Uniqueness and intrinsic properties of non-commutative Koszul brackets

Manetti, Marco

doi:10.1007/s40062-016-0136-0

Cited by 3 publications

(5 citation statements)

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“…In this case, our BV ∞ -operators coincide with the homotopy BV-operators of Kravchenko [28]. See also [40,35,33,5,49,21,4,42,2].…”

Section: With Such a Choice Of Berezinian Volume Form We May Identify...mentioning

confidence: 78%

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Quantization of (-1)-Shifted Derived Poisson Manifolds

Behrend¹,

Peddie²,

Xu³

2022

Preprint

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We investigate the quantization problem of (−1)-shifted derived Poisson manifolds in terms of BV∞-operators on the space of Berezinian half-densities. We prove that quantizing such a (−1)-shifted derived Poisson manifold is equivalent to the lifting of a consecutive sequences of Maurer-Cartan elements of short exact sequences of differential graded Lie algebras, where the obstruction is a certain class in the second Poisson cohomology. Consequently, a (−1)-shifted derived Poisson manifold is quantizable if the second Poisson cohomology group vanishes. We also prove that for any L∞-algebroid L, its corresponding linear (−1)-shifted derived Poisson manifold L ∨ [−1] admits a canonical quantization. Finally, given a Lie algebroid A and a one-cocycle s ∈ Γ A ∨ , the (−1)-shifted derived Poisson manifold corresponding to the derived intersection of coisotropic submanifolds determined by the graph of s and the zero section of the Lie Poisson A ∨ is shown to admit a canonical quantization in terms of Evens-Lu-Weinstein module.

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“…In this case, our BV ∞ -operators coincide with the homotopy BV-operators of Kravchenko [28]. See also [40,35,33,5,49,21,4,42,2].…”

Section: With Such a Choice Of Berezinian Volume Form We May Identify...mentioning

confidence: 78%

“…Remark 2.27. One may compare (28) with the multi-brackets constructed in [4,42,5,35,33]. Note that the -modification is needed in order to absorb the modified commutator.…”

Section: With Such a Choice Of Berezinian Volume Form We May Identify...mentioning

confidence: 99%

Quantization of (-1)-Shifted Derived Poisson Manifolds

Behrend¹,

Peddie²,

Xu³

2022

Preprint

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“…and the same proof of Corollary 5.5 works in this case as well. Further properties of this noncommutative extension of Koszul brackets are studied in [24], where it is proved in particular that they coincide with the brackets defined by Bering and Bandiera, while they are different from the symmetrization of Börjeson's brackets, cf. [26].…”

Section: Proofmentioning

confidence: 98%

“…Moreover, as additional byproduct, some of the proved universal formulas for higher antibrackets, when applied to operators in noncommutative algebras, give hierarchies of higher antibrackets satisfying generalized Jacobi identities, cf. [24].…”

Section: Introductionmentioning

confidence: 99%

“…In fact, also in this more general situation the brackets defined in Corollary 2.2 satisfy the generalized Jacobi identities (see Remark 5.6), while neither the symmetrization of (2.1) nor the symmetrization of (2.2) do. Extensions of higher antibrackets to noncommutative associative algebras were discussed in [2,6], and a complete description has been given, with different approaches, in [4,5,13,24].…”

Section: Introductionmentioning

confidence: 99%

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Universal Lie Formulas for Higher Antibrackets

Manetti

Ricciardi

2016

SIGMA

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Abstract. We prove that the hierarchy of higher antibrackets (aka higher Koszul brackets, aka Koszul braces) of a linear operator ∆ on a commutative superalgebra can be defined by some universal formulas involving iterated Nijenhuis-Richardson brackets having as arguments ∆ and the multiplication operators. As a byproduct, we can immediately extend higher antibrackets to noncommutative algebras in a way preserving the validity of generalized Jacobi identities.

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