We need the following easy lemma. Lemma 4.1. Let G ⊂ H be grou... |

5 |
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Citation Context ...Drensky [19] or the book [23] for further application to the theory of PI-algebras. It is known, see Gerritzen [40], that in this case the algebra of constants is free, see also Drensky and Kasparian =-=[28]-=- for an explicit basis. (The freedom of the algebra of constants of the partial derivatives of K[X] does not follow immediately from the result of Lane [53] and Kharchenko [45]. The derivations ∂/∂xj ... |

5 |
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Citation Context ...th zero diagonal. Another important application of locally nilpotent derivations is the construction of candidates for wild automorphisms of polynomial algebras, see e.g. the survey of Drensky and Yu =-=[31]-=-. Typical example is the following. If δ is a Weitzenböck derivation of K[x1, . . .,xm] and 0 ̸= w ∈ K[x1, . . .,xm] δ , then ∆ = wδ is also a locally nilpotent derivation of K[x1, . . .,xm] with the ... |

5 | Noncommutative invariants of finite groups and Noetherian varieties - Kharchenko - 1984 |

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Citation Context ...algebra is equal to 3). .12 VESSELIN DRENSKY AND C. K. GUPTA Example 5.1. Let L2 be the variety of associative algebras defined by the identity [[x, y], z] = 0. By the theorem of Krakowski and Regev =-=[51]-=- L2 coincides with the variety generated by the infinite dimensional Grassmann algebra. The Sncocharacter sequence of L2 is equal to n∑ χn(L2) = χ (k,1n−k), n ≥ 1, k=1 see [51]. In virtue of the corre... |

5 |
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Citation Context ...nitely generated for the linear transformation g defined by g(x1) = −x1, g(x2) = x2. If we consider the finite generation of Fm(W) G for the class all reductive groups G, then the results of Vonessen =-=[74]-=-, Domokos and Drensky [17] give that Fm(W) G is finitely generated for all reductive G if and only if the finitely generated algebras in W are one-side noetherian. For unitary algebras this means that... |

4 |
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Citation Context ...e dimensional Grassmann algebra. The Sncocharacter sequence of L2 is equal to n∑ χn(L2) = χ (k,1n−k), n ≥ 1, k=1 see [51]. In virtue of the correspondence between cocharacters and Hilbert series, see =-=[6]-=- and [18] (or the book [23]) the Hilbert series of the relatively free algebra Fm(L2) is equal to H(Fm(L2), t1, . . . , tm) = 1 + ∑ It is well known that Fm(L2) has a basis x a1 1 m−1 ∑ k≥1 l=0 S (k,1... |

4 |
automorphisms of free P.I. algebras and some new identities
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(Show Context)
Citation Context ... performed in the opposite direction. The automorphisms of F2(var M2(K)) and of the trace algebra have been used to produce automorphisms of the polynomial algebra in five variables, see e.g. Bergman =-=[7]-=-, Alev and Le Bruyn [1], Drensky and Gupta [26]. Finally, we obtain also some partial results on relatively free Lie algebras.4 VESSELIN DRENSKY AND C. K. GUPTA 2. Survey 2.1. Motivation from Commuta... |

4 |
Commutative and noncommutative invariant theory. Math. and Education
- Drensky
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(Show Context)
Citation Context ... algebra Lm = L(X) and relatively free algebras Lm(V) in varieties of Lie algebras V. For more detailed exposition we refer to the surveys on noncommutative invariant theory by Formanek [35], Drensky =-=[22]-=- and the survey on algorithmic methods for relatively free semigroups, groups and algebras by Kharlampovich and Sapir [47]. 2.2.1. Free Associative Algebras. By a theorem of Lane [53] and Kharchenko [... |

4 |
New stably tame automorphisms of polynomial algebras
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Citation Context ...udenburg (obtained with his local slice construction [38]) and the automorphisms of Drensky and Gupta (obtained by methods of noncommutative algebra, [26]). Later, Drensky, van den Essen and Stefanov =-=[24]-=- have shown that the automorphisms from [26] also can be obtained in terms of locally nilpotent derivations and are stably tame. 2.2. Noncommutative Invariant Theory. An important part of noncommutati... |

4 |
New automorphisms of generic matrix algebras and polynomial algebras
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(Show Context)
Citation Context ...morphisms of F2(var M2(K)) and of the trace algebra have been used to produce automorphisms of the polynomial algebra in five variables, see e.g. Bergman [7], Alev and Le Bruyn [1], Drensky and Gupta =-=[26]-=-. Finally, we obtain also some partial results on relatively free Lie algebras.4 VESSELIN DRENSKY AND C. K. GUPTA 2. Survey 2.1. Motivation from Commutative Algebra. Locally nilpotent derivations of ... |

4 |
Weak polynomial identities for a vector space with a symmetric bilinear form
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Citation Context ...y 1, x0, y0, [x0, y0].22 VESSELIN DRENSKY AND C. K. GUPTA The defining relations of the algebra generated by the 2 ×2 traceless matrices x0 and y0 are [x 2 0, y0] = [y 2 0, x0] = 0, see e.g. [55] or =-=[29]-=- for the case of characteristic 0 and [50] for the case of an arbitrary infinite base field. More generally, the defining relations of the algebra generated by m generic 2 × 2 traceless matrices y1, .... |

