Spiders for rank 2 Lie algebras

Abstract

A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. We define certain combinatorial spiders by generators and relations that are isomorphic to the representation theories of the three rank two simple Lie algebras, namely A2, B2, and G2. They generalize the widely-used Temperley-Lieb spider for A1. Among other things, they yield bases for invariant spaces which are probably related to Lusztig's canonical bases, and they are useful for computing quantities such as generalized 6j-symbols and quantum link invariants.

33 pages. Has color figures

References in corpus (1)

• Spherical Categories

Cited by in corpus (176)

• Non-Abelian Anyons and Topological Quantum Computation
• A functor-valued invariant of tangles
• A Class of $P,T$-Invariant Topological Phases of Interacting Electrons
• Quantum Picturalism
• sl(3) link homology
• Topological Defects on the Lattice: Dualities and Degeneracies
• The classification of subfactors of index at most 5
• Realizing non-Abelian statistics
• Monoidal supercategories
• The universal sl3-link homology
• Skein theory for the D_{2n} planar algebras
• How hard is it to approximate the Jones polynomial?
• Topological order from quantum loops and nets
• Fermion condensation and super pivotal categories
• Constructing the extended Haagerup planar algebra
• Super $q$-Howe duality and web categories
• Confluence Theory for Graphs
• A Diagrammatic Category for the Representation Theory of U_q(sl_n)
• On Khovanov's cobordism theory for su(3) knot homology
• The Blanchet-Khovanov algebras
• Integral presentations of quantum lattice Heisenberg algebras
• Cellular structures using ${\text{U}}_q$-tilting modules
• Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m)
• The sl_3 web algebra
• Skein theory for SU(n)-quantum invariants
• The Karoubi envelope and Lee's degeneration of Khovanov homology
• The ${\mathfrak{sl}}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality
• Higher laminations, webs and N=2 line operators
• Symbol Alphabets from Tensor Diagrams
• Invariant tensors and cellular categories
• The colored HOMFLYPT function is $q$-holonomic
• The foam and the matrix factorization sl3 link homologies are equivalent
• Invariant tensors for the spin representation of so(7)
• Functoriality of colored link homologies
• Trace as an alternative decategorification functor
• Webs on surfaces, rings of invariants, and clusters
• Ocneanu Cells and Boltzmann Weights for the SU(3) ADE Graphs
• Dimers, webs, and positroids
• The Nakayama automorphism of the almost Calabi-Yau algebras associated to SU(3) modular invariants
• A closed formula for the evaluation of ${\mathfrak{sl}}_N$-foams
• A Formula for the Jones-Wenzl Projections
• Clasp technology to knot homology via the affine Grassmannian
• Symmetric webs, Jones-Wenzl recursions and $q$-Howe duality
• Semisimplicity of Hecke and (walled) Brauer algebras
• SU(3)-Goodman-de la Harpe-Jones subfactors and the realisation of SU(3) modular invariants
• Comments about quantum symmetries of SU(3) graphs
• The finiteness conjecture for skein modules
• Enumeration of non-positive planar trivalent graphs
• Tutte chromatic identities from the Temperley-Lieb algebra
• Functoriality for the su(3) Khovanov homology
• Thick Soergel calculus in type A
• sl3-foam homology calculations
• Graphical Calculus on Representations of Quantum Lie Algebras
• Infinitely many Lagrangian fillings
• A Categorification of Quantum sl_3 Projectors and the sl_3 Reshetikhin-Turaev Invariant of Tangles
• Ward Identities for Invariant Group Integrals
• Tensor categories: A selective guided tour
• Promotion of Kreweras words
• ${\mathfrak{gl}}_n$-webs, categorification and Khovanov-Rozansky homologies
