Higher derived brackets and homotopy algebras

Voronov, Theodore

doi:10.1016/j.jpaa.2005.01.010

Cited by 223 publications

(214 citation statements)

References 32 publications

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“…Observe that λ i is a multiderivation of degree 2 − i. As shown in [42] (this is actually Example 2.3 there), the operations λ i define the structure of an L ∞ -algebra on A. In [6] such a structure is called a P ∞ -algebra (P for Poisson) since the λ i 's are multiderivations.…”

Section: 2mentioning

confidence: 96%

“…For the appearance of Gerstenhaber algebras in the theory of Lie bialgebras and Lie bialgebroids, see [18,27,26,31,36,42].…”

Section: 2mentioning

confidence: 99%

“…Graded symplectic manifolds. The construction of Example 1.3 can be extended to graded symplectic manifolds [1,11,14,38,42]. Recall that a symplectic structure of degree n on a graded manifold N is a closed nondegenerate two-form ω such that L E ω = nω where L E is the Lie derivative with respect to the Euler field of N (see [35] for details).…”

Section: 2mentioning

confidence: 99%

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Graded Poisson Algebras

Cattaneo

Fiorenza

Longoni

2006

Encyclopedia of Mathematical Physics

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1. Definitions 1.1. Graded vector spaces. By a Z-graded vector space (or simply, graded vector space) we mean a direct sum A = ⊕ i∈Z A i of vector spaces over a field k of characteristic zero. The A i are called the components of A of degree i and the degree of a homogeneous element a ∈ A is denoted by |a|. We also denote by A[n] the graded vector space with degree shifted by n, namely,The tensor product of two graded vector spaces A and B is again a graded vector space whose degree r component is given byThe symmetric and exterior algebra of a graded vector space A are defined respectively as S(A) = T (A)/I S and (A) = T (A)/I ∧ , where T (A) = ⊕ n≥0 A ⊗n is the tensor algebra of A and I S (resp. I ∧ ) is the two-sided ideal generated by elements of the form a ⊗ b − (−1) |a| |b| b ⊗ a (resp. a ⊗ b + (−1) |a| |b| b ⊗ a), with a and b homogeneous elements of A. The images of A ⊗n in S(A) and (A) are denoted by S n (A) and n (A) respectively. Notice that there is a canonical decalage isomorphism1.2. Graded algebras and graded Lie algebras. We say that A is a graded algebra (of degree zero) if A is a graded vector space endowed with a degree zero bilinear associative product · : A⊗A → A. A graded algebra is graded commutative if the product satisfies the conditionfor any two homogeneous elements a, b ∈ A of degree |a| and |b| respectively. A graded Lie algebra of degree n is a graded vector space A endowed with a graded Lie bracket on A[n]. Such a bracket can be seen as a degree −n Lie bracket on A, i.e., as bilinear operation {·, ·} : A ⊗ A → A[−n] satisfying graded antisymmetry and graded Jacobi relations: {a, b} = −(−1) (|a|+n)(|b|+n) {b, a} {a, {b, c}} = {{a, b}, c} + (−1) (|a|+n)(|b|+n) {a{b, c}}

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Section: 2mentioning

confidence: 96%

“…For the appearance of Gerstenhaber algebras in the theory of Lie bialgebras and Lie bialgebroids, see [18,27,26,31,36,42].…”

Section: 2mentioning

confidence: 99%

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Graded Poisson Algebras

Cattaneo

Fiorenza

Longoni

2006

Encyclopedia of Mathematical Physics

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“…We refer to [18, Section 6.1] for additional details on this relation. Note that T. Voronov has provided an analogous definition in Z 2 -settings, which gives an equivalent result for supermanifolds [23,24].…”

Section: Remark 311mentioning

confidence: 99%

Introduction to graded geometry

Fairon

2017

European Journal of Mathematics

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This paper aims at setting out the basics of Z-graded manifolds theory. We introduce Z-graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric algebra to define functions is made clear. Moreover, we define vector fields and exhibit their graded local basis. The paper also reviews some correspondences between differential Z-graded manifolds and algebraic structures.

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“…In fact, this formula is the main creative input for this work: all the other formulas were "reverse-engineered" from this one and then shown to be valid in the general case. This construction should be compared to Voronov's "higher derived brackets" [25].…”

Section: The Aim and Content Of This Papermentioning

confidence: 99%

Courant–Dorfman Algebras and their Cohomology

Roytenberg

2009

Lett Math Phys

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Abstract. We introduce a new type of algebra, the Courant-Dorfman algebra. These are to Courant algebroids what Lie-Rinehart algebras are to Lie algebroids, or Poisson algebras to Poisson manifolds. We work with arbitrary rings and modules, without any regularity, finiteness or non-degeneracy assumptions. To each Courant-Dorfman algebra (R, E) we associate a differential graded algebra C(E, R) in a functorial way by means of explicit formulas. We describe two canonical filtrations on C(E, R), and derive an analogue of the Cartan relations for derivations of C(E, R); we classify central extensions of E in terms of H 2 (E, R) and study the canonical cocycle ∈ C 3 (E, R) whose class [ ] obstructs re-scalings of the Courant-Dorfman structure. In the nondegenerate case, we also explicitly describe the Poisson bracket on C(E, R); for Courant-Dorfman algebras associated to Courant algebroids over finite-dimensional smooth manifolds, we prove that the Poisson dg algebra C(E, R) is isomorphic to the one constructed in Roytenberg (On the structure of graded symplectic supermanifolds and Courant algebroids. American Mathematical Society, Providence, 2002) using graded manifolds.

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Higher derived brackets and homotopy algebras

Cited by 223 publications

References 32 publications

Graded Poisson Algebras

Graded Poisson Algebras

Introduction to graded geometry

Courant–Dorfman Algebras and their Cohomology

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