Quadratic Algebras

Polishchuk, Alexander; Positselski, Leonid

doi:10.1090/ulect/037

Cited by 337 publications

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“…is a Hopf algebra and was able to realize well known quantum groups as black products of an algebra with its Koszul dual. For more properties of Manin's products for quadratic algebras, we refer the reader to the book of A. Polishchuk and L. Positselski [PP05].…”

Section: Introductionmentioning

confidence: 99%

Manin products, Koszul duality, Loday algebras and Deligne conjecture

Vallette¹

2008

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

136

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Abstract. In this article we give a conceptual definition of Manin products in any category endowed with two coherent monoidal products. This construction can be applied to associative algebras, non-symmetric operads, operads, colored operads, and properads presented by generators and relations. These two products, called black and white, are dual to each other under Koszul duality functor. We study their properties and compute several examples of black and white products for operads. These products allow us to define natural operations on the chain complex defining cohomology theories. With these operations, we are able to prove that Deligne's conjecture holds for a general class of operads and is not specific to the case of associative algebras. Finally, we prove generalized versions of a few conjectures raised by M. Aguiar and J.-L. Loday related to the Koszul property of operads defined by black products. These operads provide infinitely many examples for this generalized Deligne's conjecture.

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Section: Introductionmentioning

confidence: 99%

Manin products, Koszul duality, Loday algebras and Deligne conjecture

Vallette¹

2008

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

136

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“…, L r ) is ‫.ޚ‬ Recall that a graded ring R is Koszul if the ideal generated by elements of positive degree has a linear resolution as an R-module. See [Polishchuk and Positselski 2005] for background on Koszul rings and further details. Theorem 1.4.…”

Section: Introductionmentioning

confidence: 99%

Frobenius splittings of toric varieties

Payne

2009

ANT

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“…Self-consistency of nonhomogeneous quadratic relations of the type (1) is studied in the paper [31] and the book [34,Chapter 5]. The main result, called "the Poincaré-Birkhoff-Witt theorem for nonhomogeneous quadratic algebras", claims that it suffices to perform computations with expressions of degree 3 when checking self-consistency of relations of the type (1) with Koszul quadratic principal parts (3).…”

Section: Self-consistency Of Nonhomogeneous Quadratic Relationsmentioning

confidence: 99%

“…The condition of injectivity of this map in degree 3 means, basically, that syzygies of degree 3 between the homogeneous quadratic parts (3) of nonhomogeneous quadratic corelations in C do not lead to non-self-consistencies (as it happens in the example with the relation (8)). In this sense, Theorem 1.3 can be viewed as an analogue for relations of the type (2) of the Poincaré-Birkhoff-Witt theorem for relations of the type (1) [31,34] (cf. [23]).…”

Section: Self-consistency Of Nonhomogeneous Quadratic Relationsmentioning

confidence: 99%

Koszulity of cohomology = K(π,1)-ness + quasi-formality

Positselski

2017

Journal of Algebra

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Abstract. This paper is a greatly expanded version of [37, Section 9.11]. A series of definitions and results illustrating the thesis in the title (where quasi-formality means vanishing of a certain kind of Massey multiplications in the cohomology) is presented. In particular, we include a categorical interpretation of the "Koszulity implies K(π, 1)" claim, discuss the differences between two versions of Massey operations, and apply the derived nonhomogeneous Koszul duality theory in order to deduce the main theorem. In the end we demonstrate a counterexample providing a negative answer to a question of Hopkins and Wickelgren about formality of the cochain DG-algebras of absolute Galois groups, thus showing that quasi-formality cannot be strengthened to formality in the title assertion.

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Quadratic Algebras

Cited by 337 publications

References 0 publications

Manin products, Koszul duality, Loday algebras and Deligne conjecture

Manin products, Koszul duality, Loday algebras and Deligne conjecture

Frobenius splittings of toric varieties

Koszulity of cohomology = K(π,1)-ness + quasi-formality

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