Koszul Calculus

Berger, Roland; Lambre, Thierry; Solotar, Andréa

doi:10.1017/s0017089517000167

Cited by 7 publications

(62 citation statements)

References 18 publications

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“…if p = 2p ′ + 1. If N = 2, we recover formulas of the quadratic case [9]. If A is Koszul, then HK • (A, M ), resp.…”

Section: Koszul Homology and Cohomology Of Amentioning

confidence: 88%

“…For any A-bimodule M , the left derived functor M L ⊗ A e − allows us to show, as in the quadratic case with the same proof [9], that the Koszul homology is isomorphic to the following Hochschild hyperhomology…”

Section: Koszul Homology and Cohomology Of Amentioning

confidence: 95%

“…Koszul calculus is more flexible and more symmetric since Formula (1.1) holds for any A-bimodule and exists in homology. Moreover, higher Koszul calculus defined in [9] reveals some Koszul analogues of results contained in Tamarkin-Tsygan calculus, providing a new information about the quadratic algebra A when A is not Koszul.…”

Section: Introductionmentioning

confidence: 94%

“…In [9], Koszul calculus was introduced as a new homological tool for studying quadratic algebras. Koszul calculus consists in Koszul (co)homology, together with Koszul cup and cap products.…”

Section: Introductionmentioning

confidence: 99%

“…If the quadratic algebra A is Koszul, Koszul calculus is reduced to Hochschild calculus, but if A is not Koszul, Koszul calculus is a new invariant of Manin's category. An application of Koszul calculus to Koszul duality is given in [9], showing that the true nature of Koszul duality Theorem does not depend on any Koszulity assumption on A.…”

Section: Introductionmentioning

confidence: 99%

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Koszul calculus for N-homogeneous algebras

Berger

2019

Journal of Algebra

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We extend Koszul calculus defined on quadratic algebras by Berger, Lambre, Solotar (Koszul calculus, Glasg. Math. J.) to N -homogeneous algebras for any N ≥ 2, quadratic algebras corresponding to N = 2. We emphasize that N -homogeneous algebras are considered in full generality, with no Koszulity assumption. Koszul cup and cap products are introduced and are reduced to usual cup and cap products if N = 2, but if N > 2, they are defined by very specific expressions. These specific expressions are compatible with the Koszul differentials and provide associative products on classes.There is no associativity in general on chains-cochains, suggesting that Koszul cochains should constitute an A∞-algebra, acting as an A∞-bimodule on Koszul chains. 2010 MSC: 16S37, 16S38, 16E40, 16E45. Keywords: N -homogeneous algebras, N -Koszul algebras, Koszul (co)homology, Hochschild (co)homology, cup and cap products.by Priddy when N = 2 [36]. Since then, N -homogeneous algebras and N -Koszul algebras have been connected to the following domains.

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“…if p = 2p ′ + 1. If N = 2, we recover formulas of the quadratic case [9]. If A is Koszul, then HK • (A, M ), resp.…”

Section: Koszul Homology and Cohomology Of Amentioning

confidence: 88%

Section: Koszul Homology and Cohomology Of Amentioning

confidence: 95%

Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Koszul calculus for N-homogeneous algebras

Berger

2019

Journal of Algebra

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Calabi-Yau algebras and the shifted noncommutative symplectic structure

Chen

Eshmatov

2020

Advances in Mathematics

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In this paper we show that for a Koszul Calabi-Yau algebra, there is a shifted bi-symplectic structure in the sense of Crawley-Boevey-Etingof-Ginzburg [14], on the cobar construction of its co-unitalized Koszul dual coalgebra, and hence its DG representation schemes, in the sense of Berest-Khachatryan-Ramadoss [3], have a shifted symplectic structure in the sense of Pantev-Toën-Vaquié-Vezzosi [28].1 2 XIAOJUN CHEN AND FARKHOD ESHMATOV justify this question, let us remind a version of noncommutative symplectic structure introduced by Crawley-Boevey, Etingof and Ginzburg in [14], which they called the bi-symplectic structure in the paper. For an associative algebra, a bi-symplectic structure on it is a closed 2-form in its Karoubi-de Rham complex that induces an isomorphism between the space of noncommutative vector fields (more precisely, the space of double derivations) and the space of noncommutative 1-forms. In [38, Appendix], Van den Bergh showed that any bi-symplectic structure naturally gives rise to a double Poisson structure, which is completely analogous to the classical case. However, the reverse is in general not true. Nevertheless, it is still interesting to ask if this is true in the special case of Calabi-Yau algebras.The second motivation of the paper comes from the 2012 paper [28] of Pantev, Toën, Vaquié and Vezzosi, where they introduced the notion of shifted symplectic structure for derived stacks. This not only generalizes the classical symplectic geometry to a much broader context, but also reveals many new features on a lot of geometric spaces, especially on the various moduli spaces that mathematicians are now studying. As remarked by the authors, the shifted symplectic structure, if it exists, always comes from the Poincaré duality of the corresponding source spaces; they also outlined how to generalize the shifted symplectic structure to noncommutative spaces, such as Calabi-Yau categories.Note that Calabi-Yau algebras are highly related to Calabi-Yau categories. For example, a theorem of Keller (see [21, Lemma 4.1]) says that the bounded derived category of a Calabi-Yau algebra is a Calabi-Yau category. Applying the idea of [28], we would expect that the noncommutative Poincaré duality of a Calabi-Yau algebra shall also play a role in the corresponding shifted noncommutative symplectic structure if it exists.The main results of the current paper may be summarized as follows. Let A be a Calabi-Yau algebra of dimension n. Assume A is also Koszul, and denote its Koszul dual coalegbra by A ¡ . LetR = Ω(Ã ¡ ) be the cobar construction of the co-unitalizationÃ ¡ of A ¡ . In this paper, we show thatR, rather than Ω(A ¡ ) as studied in [1,9], has a (2 − n)-shifted bi-symplectic structure. Such bi-symplectic structure comes from the volume form of the noncommutative Poincaré duality of A, and naturally induces a (2 − n)-shifted symplectic structure on the DG representation schemes Rep V (R), for all vector spaces V (see Theorem 4.6). By taking the corresponding trace maps we obtain a commutative di...

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