String diagrams and categorification

Savage, Alistair

doi:10.48550/arxiv.1806.06873

Cited by 3 publications

(5 citation statements)

References 9 publications

(12 reference statements)

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“…We easily prove following [31] that this linear (3, 2)-polygraph is a presentation of AOB. To study this linear (3, 2)-polygraph modulo, we consider its convergent subpolygraph E defined by E i = AOB i for i = 0, 1, E 2 contains the last six generating 2-cells in 5 and E 3 contains exactly the isotopy 3-cells (6).…”

Section: Affine Oriented Brauer Categorymentioning

confidence: 91%

“…Throughout this paper, 2-cells in 2-categories are represented using the classical representation by string diagrams, see [28,31] for surveys on the correspondance between 2-cells and diagrams. The ⋆ 0 composition of 2-cells is depicted by placing two diagrams next to each other, the ⋆ 1 -composition is vertical concatenation of diagrams.…”

Section: Preliminariesmentioning

confidence: 99%

“…We recall from [31] the natural presentation of the affine oriented Brauer category from the degenerate affine Hecke monoidal category. Following [31], End AH deg ⊗n is isomorphic to the degenerate affine Hecke algebra of degree n.…”

Section: Affine Oriented Brauer Categorymentioning

confidence: 99%

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Rewriting modulo isotopies in pivotal linear $(2,2)$-categories

Dupont

2019

Preprint

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In this paper, we study rewriting modulo a set of algebraic axioms in categories enriched in linear categories, called linear (2, 2)-categories. We introduce the structure of linear (3, 2)polygraph modulo as a presentation of a linear (2, 2)-category by a rewriting system modulo algebraic axioms. We introduce a symbolic computation method in order to compute linear bases for the vector spaces of 2-cells of these categories. In particular, we study the case of pivotal 2-categories using the isotopy relations given by biadjunctions on 1-cells and cyclicity conditions on 2-cells as axioms for which we rewrite modulo. By this constructive method, we recover the bases of normally ordered dotted oriented Brauer diagrams in te affine oriented Brauer linear (2, 2)-category.

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Section: Affine Oriented Brauer Categorymentioning

confidence: 91%

Section: Preliminariesmentioning

confidence: 99%

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Rewriting modulo isotopies in pivotal linear $(2,2)$-categories

Dupont

2019

Preprint

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“…Throughout this paper, 2-cells in 2-categories are represented using the classical representation by string diagrams, see [23,30] for surveys on the correspondance between 2-cells and diagrams. The ⋆ 0 composition of 2-cells is depicted by placing two diagrams next to each other, the ⋆ 1 -composition is vertical concatenation of diagrams.…”

Section: Preliminariesmentioning

confidence: 99%

Rewriting modulo isotopies in Khovanov-Lauda-Rouquier's categorification of quantum groups

Dupont

2019

Preprint

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We study a presentation of Khovanov -Lauda -Rouquier's candidate 2-categorification of a quantum group using algebraic rewriting methods. We use a computational approach based on rewriting modulo the isotopy axioms of its pivotal structure to compute a family of linear bases for all the vector spaces of 2-cells in this 2-category. We show that these bases correspond to Khovanov and Lauda's conjectured generating sets, proving the non-degeneracy of their diagrammatic calculus. This implies that this 2-category is a categorification of Lusztig's idempotent and integral quantum group U q (g) associated to a symmetrizable simply-laced Kac-Moody algebra g.

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“…Just as one can define associative algebras via generators and relations, one can also define strict k-linear monoidal categories in this way. We follow the definition given in [Sav18,§3].…”

Section: String Diagramsmentioning

confidence: 99%

Presentations of diagram categories

Hu¹

2019

Preprint

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We describe the planar rook category, the rook category, the rook-Brauer category, and the Motzkin category in terms of generators and relations. We show that the morphism spaces of these categories have linear bases given by planar rook diagrams, rook diagrams, rook-Brauer diagrams, and Motzkin diagrams. Contents 1. Introduction 1 2. Strict k-linear monoidal categories and string diagrams 3 3. The partition category Par(t) 5 4. The planar rook category PR(t) 9 5. The rook category R(t) 12 6. The rook-Brauer category RB(t) 17 7. The Motzkin category M(t) 19 References 22 2010 Mathematics Subject Classification. 18D10.

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String diagrams and categorification

Cited by 3 publications

References 9 publications

Rewriting modulo isotopies in pivotal linear $(2,2)$-categories

Rewriting modulo isotopies in pivotal linear $(2,2)$-categories

Rewriting modulo isotopies in Khovanov-Lauda-Rouquier's categorification of quantum groups

Presentations of diagram categories

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