Algebraic structures in comodule categories over weak bialgebras AMS Western Sectional Meeting Virtual (formerly at University of Utah) Robert Won University of Washington October 25, 2020 1 / 20 Joint work with Chelsea Walton (Rice University) Elizabeth Wicks (Microsoft) arXiv:1911.12847 October 25, 2020 2 / 20 Big picture • Fix a field k. Goal. Study the symmetries of k-algebras A. A = (A; m : A ⊗k A ! A; u : k ! A) satisfying associativity and unit constraints. • Classically, study group actions of G on A such that g · (ab) = (g · a)(g · b) and g · 1A = 1A; i.e., A is a G-module and m and u are G-module morphisms. October 25, 2020 Symmetry 3 / 20 Bialgebras • A bialgebra H over k is a k-algebra (H; m; u) and a k-coalgebra (H; ∆;") such that (a) m and u are coalgebra morphisms ∆(ab) = ∆(a)∆(b) and ∆(1) = 1 ⊗ 1 (a’) ∆ and " are algebra morphisms ∆(ab) = ∆(a)∆(b) and "(ab) = "(a)"(b): • Sweedler Notation. ∆(a) = a1 ⊗ a2. • A Hopf algebra is a bialgebra with a k-linear antipode S S(a1)a2 = a1S(a2) = "(a): October 25, 2020 Symmetry 4 / 20 Quantum symmetries • Bialgebra (co)actions capture quantum symmetries. • Let HM be the category of left H-modules. • A is called a left H-module algebra if A 2 HM such that h · (ab) = (h1 · a)(h2 · b) h · 1A = "(h)1A: Theorem. [Etingof–Walton, 2013] Let H be a semisimple Hopf algebra over an algebraically closed field of characteristic zero. If A is a commutative domain that is an H-module algebra, then the H-action factors through a group action. October 25, 2020 Symmetry 5 / 20 Hopf algebras as quantum symmetries Theorem. [Etingof–Walton] Semisimple Hopf actions on commutative domains are captured by group actions. Motivating question. How do we enlarge this picture? Path algebras of quivers? October 25, 2020 Symmetry 6 / 20 Symmetries as objects in categories • A an associative unital k-algebra. Symmetries captured by: • G a group g · (ab) = (g · a)(g · b) and g · 1A = 1A: A an object, m : A ⊗ A ! A and u : k ! A morphisms in the monoidal category kGM. • H a bialgebra (or Hopf algebra) h · (ab) = (h1 · a)(h2 · b) and h · 1A = "(h)1A: A an object, m and u morphisms in the monoidal category HM. October 25, 2020 Symmetry 7 / 20 Symmetries as objects in categories • (C; ⊗; 1) a monoidal category. An algebra object in C is (A; m : A ⊗ A ! A; u : 1 ! A) where A is an object, m and u are morphisms satisfying: αA;A;A Id ⊗m (A ⊗ A) ⊗ A / A ⊗ (A ⊗ A) / A ⊗ A m⊗IdA m ⊗ A A m / A u⊗Id Id ⊗u 1 ⊗ A / A ⊗ A o A ⊗ 1 m r lA % A y • Can define a coalgebra object by flipping arrows. October 25, 2020 Symmetry 8 / 20 Symmetries as objects in categories H H • If H is a bialgebra, then HM, MH, M , and M are all monoidal. • A right H-comodule algebra is an algebra object in MH. • Equivalently, A 2 MH ρ : A ! A ⊗ H; ρ(a) = a[0] ⊗ a[1] such that ρ(ab) = a[0]b[0] ⊗ a[1]b[1] and ρ(1A) = 1A ⊗ 1H: October 25, 2020 Symmetry 9 / 20 A more general framework? H Look for H with HM or M monoidal. Many generalizations of Hopf algebras: 1. Weak Hopf algebras 2. Quasi-bialgebras with antipode 3. Hopf algebroids 4. Hopf bimonoids in duoidal categories 5. Hopfish algebras October 25, 2020 Symmetry 10 / 20 Weak Hopf algebras • A weak bialgebra H over k is a k-algebra (H; m; u) and a k-coalgebra (H; ∆;") such that (1) ∆(ab) = ∆(a)∆(b), (2) (∆ ⊗ Id) ◦ ∆ = (∆(1) ⊗ 1)(1 ⊗ ∆(1)) = (1 ⊗ ∆(1))(∆(1) ⊗ 1), (3) "(abc) = "(ab1)"(b2c) = "(ab2)"(b1c). • Bialgebra if and only if ∆(1) = 1 ⊗ 1 if and only if "(ab) = "(a)"(b). • A weak