Brackets in representation algebras of Hopf algebras

Massuyeau, Gwénaël; Turaev, Vladimir

doi:10.48550/arxiv.1508.07566

Cited by 3 publications

(4 citation statements)

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“…Finally, in Section 6.1 we investigate brackets on representation algebras induced by the modified double Poisson brackets. We show that some recent results of G. Massuyeau and V. Turaev [MT15] can be extended beyond skew-symmetric case as well.…”

Section: Introductionsupporting

confidence: 57%

Modified double Poisson brackets

Arthamonov

2017

Journal of Algebra

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Section: Introductionsupporting

confidence: 57%

Modified double Poisson brackets

Arthamonov

2017

Journal of Algebra

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“…The result essentially equivalent to the first part of this Theorem was independently obtained by Massuyeau-Turaev, see [28]. They used that both functors [k(G)] and k[Γ] are monoidal, which turns the tensor product over H into a quotient of the tensor algebra of k(G) ⊗ k[Γ].…”

Section: Introductionmentioning

confidence: 85%

“…Remark 4.9. Theorem 4.1 essentially appears in [28]. However, Massuyeau-Turaev do not work with tensor products over a category and instead express it in terms of quotient of the tensor algebra over the product.…”

Section: Corollary 44 There Is An Isomorphism Of Commutative Algebrasmentioning

confidence: 99%

Character varieties as a tensor product

Kassabov¹,

Patotski²

2018

Journal of Algebra

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In this short note we show that representation and character varieties of discrete groups can be viewed as tensor products of suitable functors over the PROP of cocommutative Hopf algebras. Such view point has several interesting applications. First, it gives a straightforward way of deriving the functor sending a discrete group to the functions on its representation variety, which leads to representation homology. Second, using a suitable deformation of the functors involved in this construction, one can obtain deformations of the representation and character varieties for the fundamental groups of 3-manifolds, and could lead to better understanding of quantum representations of mapping class groups.

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“…This class of algebras includes the representation algebras considered here and associated with the general linear groups. For more on this, see [MT2]. The bibrackets arising below in the geometric context are reducible.…”

Section: Bibrackets In Hopf Categoriesmentioning

confidence: 99%

Brackets in the Pontryagin Algebras of Manifolds

Massuyeau¹,

Turaev²

2017

Mémoires Soc. Math. France

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Given a smooth oriented manifold M with non-empty boundary, we study the Pontryagin algebra A = H * (Ω) where Ω is the space of loops in M based at a distinguished point of ∂M . Using the ideas of string topology of Chas-Sullivan, we define a linear map { {−, −} } : A ⊗ A → A ⊗ A which is a double bracket in the sense of Van den Bergh satisfying a version of the Jacobi identity. For dim(M ) ≥ 3, the double bracket { {−, −} } induces Gerstenhaber brackets in the representation algebras associated with A. This extends our previous work on the case dim(M ) = 2 where A = H 0 (Ω) is the group algebra of the fundamental group π 1 (M ) and the double bracket { {−, −} } induces the standard Poisson brackets on the moduli spaces of representations of π 1 (M ). Contents Introduction Chapter 1. Algebras, brackets, and bibrackets 1.1. Algebras and brackets 1.2. Bibrackets 1.3. Equivariance 1.4. The associated pairing and the trace Chapter 2. Bibrackets in unital algebras and in categories 2.1. Bibrackets in unital algebras 2.2. Bibrackets in categories 2.3. Bibrackets in Hopf categories 2.4. Hamiltonian reduction of bibrackets Chapter 3. Face homology 3.1. Manifolds with faces and partitions 3.2. Polychains, polycycles, and face homology 3.3. Face homology versus singular homology 3.4. Smooth polychains Chapter 4. Operations on polychains 4.1. Transversality in path spaces 4.2. Intersection of polychains 4.3. The operation Υ 4.4. The operation Υ Chapter 5. The intersection bibracket 5.1. Construction of the intersection bibracket 5.2. The Jacobi identity 5.3. Computations and examples Chapter 6. Properties of the intersection bibracket 6.1. The scalar intersection form 6.2. The reducibility 6.3. The string bracket 6.4. Moment maps and Hamiltonian reduction Bibliography IndexConventions. Throughout the memoir, the letter K denotes a commutative ring which serves as the ground ring of all modules and algebras. Thus, by a module (respectively, an algebra, a linear map) we mean a K-module (respectively, a Kalgebra, a K-linear map). By the singular homology of a topological space we mean singular homology with coefficients in K.Given a smooth oriented manifold M and a smooth orientable submanifold N ⊂ M , an orientation of the normal bundle of N in M determines an orientation of N , and vice versa, via the following rule: a positive frame in the normal bundle of N followed by a positive frame in the tangent bundle of N is a positive frame in the tangent bundle of M . If ∂M = ∅, then the orientation of M induces an orientation of ∂M using the "outward vector first" rule. CHAPTER 1Algebras, brackets, and bibrackets Algebras and bracketsWe start by recalling some standard terminology.1.1.1. Graded modules and graded algebras. By a graded module we mean a Z-graded module A = ⊕ p∈Z A p (over K). An element a of A is homogeneous if a ∈ A p for some p; we write then |a| = p and call |a| the degree of a. By definition, the degree of 0 ∈ A is an arbitrary integer. For any d ∈ Z, the d-degreeA graded algebra is a graded module A endowed wi...

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Brackets in representation algebras of Hopf algebras

Cited by 3 publications

References 15 publications

Modified double Poisson brackets

Modified double Poisson brackets

Character varieties as a tensor product

Brackets in the Pontryagin Algebras of Manifolds

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