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The algebra of a two-object discrete groupoid is weak Hopf

Statement

Let $H = C e_{0} \oplus C e_{1}$ with $e_{i} e_{j} = δ_{i j} e_{i}$ , unit $1 = e_{0} + e_{1}$ , coproduct $Δ (e_{i}) = e_{i} \otimes e_{i}$ , counit $ε (e_{i}) = 1$ , and antipode $S (e_{i}) = e_{i}$ . Then $H$ is a weak Hopf algebra with $H_{t} = H_{s} = H$ and $Δ (1) = e_{0} \otimes e_{0} + e_{1} \otimes e_{1} \neq 1 \otimes 1.$ For any left $H$ -modules $V, W$ , give $V$ the right $H_{t}$ -action $v ◃ x = ε_{s}^{'} (x) v$ and let $H_{t}$ act on $W$ by restriction. Here $V \otimes_{H_{t}} W$ is the quotient of $V \otimes_{k} W$ by the span of $(v ◃ x) \otimes w - v \otimes x w$ . The induced diagonal action is $e_{i} \cdot [v \otimes w] = [e_{i} v \otimes e_{i} w]$ , and this relative tensor product is naturally isomorphic as a left $H$ -module to $e_{0} V \otimes e_{0} W \oplus e_{1} V \otimes e_{1} W = Δ (1) (V \otimes W) .$ The counit is not multiplicative, so this weak Hopf algebra is not an ordinary bialgebra.

Diagram

The underlying discrete groupoid has two objects and only their identity arrows; no arrow connects the two objects.

Facts & Assumptions

Given: The algebra $H = C e_{0} \oplus C e_{1}$ , the structure maps in the Statement, and unital left $H$ -modules $V, W$ .

[F1]

Weak bialgebras, weak Hopf algebras, and counital maps: the weak unit, weak counit, and weak antipode identities, and the formulas for $ε_{t}$ , $ε_{s}$ , and $ε_{s}^{'}$ .

[F2]

Unital left and right modules over a ring; unqualified module means left module: a left module action is unital and associative.

[F3]

Module homomorphism and isomorphism, kernel, image and cokernel: an $H$ -linear map preserves addition and the left $H$ -action.

[F4]

Bialgebras and bialgebra maps: an ordinary bialgebra has multiplicative counit $ε$ .

Verification

1.1givenF1algebra

Multiplication is associative and unital. The coproduct is coassociative, and $(ε \otimes i d) Δ (e_{i}) = e_{i} = (i d \otimes ε) Δ (e_{i})$ . Also $Δ (e_{i}) Δ (e_{j}) = δ_{i j} e_{i} \otimes e_{i} = Δ (e_{i} e_{j})$ , so $Δ$ is multiplicative.

2.1givenF1step 1.1algebra

Put $E = Δ (1) = \sum_{i} e_{i} \otimes e_{i}$ . Both iterated coproducts of $E$ equal $\sum_{i} e_{i} \otimes e_{i} \otimes e_{i}$ , and $(E \otimes 1) (1 \otimes E) = (1 \otimes E) (E \otimes 1) = \sum_{i} e_{i} \otimes e_{i} \otimes e_{i}$ by orthogonality. For basis triples $e_{i}, e_{j}, e_{k}$ , each weak-counit product equals $1$ exactly when $i = j = k$ and is $0$ otherwise; trilinearity proves both weak-counit identities. The counital-map formulas give $ε_{t} (e_{i}) = e_{i} = ε_{s} (e_{i})$ . Finally, $S (e_{i}) = e_{i}$ gives $e_{i} S (e_{i}) = ε_{t} (e_{i})$ , $S (e_{i}) e_{i} = ε_{s} (e_{i})$ , and $S (e_{i}) e_{i} S (e_{i}) = S (e_{i})$ . Thus all weak Hopf axioms hold by linearity.

3.1F1F4step 2.1algebra

The counital formulas also give $ε_{s}^{'} (e_{i}) = \sum_{j} e_{j} ε (e_{j} e_{i}) = e_{i}$ , hence $H_{t} = H_{s} = H$ . Since $E$ has no off-diagonal terms whereas $1 \otimes 1$ has them, $Δ (1) \neq 1 \otimes 1$ . Moreover $ε (e_{0} e_{1}) = 0$ but $ε (e_{0}) ε (e_{1}) = 1$ , so the structure is not an ordinary bialgebra.

4.1F1F2step 3.1algebra

Orthogonal idempotents summing to $1$ give direct sums $V = e_{0} V \oplus e_{1} V$ and $W = e_{0} W \oplus e_{1} W$ by the left module axioms. On each elementary summand $e_{i} V \otimes e_{j} W$ , $E$ acts as multiplication by $δ_{i j}$ , so its image is $(e_{0} V \otimes e_{0} W) \oplus (e_{1} V \otimes e_{1} W)$ . Also $v ◃ e_{i} = e_{i} v$ . The balancing relation for $x = e_{i}$ kills $e_{i} V \otimes e_{j} W$ when $i \neq j$ , while for $i = j$ the left and right actions agree. For $v_{i} = e_{i} v$ and $w_{j} = e_{j} w$ , the relation for $x = a_{0} e_{0} + a_{1} e_{1}$ is $(a_{i} - a_{j}) v_{i} \otimes w_{j}$ . It vanishes when $i = j$ ; when $i \neq j$ , choosing $x = e_{i}$ generates the whole summand $e_{i} V \otimes e_{j} W$ . Thus the relation subspace is exactly the sum of the two off-diagonal summands.

5.1F1F3step 4.1algebra∎

The quotient and image are both the direct sum of the diagonal summands by step 4.1, and the quotient map and its inverse are the identity on each such summand. On each diagonal summand $e_{j} V \otimes e_{j} W$ , both the image and quotient actions of $e_{i}$ are multiplication by $δ_{i j}$ ; hence the isomorphism is $H$ -linear. If $f : V \to V^{'}$ and $g : W \to W^{'}$ are $H$ -linear, then [F3] gives $f (e_{i} v) = e_{i} f (v)$ and $g (e_{i} w) = e_{i} g (w)$ , so $f \otimes g$ preserves each diagonal summand and the isomorphism is natural in $V, W$ .

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Sources

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories (standard reference, not scraped)
P. Schauenburg, Weak Hopf Algebras and Quantum Groupoids (standard reference, not scraped)