Gröbner Bases in Ring Theory

Li, Huishi

doi:10.1142/9789814365147

Cited by 21 publications

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“…We filter H λ,κ by degree, assigning deg v = 1 and deg g = 0 for each v in V and g in G. Then the associated graded algebra gr H λ,κ is a quotient of its homogeneous version S(V )#G obtained by crossing out all but the highest homogeneous part of each generating relation defining H λ,κ (see Li [18,Theorem 3.2] or Braverman and Gaitsgory [5]). Recall that a "PBW property" on a filtered algebra indicates that the homogeneous version (with respect to some generating set of relations) and its associated graded algebra coincide.…”

Section: Filtered Quadratic Algebrasmentioning

confidence: 99%

PBW Deformations of Skew Group Algebras in Positive Characteristic

Shepler

Witherspoon

2014

Algebr Represent Theor

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Abstract. We investigate deformations of a skew group algebra that arise from a finite group acting on a polynomial ring. When the characteristic of the underlying field divides the order of the group, a new type of deformation emerges that does not occur in characteristic zero. This analogue of Lusztig's graded affine Hecke algebra for positive characteristic can not be forged from the template of symplectic reflection and related algebras as originally crafted by Drinfeld. By contrast, we show that in characteristic zero, for arbitrary finite groups, a Lusztig-type deformation is always isomorphic to a Drinfeldtype deformation. We fit all these deformations into a general theory, connecting Poincaré-Birkhoff-Witt deformations and Hochschild cohomology when working over fields of arbitrary characteristic. We make this connection by way of a double complex adapted from Guccione, Guccione, and Valqui, formed from the Koszul resolution of a polynomial ring and the bar resolution of a group algebra.

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Section: Filtered Quadratic Algebrasmentioning

confidence: 99%

PBW Deformations of Skew Group Algebras in Positive Characteristic

Shepler

Witherspoon

2014

Algebr Represent Theor

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“…We quote three elementary lemmas, which are special cases of ones in [16]. The analogous results and proofs can be found in [9,12]. Lemma 4.6.…”

Section: Nakayama Automorphismsmentioning

confidence: 97%

Homogeneous PBW Deformation for Artin–schelter Regular Algebras

Shen

Zhou

2014

Bull. Aust. Math. Soc.

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“…(i) Definition 1.2.1 is indeed borrowed from the theory of Gröbner bases for general finitely generated K-algebras, in which the algebras considered may be noncommutative, may have divisors of zero, and the K-bases used may not be a PBW basis, but with a (one-sided, two-sided) monomial ordering such algebras may theoretically have a (one-sided, two-sided) Gröbner basis theory. For more details on this topic, one may referrer to [Li2,Section 3.1 of Chapter 3 and Section 8.3 of Chapter 8]. Also, to see the essential difference between Definition 1.2.1 and the classical definition of a monomial ordering in the commutative case, one may refer to Definition 1.4.1 and the proof of Theorem 1.4.6 given in [AL2].…”

Section: Definitionmentioning

confidence: 99%

“…It is known that Gröbner bases for ungraded ideals in both a commutative polynomial algebra and a noncommutative free algebra can be obtained via computing homogeneous Gröbner bases for graded ideals in the corresponding homogenized (graded) algebras (cf. [Fröb], [LS], [Li2]). Similarly for an N-filtered solvable polynomial algebra A with respect to a positive-degree function d( ), by using a (de)homogenization-like trick with respect to the central regular element Z in A, the discussion on A and L presented in Subsection 3.1 indeed enables us to obtain left Gröbner bases of submodules (left ideals) in L (in A) via computing homogeneous left Gröbner bases of graded submodules (graded left ideals) in L (in A).…”

Section: Lemmamentioning

confidence: 99%

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On computation of minimal free resolutions over solvable polynomial algebras

Li¹

2015

Commentationes Mathematicae Universitatis Carolinae

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It is shown that the methods and algorithms, developed in (A. Capani et al., Computing minimal finite free resolutions, Journal of Pure and Applied Algebra, (117& 118)(1997), 105 -117; M. Kreuzer and L. Robbiano, Computational Commutative Algebra 2, Springer, 2005.) for computing minimal homogeneous generating sets of graded submodules and graded quotient modules of free modules over a commutative polynomial algebra, can be adapted for computing minimal homogeneous generating sets of graded submodules and graded quotient modules of free modules over a weighted N-graded solvable polynomial algebra, where solvable polynomial algebras are in the sense of (A. Kandri-Rody and V. Weispfenning, Non-commutative Gröbner bases in algebras of solvable type. J. Symbolic Comput., 9(1990), 1-26).Consequently, algorithmic procedures for computing minimal finite graded free resolutions over weighted N-graded solvable polynomial algebras are achieved.

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Gröbner Bases in Ring Theory

Cited by 21 publications

References 0 publications

PBW Deformations of Skew Group Algebras in Positive Characteristic

PBW Deformations of Skew Group Algebras in Positive Characteristic

Homogeneous PBW Deformation for Artin–schelter Regular Algebras

On computation of minimal free resolutions over solvable polynomial algebras

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