Deformations of path algebras of quivers with relations

Barmeier, Severin; Wang, Zhengfang

doi:10.48550/arxiv.2002.10001

Cited by 6 publications

(33 citation statements)

References 108 publications

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“…The Koszul complex K •+1 is isomorphic to the (shifted) complex of polyvector fields (with polynomial coefficients) which admits a natural graded Lie algebra structure given by the Schouten-Nijenhuis bracket [−,−] SN , and Maurer-Cartan elements of (K •+1 , [−,−] SN ) ⊗ (t) are precisely formal Poisson structures. The binary bracket −,− of the L ∞ algebra structure on K •+1 coincides with the Schouten-Nijenhuis bracket [2]. However, when viewed as a minimal model of (Hom(A •+1 , A), d, [−,−]), the Koszul complex K •+1 carries nontrivial n-ary brackets for all n ≥ 2.…”

Section: Combinatorial Star Productsmentioning

confidence: 98%

“…In this context, the operation of replacing x j x i by x i x j + ϕ(x j x i ) is called a reduction. Indeed, the deformation theory of any finitely generated algebra is equivalent to the deformation theory of any suitable reduction system [2].…”

Section: Combinatorial Star Productsmentioning

confidence: 99%

“…In order to obtain explicit formulae of formal star products which can be used in convergence and continuity results and then be extended to strict deformation quantizations, we work with the combinatorial star products introduced in [2]. We give a brief review of these star product and also consider variants which are compatible with * -involutions.…”

Section: Combinatorial Star Productsmentioning

confidence: 99%

“…. , x d ), starting from the right, by replacing each occurrence of x j x i for i < j by x i x j + ϕ(x j x i ) 2 , where the terms appearing in each ϕ n (x j x i ) for n ≥ 1 are expressed in the form x I1 1 x I2 2 . .…”

Section: Combinatorial Star Productsmentioning

confidence: 99%

“…We work with the combinatorial star products introduced in [2] via natural higher structures on the Koszul complex, which can be used to produce explicit formulae for quantizations of polynomial Poisson structures (see §2.2) for which convergence on polynomials can be shown directly (see §2.3). These formulae can then be used in continuity estimates and the star product can be extended from the space of polynomial functions to larger function spaces (see §3).…”

Section: Introductionmentioning

confidence: 99%

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Strict quantization of polynomial Poisson structures

Barmeier,

Schmitt

2022

Preprint

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We show how combinatorial star products can be used to obtain strict deformation quantizations of polynomial Poisson structures on R d , generalizing known results for constant and linear Poisson structures to polynomial Poisson structures of arbitrary degree. We give several examples of nonlinear Poisson structures and construct explicit formal star products whose deformation parameter can be evaluated to any real value of , giving strict quantizations on the space of analytic functions on R d with infinite radius of convergence.We also address further questions such as continuity of the classical limit → 0, compatibility with * -involutions, and the existence of positive linear functionals. The latter can be used to realize the strict quantizations as * -algebras of operators on a pre-Hilbert space which we demonstrate in a concrete example.

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Section: Combinatorial Star Productsmentioning

confidence: 98%

Section: Combinatorial Star Productsmentioning

confidence: 99%

Section: Combinatorial Star Productsmentioning

confidence: 99%

Section: Combinatorial Star Productsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Strict quantization of polynomial Poisson structures

Barmeier,

Schmitt

2022

Preprint

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Deformations of categories of coherent sheaves via quivers with relations

Barmeier,

Wang

2024

Alg. Geom.

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We give an explicit description of the deformation theory of the Abelian category of (quasi-)coherent sheaves on any separated Noetherian scheme X via the deformation theory of path algebras of quivers with relations, by using any affine open cover of X, or any tilting bundle on X, if available.We also give sufficient criteria for obtaining algebraizations of formal deformations, in which case the deformation parameters can be evaluated to a constant and the deformations can be compared to the original Abelian category on equal terms. We give concrete examples as well as applications to the study of noncommutative deformations of singularities. IntroductionA momentous result due to P. Gabriel [Gab62] and A. L. Rosenberg [Ros98,Bra18] shows that any quasi-separated scheme can be reconstructed from its Abelian category of quasi-coherent sheaves. Abelian categories which are "close to" categories of quasi-coherent sheaves on a variety or scheme can thus be viewed as generalizations of commutative schemes and are of central importance in noncommutative algebraic geometry.In [LVdB05, LVdB06] W. Lowen and M. Van den Bergh developed a deformation theory for abstract Abelian categories, controlled by a version of Hochschild cohomology H • Ab for Abelian categories, with first-order deformations parametrized by H 2 Ab and obstructions lying in H 3 Ab . For a separated Noetherian scheme X over an algebraically closed field k of characteristic 0, Lowen and Van den Bergh showed that the Hochschild cohomology of the Abelian categories -Mod(O X ) of all sheaves of O X -modules

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Tangent complexes and the Diamond Lemma

Dotsenko¹,

Tamaroff²

2020

Preprint

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The celebrated Diamond Lemma of Bergman gives an effectively verifiable criterion of uniqueness of normal forms for term rewriting in associative algebras. We present a new way to interpret and prove this result from the viewpoint of homotopical algebra. Our main result states that every multiplicative free resolution of an algebra with monomial relations gives rise to its own Diamond Lemma, so that Bergman's condition of "resolvable ambiguities" becomes the first non-trivial component of the Maurer-Cartan equation in the corresponding tangent complex. The same approach works for many other algebraic structures, emphasizing the relevance of computing multiplicative free resolutions of algebras with monomial relations.

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Deformations of path algebras of quivers with relations

Cited by 6 publications

References 108 publications

Strict quantization of polynomial Poisson structures

Strict quantization of polynomial Poisson structures

Deformations of categories of coherent sheaves via quivers with relations

Tangent complexes and the Diamond Lemma

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