• Quantum symmetries for exceptional SU(4) modular invariants associated with conformal embeddings
• The colored HOMFLY polynomial is q-holonomic
• Categorification of tensor powers of the vector representation of $U_q(\mathfrak{gl}(1|1))$
• The sl(N)-web algebras and dual canonical bases
• Skein Modules from Skew Howe Duality and Affine Extensions
• Skein and cluster algebras of unpunctured surfaces for ${\mathfrak{sl}}_3$
• A $q$-series identity via the ${\mathfrak{sl}}_3$ colored Jones polynomials for the $(2,2m)$-torus link
• $U_q({\mathfrak{sl}}_n)$ web models and ${\mathbb{Z}}_n$ spin interfaces
• Tait colorings, and an instanton homology for webs and foams
• Skew characters and cyclic sieving
• The half-twist for U_q(g) representations
• A web basis of invariant polynomials from noncrossing partitions
• Generic ${\mathfrak{gl}}_2$-foams, web and arc algebras
• A characterisation of indecomposable web-modules over Khovanov-Kuperberg Algebras
• A large family of indecomposable projective modules for the Khovanov-Kuperberg algebra of $s{l}_3$-webs
• A magnetic model with a possible Chern-Simons phase
• Web bases for the general linear groups
• Extending and quantising the Vogel plane
• Triangular decomposition of $S{L}_3$ skein algebras
• Iterated torus knots and double affine Hecke algebras
• Tropical Fock-Goncharov coordinates for ${\mathrm{SL}}_3$-webs on surfaces I: construction
• Fusion hierarchies, $T$-systems and $Y$-systems for the $A_2^{(1)}$ models
• Jones-Wenzl idempotents For Rank 2 Simple Lie algebras
• The algebra of knotted trivalent graphs and Turaev's shadow world
• Notes on TQFT Wire Models and Coherence Equations for SU(3) Triangular Cells
• Monoidal categories, representation gap and cryptography
• Extremal weight projectors II
• ${\mathfrak{sl}}_3$-web bases, intermediate crystal bases and categorification
• Spectral Measures and Generating Series for Nimrep Graphs in Subfactor Theory II: SU(3)
• A Gauge-Independent Mechanism for Confinement and Mass Gap: Part I -- The General Framework
• A2-Planar Algebras II: Planar Modules
• Skein Modules and Lattice Gauge Field Theory
• Rational R-matrices, centralizer algebras and tensor identities for e_6 and e_7 exceptional families of Lie algebras
• Anchored foams and annular homology
• The quantum $\mathfrak{sl}(n,\mathbb{C})$ representation theory and its applications
• Tableau evacuation and webs
• A simple bijection between standard (n,n,n) tableaux and irreducible webs for sl_3
• Quantum diagrammatics for $F_4$
• Entanglement entropy from SU(2) Chern-Simons theory and symmetric webs
• Promotion and growth diagrams for fans of Dyck paths and vacillating tableaux
• The ${\mathfrak{sl}}_3$ colored Jones polynomials for $2$-bridge links
• A Gauge-Independent Mechanism for Confinement and Mass Gap: Part II -- G=SU(2) and D=3
• Cross products, invariants, and centralizers
• Skein theory for the ADE planar algebras
• Web bases in degree two from hourglass plabic graphs
• Braided Subfactors, Spectral Measures, Planar algebras and Calabi-Yau algebras associated to SU(3) modular invariants
• $K$-theory of AF-algebras from braided C*-tensor categories
• Diagrammatic morphisms between indecomposable modules of ${\bar{U}}_q({\mathfrak{sl}}_2)$
• Tensor diagrams and Chebyshev polynomials
• Points of quantum ${\mathrm{SL}}_n$ coming from quantum snakes
• Naturality of ${\mathrm{SL}}_3$ quantum trace maps for surfaces
• Exchange Relation in sl3 WZNW model in Semiclassical Limit
• Diagrammatics for $F_4$
• $U_{\mathfrak{q}}({\mathfrak{sl}}_3)$ web models: Locality, phase diagram and geometrical defects
• Trace Diagrams, Matrix Minors, and Determinant Identities
• Quantum SL(3,C)'s with classical representation theory
• The braid group surjects onto $G_2$ tensor space
• Group categories and their field theories
