Structure of Zariski-closed algebras

Belov-Kanel, Alexei; Rowen, Louis; Vishne, Uzi

doi:10.1090/s0002-9947-10-04993-7

Cited by 15 publications

(20 citation statements)

References 9 publications

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“…By the Braun-Kemer-Razmyslov theorem, J ℓ 0 = 0 for some ℓ. Hence x 1 · · · x ℓ is an identity of J 0 , and thus of its Zariski closure, which is J by [9,Proposition 3.21]. In other words, J ℓ = 0.…”

Section: Review Of Zariski Closed Algebrasmentioning

confidence: 96%

“…In this section we review Zariski closed algebras, studied in [9], which form the foundation for the study of representable PI-rings. Suppose F ⊆ K is an algebraically closed field.…”

Section: Review Of Zariski Closed Algebrasmentioning

confidence: 99%

“…When u = 0, i.e., when this automorphism is the identity, we say there is no Frobenius twist, and call the gluing identical gluing. [9,Theorem 5.14] (given below as Theorem 2.12) describes all gluing along the diagonal blocks.…”

Section: Introductionmentioning

confidence: 99%

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Full quivers of representations of algebras

Belov-Kanel¹,

Rowen²,

Vishne³

2012

Trans. Amer. Math. Soc.

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We introduce the notion of the full quiver of a representation of an algebra, which is a cover of the (classical) quiver, but which captures properties of the representation itself. Gluing of vertices and of arrows enables one to study subtle combinatorial aspects of algebras which are lost in the classical quiver. Full quivers of representations apply especially well to \Zcd\ algebras, which have properties very like those of finite dimensional algebras over fields. By choosing the representation appropriately, one can restrict the gluing to two main types: {\it Frobenius} (along the diagonal) and, more generally {\it proportional} Frobenius gluing (above the diagonal), and our main result is that any representable algebra has a faithful representation described completely by such a full quiver. Further reductions are considered, which bear on the polynomial identities.Comment: 44 p

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Section: Review Of Zariski Closed Algebrasmentioning

confidence: 96%

“…In this section we review Zariski closed algebras, studied in [9], which form the foundation for the study of representable PI-rings. Suppose F ⊆ K is an algebraically closed field.…”

Section: Review Of Zariski Closed Algebrasmentioning

confidence: 99%

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Full quivers of representations of algebras

Belov-Kanel¹,

Rowen²,

Vishne³

2012

Trans. Amer. Math. Soc.

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“…It implies Sμ(A) = Sμ(Γ), that contradicts to the properties of A. Thereforeg j | (17) = 0. Thus in any case hμ| (17)…”

Section: Lemma 43 Letmentioning

confidence: 99%

“…If a graded pure form polynomials j in (16) depends essentially on Z then γ−1+μ m=1 ϑ∈ G A Z ϑ m s jgj | (17) = 0, since the forms are zero on radical elements (see (12)). Ifs j does not depend on Z then (17) = 0 in A i then one of the multihomogeneous on degrees components ofg j is aμ-boundary polynomial for A i . And it is not aμ-boundary polynomial for Γ, because it belongs to Γ.…”

Section: Lemma 43 Letmentioning

confidence: 99%

Finitely generated algebras with involution and their identities

Sviridova

2013

Journal of Algebra

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We consider associative algebras with involution graded by a finite abelian group G over a field of characteristic zero. Suppose that the involution is compatible with the grading. We represent conditions permitting PI-representability of such algebras. Particularly, it is proved that a finitely generated (Z/qZ)graded associative PI-algebra with involution satisfies exactly the same graded identities with involution as some finite dimensional (Z/qZ)-graded algebra with involution for any prime q or q = 4. This is an analogue of the theorem of A.Kemer for ordinary identities [31], and an extension of the result of the author for identities with involution [42]. The similar results were proved also for graded identities [1], [41]. MSC: Primary 16R50; Secondary 16W20, 16W50, 16W10The first results on PI-representability of associative algebras belong to A.Kemer. He proves that any finitely generated associative (Z/2Z)-graded PI-algebra over a field of characteristic zero satisfies the same (Z/2Z)-graded identities as some finite dimensional (Z/2Z)-graded algebra over the same field [31], [34]. Later, he proves also that a finitely generated associative PI-algebra over an infinite field satisfies the same ordinary (non-graded) polynomial identities as some finite dimensional algebra [32]. PI-representability of finitely generated associative algebras over a commutative associative Noetherian ring with respect to ordinary polynomial identities was studied in [14]- [22].The series of results were obtained also for graded identities and identities with involution of associative algebras over a field of characteristic zero. If G is a finite abelian group then a finitely generated G-graded associative PI-algebra over an algebraically closed field of characteristic zero satisfies the same graded identities as some finite dimensional G-graded algebra over the same field [41]. For more general case of a finite (not necessarily abelian) group G it was proved that a finitely generated G-graded associative PI-algebra over a field of characteristic zero satisfies the same graded identities as some finite dimensional G-graded algebra over some extension of the base field [1]. As the direct consequences of [1], [41] we have also the similar results for G-identities if G is a finite abelian group of automorphisms of an associative algebra.Recently, PI-representability was proved also for identities with involution [42]. A finitely generated associative PI-algebra with involution over a field of characteristic zero satisfies the same identities with involution as some finite dimensional algebra with involution over the same field.The special interest to graded identities in the case of characteristic zero is explained by the super-trick and relation with the Specht problem [31]. Therefore the problem of PI-representability of graded algebras with involution is of current interest also.We consider associative algebras over a field F of characteristic zero. Further they will be called algebras.An F -algebra A is graded by a group G (G...

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The Jacobian Conjecture, Together with Specht and Burnside-Type Problems

Kanel-Belov

Bokut

Rowen

et al. 2014

Springer Proceedings in Mathematics &Amp; Statistics

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Dedicated to the memory of A.V. Yagzhev Abstract. We explore an approach to the celebrated Jacobian Conjecture by means of identities of algebras, initiated by the brilliant deceased mathematician, Alexander Vladimirovich Yagzhev , only partially published. This approach also indicates some very close connections between mathematical physics, universal algebra and automorphisms of polynomial algebras.

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Structure of Zariski-closed algebras

Cited by 15 publications

References 9 publications

Full quivers of representations of algebras

Full quivers of representations of algebras

Finitely generated algebras with involution and their identities

The Jacobian Conjecture, Together with Specht and Burnside-Type Problems

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