4 |
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(Show Context)
Citation Context ... by R and ¯ C. It is well known that ¯ C is generated by y21 tr(x), tr(y), det(x), det(y), tr(xy) and is isomorphic to the polynomial algebra in five variables. Proposition 7.1. (Formanek, Halpin, Li =-=[37]-=-) The vector subspace of C consisting of all polynomials without constant term is a free ¯ C-module generated by [x, y] 2 . For our purposes it is more convenient to replace in T (as in [1]) the gener... |

4 |
Local slice construction in k[X
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(Show Context)
Citation Context ...exceptions of locally nilpotent derivations and their exponents which do not arrise immediately from triangular derivations: the derivations of Freudenburg (obtained with his local slice construction =-=[38]-=-) and the automorphisms of Drensky and Gupta (obtained by methods of noncommutative algebra, [26]). Later, Drensky, van den Essen and Stefanov [24] have shown that the automorphisms from [26] also can... |

4 |
Trace rings of generic 2×2
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Citation Context ...erated by 1, x0, y0, [x0, y0].22 VESSELIN DRENSKY AND C. K. GUPTA The defining relations of the algebra generated by the 2 ×2 traceless matrices x0 and y0 are [x 2 0, y0] = [y 2 0, x0] = 0, see e.g. =-=[55]-=- or [29] for the case of characteristic 0 and [50] for the case of an arbitrary infinite base field. More generally, the defining relations of the algebra generated by m generic 2 × 2 traceless matric... |

4 |
Polynomial Derivations and their
- Nowicki
- 1994
(Show Context)
Citation Context ...the polynomial algebra K[X] = K[x1, . . . , xm] have been studied for many decades and have had significant impact on different branches of algebra and invariant theory, see e.g. the books by Nowicki =-=[61]-=- and van den Essen [33]. Let G be a subgroup of GLm and let K[X] G = K[x1, . . .,xm] G be the algebra of G-invariants. The problem for finite generation of K[X] G was the main motivation for the famou... |

4 |
On Engel Lie algebras
- Zelmanov
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(Show Context)
Citation Context ...the form [x1, x2] · · · [x2k−1, x2k] = 0. Applying this result to F2(W) we obtain that F2(W) is solvable as a Lie algebra, and, by a theorem of Higgins [43] F2(W) is Lie nilpotent. (Actually Zelmanov =-=[76]-=- proved the stronger result that any Lie algebra over a field of characteristic zero satisfying the Engel identity is nilpotent.) By Drensky [19], for any nilpotent variety W, and for a fixed positive... |

3 |
Automorphisms of generic 2 by 2 matrices, in “Perspectives in Ring Theory
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(Show Context)
Citation Context ...ite direction. The automorphisms of F2(var M2(K)) and of the trace algebra have been used to produce automorphisms of the polynomial algebra in five variables, see e.g. Bergman [7], Alev and Le Bruyn =-=[1]-=-, Drensky and Gupta [26]. Finally, we obtain also some partial results on relatively free Lie algebras.4 VESSELIN DRENSKY AND C. K. GUPTA 2. Survey 2.1. Motivation from Commutative Algebra. Locally n... |

3 |
Rationality of Hilbert series of relatively free algebras
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(Show Context)
Citation Context ...gebra, expressing the Hilbert series of Fm(W) G in terms of the Hilbert series H(Fm(W), t1, . . .,tm). If G is finite, then H(Fm(W) G , t) involves the eigenvalues of all g ∈ G. By a theorem of Belov =-=[5]-=-, the Hilbert series of Fm(W) is always a rational function and this imlies that H(Fm(W) G , t) is also rational for G finite.WEITZENBÖCK DERIVATIONS AND UNIPOTENT TRANSFORMATIONS 7 For reductive G t... |

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A basis of a free metabelian associative algebra
- Bokut, Makar-Limanov
- 1991
(Show Context)
Citation Context ...nown that Fm(L2) has a basis x a1 1 m−1 ∑ k≥1 l=0 S (k,1 l )(t1, . . .,tm). · · · xam m [xi1, xi2] · · · [xi2p−1, xi2p], 1 ≤ i1 < i2 < · · · < i2p−1 < i2p ≤ m, see for example Bokut and Makar-Limanov =-=[8]-=- or the book [23]. The commutators [xi, xj] are in the centre of Fm(L2) and satisfy the relations [x σ(1), x σ(2)] · · · [x σ(2p−1), x σ(2p)] = (signσ)[x1, x2] · · · [x2p−1, x2p], σ ∈ S2p. Let m = 2. ... |

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The kernel of a derivation
- Derksen
- 1993
(Show Context)
Citation Context ...roup G. Today, most of the known counterexamples have been obtained (or can be obtained) as algebras of constants of some derivations. This includes the original counterexample of Nagata, see Derksen =-=[14]-=- who was the first to recognize the connection between the Hilbert 14-th problem and constants of derivations (but his derivations were not always locally nilpotent) and the counterexample of Daigle a... |