Algebras, bialgebras, quantum groups, and algebraic deformations

Gerstenhaber, Murray; Schack, Samuel D.

doi:10.1090/conm/134/1187279

Cited by 94 publications

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“…The Lie-Rinehart structures on the pair (B, g) bijectively correspond to the Gerstenhaber algebra structures on the exterior algebra B (g) [10,15,16,20].…”

Section: 2mentioning

confidence: 99%

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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“…The Lie-Rinehart structures on the pair (B, g) bijectively correspond to the Gerstenhaber algebra structures on the exterior algebra B (g) [10,15,16,20].…”

Section: 2mentioning

confidence: 99%

Graded Poisson Algebras

Cattaneo

Fiorenza

Longoni

2006

Encyclopedia of Mathematical Physics

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“…Definition (cf. [3,4,5,2]). A linear (right) pre-operad (composition system) with coefficients in K is a sequence C := {C n } n∈N of unital K-modules (an N-graded K-module), such that the following conditions hold.…”

Section: Pre-operad (Composition System)mentioning

confidence: 99%

“…Above we modified the Gerstenhaber comp algebra defining relations [2,5] by introducing the sign (−1) |f ||g| in the defining relations of the pre-operad. The modification enables us to keep track of (control) sign changes more effectively.…”

Section: Remarkmentioning

confidence: 99%

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Invitation to composition

Kluge

Paal

Stasheff³

2000

Communications in Algebra

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Abstract. In 1963 [Ann. of Math. 78, 267-288], Gerstenhaber invented a comp(osition) calculus in the Hochschild complex of an associative algebra. In this paper, the first steps of the Gerstenhaber theory are exposed in an abstract (comp system) setting. In particular, as in the Hochschild complex, a graded Lie algebra and a pre-coboundary operator can be associated to every comp system. A derivation deviation of the pre-coboundary operator over the total composition is calculated in two ways, (the long) one of which is essentially new and can be seen as an example and elaboration of the auxiliary variables method proposed by Gerstenhaber in the early days of the comp calculus.Classification. 18D50 (MSC2000).

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“…Nonetheless the algebra Z with product • z provides an Abelian deformation of the algebra Z 0 and this also is interesting per se because it gives an example of a non-trivial Abelian deformation, however generalized and therefore not necessarily classified by the Harrison cohomology (defined on the sub-complex of the Hochschild complex consisting of symmetric cochains [12], [13]). …”

Section: Zariski Productmentioning

confidence: 99%

Deformation quantization and Nambu Mechanics

et al. 1997

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Starting from deformation quantization (star-products), the quantization problem of Nambu Mechanics is investigated. After considering some impossibilities and pushing some analogies with field quantization, a solution to the quantization problem is presented in the novel approach of Zariski quantization of fields (observables, functions, in this case polynomials). This quantization is based on the factorization over R of polynomials in several real variables. We quantize the infinite-dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is Abelian, associative and distributive. This procedure is then adapted to derivatives (needed for the Nambu brackets), which ensures the validity of the Fundamental Identity of Nambu Mechanics also at the quantum level. Our construction is in fact more general than the particular case considered here: it can be utilized for quite general defining identities and for much more general star-products.

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Algebras, bialgebras, quantum groups, and algebraic deformations

Cited by 94 publications

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Graded Poisson Algebras

Graded Poisson Algebras

Invitation to composition

Deformation quantization and Nambu Mechanics

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