Braided compact closed categories with applications to low dimensional topology

Freyd, Peter; Yetter, David N.

doi:10.1016/0001-8708(89)90018-2

Cited by 243 publications

(205 citation statements)

References 8 publications

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“…Categories of ribbon graphs. The category of directed ribbon graphs was introduced in [38] (also see [15,43] Let k, l be nonnegative integers. A ribbon (k, ℓ)-graph is an oriented and directed surface embedded in R 2 × [0, 1] ⊂ R 3 , which is decomposed into the union of ribbons and annuli without intersections.…”

Section: 4mentioning

confidence: 99%

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

Lehrer

Zhang

2016

Commun. Math. Phys.

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Abstract. Let U q (g) be the quantum supergroup of gl m|n or the modified quantum supergroup of osp m|2n over the field of rational functions in q, and let V q be the natural module for U q (g). There exists a unique tensor functor, associated with V q , from the category of ribbon graphs to the category of finite dimensional representations of U q (g), which preserves ribbon category structures. We show that this functor is full in the cases g = gl m|n or osp 2ℓ+1|2n . For g = osp 2ℓ|2n , we show that the space Hom Uq(g) (V ⊗r q , V ⊗s q ) is spanned by images of ribbon graphs if r + s < 2ℓ(2n + 1). The proofs involve an equivalence of module categories for two versions of the quantisation of U(g).

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Section: 4mentioning

confidence: 99%

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

Lehrer

Zhang

2016

Commun. Math. Phys.

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“…Freyd and Yetter [24] observed that the knot invariants obtained from c Q are the wellknown colouring numbers of classical knot theory. These invariants are not of finite type [13].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…These invariants are not of finite type [13]. Freyd and Yetter [24,45] also initiated the natural question of deforming set-theoretic solutions within the space of Yang-Baxter operators over a ring A, and illustrated their general approach by the simplified ansatz of diagonal deformations [24, §4]. The latter are encoded by rack cohomology, which was independently developed by Fenn and Rourke [21] from a homotopy-theoretic viewpoint via classifying spaces.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Yang–Baxter deformations and rack cohomology

Eisermann¹

2014

Trans. Amer. Math. Soc.

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ABSTRACT. Every rack Q provides a set-theoretic solution c Q of the Yang-Baxter equation by setting c Q : x ⊗ y → y ⊗ x y for all x,y ∈ Q. This article examines the deformation theory of c Q within the space of Yang-Baxter operators over a ring A, a problem initiated by Freyd and Yetter in 1989. As our main result we classify deformations in the modular case, which had previously been left in suspense, and establish that every deformation of c Q is gauge-equivalent to a quasi-diagonal one. Stated informally, in a quasi-diagonal deformation only behaviourally equivalent elements interact. In the extreme case, where all elements of Q are behaviourally distinct, Yang-Baxter cohomology thus collapses to its diagonal part, which we identify with rack cohomology. The latter has been intensively studied in recent years and, in the modular case, is known to produce non-trivial and topologically interesting Yang-Baxter deformations.

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“…Freyd and D.N. Yetter in [16]. The deformation of quandles and racks appeared as an example, but only diagonal deformations were taken into account.…”

Section: Related Workmentioning

confidence: 99%

“…Freyd and D.N. Yetter [16], we carry out this programme for racks (linearized over some ring A) and their formal deformations in the space of A-linear Yang-Baxter operators.…”

Section: Introductionmentioning

confidence: 99%

Yang–Baxter deformations of quandles and racks

Eisermann

2005

Algebr. Geom. Topol.

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Given a rack Q and a ring A, one can construct a Yang-Baxter operator c Q :y for all x, y ∈ Q. In answer to a question initiated by D.N. Yetter and P.J. Freyd, this article classifies formal deformations of c Q in the space of Yang-Baxter operators. For the trivial rack, where x y = x for all x, y , one has, of course, the classical setting of r-matrices and quantum groups. In the general case we introduce and calculate the cohomology theory that classifies infinitesimal deformations of c Q . In many cases this allows us to conclude that c Q is rigid. In the remaining cases, where infinitesimal deformations are possible, we show that higher-order obstructions are the same as in the quantum case.

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Braided compact closed categories with applications to low dimensional topology

Cited by 243 publications

References 8 publications

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

Yang–Baxter deformations and rack cohomology

Yang–Baxter deformations of quandles and racks

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