• Non-Semisimple Planar Algebras from the Representation Theory of ${\bar{U}}_q({\mathfrak{sl}}_2)$
• Linv invariant and $G_2$ web space
• On sequences associated to the invariant theory of rank two simple Lie algebras
• Web bases for sl(3) are not dual canonical
• Hurwitz' theorem on composition algebras
• On webs in quantum type $C$
• An explicit bijection between semistandard tableaux and non-elliptic sl_3 webs
• Transitioning between tableaux and spider bases for Specht modules
• Rotation-invariant web bases from hourglass plabic graphs
• Spiders and Generalized Confluence
• Quantum affine algebras and Grassmannians
• Comparing Skein and Quantum Group Representations and Their Application to Asymptotic Faithfulness
• Triangle presentations and tilting modules for ${\text{SL}}_{2k+1}$
• Tropical Fock-Goncharov coordinates for ${\mathrm{SL}}_3$-webs on surfaces II: naturality
• Defect Networks and Supersymmetric Loop Operators
• The Robinson-Schensted Correspondence and $A_2$-web Bases
• Categorification of the internal braid group action for quantum groups I: 2-functoriality
• Algebraic Structures Derived from Foams
• On geometric bases for A-polynomials II: ${\mathfrak{su}}_3$ and Kuberberg bracket
• Matrix factorizations and double line in ${\mathfrak{sl}}_n$ quantum link invariant
• Promotion and cyclic sieving via webs
• A categorification of the quantum sl(N)-link polynomials using foams
• Growth Diagrams and Minuscule Polygon Configurations in the Affine Grassmannian
• Power sum elements in the $G_2$ skein algebra
• Graphical methods and rings of invariants on the symmetric algebra
• Unbounded ${\mathfrak{sl}}_3$-laminations and their shear coordinates
• Polynomial invariants of links satisfying cubic skein relations
• Triple Clasp Formulas for $G_2$
• Complex projective structures with Schottky holonomy
• On braided Hopf structures on exterior algebras
• Link Patterns and the Catalan Tree
• Twists, Higher Dimer Covers, and Web Duality for Grassmannian Cluster Algebras
• Plabic Tangles and Cluster Promotion Maps
• Tait coloring and a moduli space
• The transition matrix between the Specht and ${\mathfrak{sl}}_3$ web bases is unitriangular with respect to shadow containment
• Decreasing subsequences and Viennot for oscillating tableaux
• Leading terms of ${\text{SL}}_3$ web invariants
• The Combinatorics of ${\mathsf{A}}_2$-webs
• $A_2$ Skein Representations of Pure Braid Groups
• Affine $G_2$ Centralizer Algebras
• Generalized Catalan Numbers and the Enumeration of Planar Embeddings
• Alternative formulation for the operator algebra over the space of paths in a ADE $SU(3)$ graph
• Categorification of the colored ${\mathfrak{sl}}_3$-invariant
• A classification of transitive links and periodic links
• Foams, iterated wreath products, field extensions and Sylvester sums
• Quantum traces for ${\mathrm{SL}}_n(\mathbb{C})$: The case $n=3$
• Riordan trees and the homotopy $s{l}_2$ weight system
• ${\mathrm{SL}}_3$-laminations as bases for ${\mathrm{PGL}}_3$ cluster varieties for surfaces
• Skein-Theoretic Methods for Unitary Fusion Categories
• The zero stability for the one-row colored ${\mathfrak{sl}}_3$ Jones polynomial
• A criterion for sharpness in tree enumeration and the asymptotic number of triangulations in Kuperberg's G2 spider
• Transverse link invariants from the deformations of Khovanov ${\mathfrak{sl}}_3$-homology
• Cyclotomic Expansion of Generalized Jones Polynomials
• Cellular subalgebras of the partition algebra
• Skein Quantization and Lattice Gauge Field Theory
• Classification of Module Categories for $SO(3{)}_{2m}$
• Web invariants for flamingo Specht modules
• On the homology of almost Calabi-Yau algebras associated to SU(3) modular invariants
• A formula to evaluate type A webs and link polynomials