Polytopal Linear Groups

Bruns, Winfried; Gubeladze, Joseph

doi:10.1006/jabr.1998.7831

Cited by 25 publications

(55 citation statements)

References 8 publications

(6 reference statements)

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“…In Section 6 we discuss the class of segmentonomial ideals, that is, ideals generated by polynomials f whose Newton polytope has dimension ≤ 1. Section 4 contains a review of the results of [BG1].…”

Section: Conjecture B a Codimension 1 Retraction Factors Through Eitmentioning

confidence: 99%

“…It follows that there exists c ∈ N with c n = cn and c n+1 = c(n + 1), and furthermore the restrictions of ϕ n and ϕ n+1 differ by an automorphism of k [c n(n+1) [BG1].) Therefore, we can assume that the restrictions of ϕ n and ϕ n+1 coincide.…”

Section: Retracts Of Dimension Twomentioning

confidence: 99%

“…and k[P ] = k[S P ] is the semigroup algebra of S P with coefficients in k (in [BGT,BG1,BG2] we have always used the notation k[S P ]). Unless specified otherwise, we will identify the sets L P and E P .…”

Section: Introductionmentioning

confidence: 99%

“…Actually, one needs the extra condition of very ampleness for the underlying polytope to get such a homogeneous coordinate ring (see [BG1,§5] for details).…”

Section: Introductionmentioning

confidence: 99%

“…As we have shown in [BG1], there is a total analogy with the linear situation for the automorphism groups in Pol k (called polytopal linear groups in [BG1]): they are generated by elementary automorphisms (generalizing elementary matrices), toric automorphisms (generalizing diagonal invertible matrices) and automorphisms of the underlying polytope; moreover, there are normal forms for such representations of arbitrary automorphisms. (In [BG2] this analogy has been extended to automorphisms of so-called polyhedral algebras.…”

Section: Introductionmentioning

confidence: 99%

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Polytopal linear retractions

Bruns

Gubeladze²

2001

Trans. Amer. Math. Soc.

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Abstract. We investigate graded retracts of polytopal algebras (essentially the homogeneous rings of affine cones over projective toric varieties) as polytopal analogues of vector spaces. In many cases we show that these retracts are again polytopal algebras and that codimension 1 retractions factor through retractions preserving the semigroup structure. We expect that these results hold in general.This paper is a part of the project started by the authors in 1999, where we investigate the graded automorphism groups of polytopal algebras. Part of the motivation comes from the observation that there is a reasonable 'polytopal' generalization of linear algebra (and, subsequently, that of algebraic K-theory).

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Section: Conjecture B a Codimension 1 Retraction Factors Through Eitmentioning

confidence: 99%

Section: Retracts Of Dimension Twomentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Actually, one needs the extra condition of very ampleness for the underlying polytope to get such a homogeneous coordinate ring (see [BG1,§5] for details).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Polytopal linear retractions

Bruns

Gubeladze²

2001

Trans. Amer. Math. Soc.

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Twisted Forms of Toric Varieties

Duncan

2016

Transformation Groups

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We consider the set of forms of a toric variety over an arbitrary field: those varieties which become isomorphic to a toric variety after base field extension. In contrast to most previous work, we also consider arbitrary isomorphisms rather than just those that respect a torus action. We define an injective map from the set of forms of a toric variety to a non-abelian second cohomology set, which generalizes the usual Brauer class of a Severi-Brauer variety. Additionally, we define a map from the set of forms of a toric variety to the set of forms of a separable algebra along similar lines to a construction of A. Merkurjev and I. Panin. This generalizes both a result of M. Blunk for del Pezzo surfaces of degree 6, and the standard bijection between Severi-Brauer varieties and central simple algebras.

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Complete toric varieties with reductive automorphism group

Nill

2006

Math. Z.

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We give equivalent and sufficient criteria for the automorphism group of a complete toric variety, respectively a Gorenstein toric Fano variety, to be reductive. In particular we show that the automorphism group of a Gorenstein toric Fano variety is reductive, if the barycenter of the associated reflexive polytope is zero. Furthermore a sharp bound on the dimension of the reductive automorphism group of a complete toric variety is proven by studying the set of Demazure roots.Comment: AMS-LaTeX, 20 pages with 1 figur

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Polytopal Linear Groups

Abstract: automorphisms are defined in terms of so-called column structures on P. 715

Cited by 25 publications

References 8 publications

Polytopal linear retractions

Polytopal linear retractions

Twisted Forms of Toric Varieties

Complete toric varieties with reductive automorphism group

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