Hopf algebra is a weak bialgebra with antipode S: S(a1)a2 = 11"(a12); a1S(a2) = "(11a)12; S(a1)a2S(a3) = S(a): October 25, 2020 Weak bialgebras 11 / 20 Why weak Hopf algebras? • Introduced by [Bohm–Nill–Szlachanyi¨ 1999], motivated by physics: study symmetries in conformal field theory. • Axioms are self-dual, so the dual of a finite-dimensional weak Hopf algebra is again a weak Hopf algebra. Example. If H; K are bialgebras, then H ⊕ K is an algebra as usual and a coalgebra under ∆(h; k) = (h1; 0) ⊗ (h2; 0) + (0; k1) ⊗ (0; k2) "(h; k) = "H(h) + "K(k) But ∆(1; 1) = (1; 0) ⊗ (1; 0) + (0; 1) ⊗ (0; 1) 6= (1; 1) ⊗ (1; 1). So H ⊕ K not a bialgebra, only a weak bialgebra. October 25, 2020 Weak bialgebras 12 / 20 Why weak Hopf algebras? • If G; H are groups, then G t H is not a group, but a groupoid. Example. G is a groupoid. kG the groupoid algebra is a weak Hopf algebra. For g 2 G: ∆(g) = g ⊗ g;"(g) = 1; S(g) = g−1: α G = 1 2 α−1 Then 1 = e1 + e2 but ∆(1) = e1 ⊗ e1 + e2 ⊗ e2 6= 1 ⊗ 1. • There is a natural coaction of a weak bialgebra H(Q) on the path algebra kQ [Hayashi]. October 25, 2020 Weak bialgebras 13 / 20 Why weak Hopf algebras? Example. [Hayashi] p q Q a finite quiver (e.g., Q = 1 2 3 ). Define a weak bialgebra H(Q): k • -basis: fxa;b j a; b 2 Q`; ` 2 Ng. ( xac;bd; if ac and bd compose • Multiplication: xa;bxc;d = 0; otherwise • ∆( ) = P ⊗ Comultiplication: xa;b c2Q` xa;c xc;b P • Unit:1 H(Q) = xi;j • Counit: "(xa;b) = δa;b. • Coacts naturally on the path algebra kQ: X a 7! b ⊗ xb;a: b2Q` October 25, 2020 Weak bialgebras 14 / 20 Why weak Hopf algebras? Theorem. [Hayashi 1999, Szlachanyi 2001] Every fusion category is equivalent to HMfd for some weak Hopf algebra H. • If (H; m; u; ∆;") is an algebra and coalgebra such that ∆(ab) = ∆(a)∆(b), then ∆ axiom ) HM and MH are monoidal, " axiom ) HM and MH are monoidal. • But not ⊗k! [Nill 1998], [Bohm–Caenepeel–Janssen¨ 2011] October 25, 2020 Weak bialgebras 15 / 20 The puzzle • (Co)module categories are monoidal, but not ⊗k. αA;A;A Id ⊗m (A ⊗ A) ⊗ A / A ⊗ (A ⊗ A) / A ⊗ A m⊗IdA m ⊗ A A m / A u⊗Id Id ⊗u 1 ⊗ A / A ⊗ A o A ⊗ 1 m r lA % A y • Algebra objects come with A ⊗ A ! A and 1 ! A. • But k-algebras come with A ⊗k A ! A and k ! A. October 25, 2020 Weak bialgebras 16 / 20 The puzzle • H a weak bialgebra. • Define AH, the category of right H-comodule algebras: • k-algebras A which are right H-comodules ρ : A ! A ⊗ H; ρ(a) = a[0] ⊗ a[1] such that (ab)[0](ab)[1] = a[0]b[0] ⊗ a[1]b[1] and ρ(1A) 2 A ⊗ Ht: • k-algebra morphisms which are H-comodule morphisms. Theorem. [Brzezinski–Caenepeel–Militaru,´ 2002] There is an isomorphism of categories between AH and algebra objects in MH. October 25, 2020 Weak bialgebras 17 / 20 Toward weak quantum symmetry Theorem. [Walton–Wicks–W] H a weak bialgebra. Define categories • CH of right H-comodule coalgebras, and • F H of right H-comodule Frobenius algebras. There are (explicitly given) isomorphisms of categories • CH and coalgebra objects in MH • F H and Frobenius algebra objects in MH. October 25, 2020 Weak bialgebras 18 / 20 Toward weak quantum symmetry • We provide new examples of “weak quantum symmetry” (weak Hopf algebra coactions) from “ordinary quantum symmetry” (Hopf algebra coactions). • Hayashi’s H(Q)-coaction on kQ makes kQ an H(Q)-comodule algebra. • In fact, H(Q) coacts universally on kQ: [Huang–Walton–Wicks–W, arXiv:2008.00606]. October 25, 2020 Weak bialgebras 19 / 20 Thank you! October 25, 2020 Weak bialgebras 20 / 20.