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, ∆, ε) such that (a) m and u are coalgebra morphisms

∆(ab) = ∆(a)∆(b) and ∆(1) = 1 ⊗ 1

(a’) ∆ and ε are algebra morphisms

∆(ab) = ∆(a)∆(b) and ε(ab) = ε(a)ε(b).

• Sweedler Notation. ∆(a) = a1 ⊗ a2. • A Hopf algebra is a bialgebra with a k-linear antipode S

S(a1)a2 = a1S(a2) = ε(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 ∈ HM such that

h · (ab) = (h1 · a)(h2 · b)

h · 1A = ε(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 = ε(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 ∈ 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, ∆, ε) such that

(1) ∆(ab) = ∆(a)∆(b), (2) (∆ ⊗ Id) ◦ ∆ = (∆(1) ⊗ 1)(1 ⊗ ∆(1)) = (1 ⊗ ∆(1))(∆(1) ⊗ 1),

(3) ε(abc) = ε(ab1)ε(b2c) = ε(ab2)ε(b1c).

• Bialgebra if and only if ∆(1) = 1 ⊗ 1 if and only if ε(ab) = ε(a)ε(b). • A weak Hopf algebra is a weak bialgebra with antipode S:

S(a1)a2 = 11ε(a12), a1S(a2) = ε(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)

ε(h, k) = εH(h) + εK(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 ∈ G: ∆(g) = g ⊗ g, ε(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: {xa,b | a, b ∈ Q`, ` ∈ N}. ( xac,bd, if ac and bd compose • Multiplication: xa,bxc,d = 0, otherwise • ∆( ) = P ⊗ Comultiplication: xa,b c∈Q` xa,c xc,b P • Unit:1 H(Q) = xi,j

• Counit: ε(xa,b) = δa,b.

• Coacts naturally on the path algebra kQ: X a 7→ b ⊗ xb,a.

b∈Q`

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, ∆, ε) is an algebra and coalgebra such that ∆(ab) = ∆(a)∆(b), then ∆ axiom ⇒ HM and MH are monoidal,

ε 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) ∈ 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