Skein Categories Juliet Cooke December 3, 2020 Universit´eCatholique de Louvain Introduction f f g g Internal Skein algebra ! Skein categories ! skein algebras # Factorisation homology or;t FV × Disc2 Catk SkV (Σ) or;t Mfld2 1 Skein Algebras The Kauffman bracket skein algebra SkAlgq(Σ) of the oriented smooth surface Σ is the Q(q) module of formal linear combinations of links up to isotopy modulo the Kauffman bracket skein relations 1 − 1 = q 2 + q 2 ; = −q − q−1: Multiplication is given by stacking. It is an invariant of framed links and it can be renormalised to give the Jones polynomial. 2 Coloured Ribbon Graphs Let V be a (strict) ribbon category: monoidal product ⊗ : V × V ! V braiding β : V × V ! V × V twist θ : V!V duals X ∗ with unit η : 1 ! V ∗ ⊗ V and counit : V ⊗ V ∗ ! 1 maps 3 Category of Coloured Ribbons A coloured ribbon diagram of the surface Σ is an embedding of a ribbon graph into Σ × [0; 1] such that unattached bases are sent to Σ × f0; 1g. f RibbonV (Σ) is the k-linear category: g Objects Finite collections of disjoint, framed, coloured, directed points in Σ Morphisms Finite k-linear combinations of coloured ribbon diagram which are compatible which the points attached to up to isotopy which preserves ribbon graph structure. 4 Evaluation Function Theorem (Turaev) There is a full surjective ribbon functor 3 eval : RibbonV ([0; 1] ) !V V V V Id V* V βV,W θV ηV εV 5 Skein Category (Walker, Johnson-Freyd) The skein category SkV (Σ) is the k-linear category of coloured ribbons RibbonV (Σ) modulo the following relation on morphisms X λi Fi ∼ 0 i if there exists an orientation preserving embedding E : [0; 1]3 ,! Σ × [0; 1] such that ! X eval λi Fi j[0;1]3 = 0; i the coloured ribbon diagrams Fi are identical outside the cube, and they only intersect the cube with strands transversely at the top and the bottom. 6 Framed En-Algebras A framed En-algebra is symmetric monoidal functor or;t ⊗ F : Discn !C : F (D) = V C⊗ is an (1; 1) monoidal category. or;t Mfldn is the symmetric monoidal (1; 1)-category: Objects smooth oriented n-dimensional manifolds or Morphisms 1-groupoid Embn ( ; ) or;t Discn is the full subcategory of finite disjoint unions of R 7 Factorisation Homology (Lurie, Ayala, Francis, Tanaka) The factorisation homology R V is the Left Kan extension or;t F ⊗ Discn C or;t Mfldn Theorem There is an equivalence of categories Z Z Z M tA N ' V ⊗R V A V M N 8 Relative Tensor Product R Let A = γ × [0; 1]. Then A := A V is a monoidal category R R Assume there are actions M V x A y N V The relative tensor product R V ⊗R R V is the colimit in C⊗ of the M A V N 2-sided bar construction: R R R R R R M V ⊗ A ⊗ A ⊗ N V M V ⊗ A ⊗ N V M V ⊗ N V 9 Characterisation Theorem (Ayala, Francis, Tanaka) R or;t ⊗ The functor V : Mfld2 !C is characterised by R 1. U V'V if U is contractible ∼ R 2. if A = Y × R for 1-manifold with corners Y then A V has canonical monoidal structure (which does not depend on choice of homeomorphism) 3. Excision Z Z Z M tA N ' V ⊗R V A V M N 10 Skein Categories as Factorisation Homology SkV (Σ) is a small k-linear category An oriented embedding of surfaces Σ ,! Π induces a k-linear functor SkV (Σ) ! SkV (Π), So have a 2-functor or;t × SkV ( ): Mfld2 ! Catk × Catk is the symmetric monoidal (2; 1)-category: Objects small k-linear categories. 1-morphisms k-linear functors 2-morphisms natural transformations 11 Skein Categories as Factorisation Homology Theorem (C.) The skein category functor is the factorisation homology or;t FV × Disc2 Catk SkV ( ) or;t Mfld2 Using characterisation of factorisation homology the hard part is proving excison: 1. The relative tensor product SkV (M) ×SkV (A) SkV (N) defined as the colimit of the 2-sided bar construction is equivalent to the standard relative tensor product of k-linear categories. 