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Algebraic Structures in Comodule Categories Over Weak Bialgebras

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Algebraic Structures in Comodule Categories Over Weak Bialgebras 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.
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