2. Using this relative tensor product SkV (M) ×SkV (A) SkV (N) ' SkV (M tA N). 12 Internal Skein Algebras (Gunningham, Jordan, Safronov) The internal skein algebra of the punctured surface Σ∗ is the functor SkAlgint (Σ∗): Vop ! Vect V f ∗ V 7! HomSkV (Σ )(P(V ); 1) g Such a functor is an object in the free-cocompletion op V := PSh(V ; Vect) 2 LFPk : It is an algebra internal to LFPk . 13 Stated Skein Algebras (L^e) ∗ The Stated skein algebra StatedSkAlgq(Σ ) of the punctured oriented smooth surface Σ∗ is the Q(q) module of formal linear combinations of stated boundary tangles up to isotopy modulo the Kauffman bracket skein relatios 1 − 1 = q 2 + q 2 ; = −q − q−1: and - 2 + -1/2 + = q -+ q + -1/2 _ = q + _ += 0 = _ 14 Temperly-Lieb Category The Temperly-Lieb category TL is a ribbon category objects [n] for n 2 Z≥0 morphisms Temperley-Lieb diagrams modulo linear relation = −q − q−1 composition of morphism is given by vertical stacking monoidal product is given by horizontal stacking 1 − 1 braiding β[1];[1] := = q 2 + q 2 duality caps and cups −3 twist θ[1] = −q 15 Relation to Stated Skein Algebras From now on we shall assume q is generic. fd In this case, the Cauchy completion TL is the category Repq (SL) of finite dimensional representations of the quantum group Uq(sl2). Theorem (Gunningham, Jordan, Safronv) ^ int ∗ ∗ T (SkAlgTL(Σ )) is the stated skein algebra StatedSkAlgq(Σ ). fd Have a functor F : TL ! Rep (SL2) sending [1] to standard 2-dimensional representation V . Compose with the forgetful functor to get a functor F : TL ! Vect then T^ : TL^ ! Vect is unique colimit preserving extension of this. 16 Reflection Equation Algebras (Majid) Let R be the R-matrix of the quantum group Uq(g) where g is the Lie algebra associated to the Lie group G. i The reflection equation algebra Oq(G) has generators u = fuj g statisfying the reflection equation R21u1Ru2 = u2R21u1R where u1 := u ⊗ 1 and u2 := 1 ⊗ u and braided versions of the relations associated to G e.g. for G = SL2 we quotient by the braided determinant 1 2 2 2 1 det(u) := u1 u2 − q u1 u2 is 1. 17 Alekseev Moduli Algebra Theorem (Ben-Zvi, Brochier, Jordan) int ∗ The internal skein algebra SkAlg fd (Σ ) is isomorphic to the Repq (SL2) Alekseev moduli algebra AΣ∗ = Oq(G)⊗~ ::: ⊗O~ q(G) | {z } number of handles where ⊗~ depends on the type of the handle. 18 Selected Referencesi David Ayala, John Francis, and Hiro Lee Tanaka. \Factorization homology of stratified spaces". In: Selecta Math. (N.S.) 23.1 (2017), pp. 293{362. A. Yu. Alekseev. \Integrability in the Hamiltonian Chern-Simons theory". In: Algebra i Analiz 6.2 (1994), pp. 53{66. D. Ben-Zvi, A. Brochier, and D. Jordan. \Integrating quantum groups over surfaces". In: J. Topol. 11.4 (2018), pp. 873{916. Francesco Costantino and Thang T. Q. Le. \Stated skein algebras of surfaces". arXiv:1912.02440. 2019. Juliet Cooke. \Excision of Skein Categories and Factorisation Homology". In: (2019). arXiv: 1910.02630. 19 Selected References ii Sam Gunningham, David Jordan, and Pavel Safronov. \The finiteness conjecture for skein modules". arXiv:1908.05233. 2019. Theo Johnson-Freyd. Heisenberg-picture quantum field theory. 2015. eprint: arXiv:1508.05908. Shahn Majid. Foundations of Quantum Group Theory. Cambridge University Press, 1995. Peter Tingley. \A minus sign that used to annoy me but now I know why it is there (two constructions of the Jones polynomial)". In: vol. 46. Proc. Centre Math. Appl. Austral. Nat. Univ. Austral. Nat. Univ., Canberra, 2017, pp. 415{427. Vladimir G. Turaev. Quantum Invariants of Knots and 3-manifolds. de Gruyter, 1994. 20 Questions? 20.