A (finite) quantum groupoid over k is a finite-dimensional k-vector space H with the structures of an associative algebra (H, m, 1) with multiplication m : H ⊗_{k} H → H and unit 1 ∈ H and a coassociative coalgebra (H, ∆, ε) with comultiplication ∆ : H → H ⊗_{k} H and counit ε : H → k such that:

- The comultiplication ∆ is a (not necessarily unit-preserving) homomorphism of algebras such that

(∆ ⊗ id)∆(1) = (∆(1) ⊗ 1) (1 ⊗ ∆(1)) = (1 ⊗ ∆(1)) (∆(1) ⊗ 1) —– (1)

2. The counit is a k-linear map satisfying the identity:

ε(fgh) = ε(fg_{(1)})ε(g_{(2)}h) = ε(f_{g(2)})ε(_{g(1)}h), (2) ∀ f, g, h ∈ H —– (2)

3. There is an algebra and coalgebra anti-homomorphism S : H → H, called an antipode, such that, ∀ h ∈ H ,

m(id ⊗ S) ∆(h) = (ε ⊗ id) ∆(1)(h ⊗ 1) —– (3)

m(S ⊗ id) ∆(h) = (id ⊗ ε)(1 ⊗ h) ∆(1) —– (4)

A quantum groupoid is a Hopf algebra iff one of the following equivalent conditions holds: (i) the comultiplication is unit preserving or (ii) the counit is a homomorphism of algebras.

A morphism of quantum groupoids is a map between them which is both an algebra and a coalgebra morphism preserving unit and counit and commuting with the antipode. The image of such a morphism is clearly a quantum groupoid. The tensor product of two quantum groupoids is defined in an obvious way.

The set of axioms is self-dual. This allows to define a natural quantum groupoid structure on the dual vector space H’ = Hom_{k} (H, k) by “reversing the arrows”:

⟨h,φ ψ⟩ = ∆(h), φ ⊗ ψ —– (5)

⟨g ⊗ h, ∆'(φ)⟩ = ⟨gh, φ⟩ —– (6)

⟨h, S'(φ)⟩ = ⟨S(h), φ⟩ —– (7)

∀ φ, ψ ∈ H’, g, h ∈ H. The unit 1ˆ ∈ H’ is ε and counit ε’ is φ → ⟨φ,1⟩. The linear endomorphisms of H defined by

h → m(id ⊗ S) ∆(h), h → m(S ⊗ id) ∆(h) —– (8)

are called the target and source counital maps and denoted ε_{t} and ε_{s}, respectively.

From axioms (3) and (4),

ε_{t}(h) = (ε ⊗ id) ∆(1)(h ⊗ 1), ε_{s}(h) = (id ⊗ ε) (1 ⊗ h)∆(1) . (9)

In the Hopf algebra case ε_{t}(h) = ε_{s}(h) = ε(h)1.

We have S ◦ ε_{s} = ε_{t} ◦ S and ε_{s} ◦ S = S ◦ ε_{t}. The images of these maps ε_{t} and ε_{s}

H_{t} = ε_{t} (H) = {h ∈ H | ∆(h) =∆(1)(h ⊗ 1)} —– (10)

H_{s} = ε_{s} (H) = {h ∈ H | ∆(h) = (1⊗h) ∆(1)} —– (11)

are subalgebras of H, called the target (respectively source) counital subalgebras. They play the role of ground algebras for H. They commute with each other and

H_{t} = {(φ ⊗ id) ∆(1)|φ ∈ H’,

H_{s} = (id ⊗ φ) ∆(1)| φ ∈ H’,

i.e., H_{t} (respectively H_{s}) is generated by the right (respectively left) tensorands of ∆(1). The restriction of S defines an algebra anti-isomorphism between H_{t} and H_{s}. Any morphism H → K of quantum groupoids preserves counital subalgebras, i.e., H_{t} ≅ K_{t} and H_{s} ≅ K_{s}.

In what follows we will use the Sweedler arrows, writing ∀ h ∈ H , φ ∈ H’:

h ⇀ φ = φ_{(1)}⟨h, φ_{(2)}⟩,

φ ↼ h = ⟨h, φ_{(1)}⟩φ_{(2)} —– (12)

∀ h ∈ H, φ ∈ H’. Then the map z → (z ⇀ ε) is an algebra isomorphism between H_{t} and H. Similarly, the map y → (ε ↼ y) is an algebra isomorphism between H and H’_{t}. Thus, the counital subalgebras of H’ are canonically anti-isomorphic to those of H. A quantum groupoid H is called connected if H_{s} ∩ Z(H) = k, or, equivalently, H_{t} ∩ Z(H ) = k, where Z(H) denotes the center of H. A k-algebra A is separable if the multiplication epimorphism m : A ⊗_{k} A → A has a right inverse as an A − A bimodule homomorphism. When the characteristic of k is 0, this is equivalent to the existence of a separability element e ∈ A ⊗_{k} A such that m(e) = 1 and (a ⊗ 1)e = e(1 ⊗ a), (1 ⊗ a)e = e(a ⊗ 1) ∀ a ∈ A. The counital subalgebras H_{t} and H_{s} are separable, with separability elements e_{t} = (S ⊗ id)∆(1) and e_{s} = (id ⊗S)∆(1), respectively. Observe that the adjoint actions of 1 ∈ H give rise to non-trivial maps

H → H : h → 1_{(1)}h_{S}(1_{(2)}) = Ad^{l}_{1}(h), h → S(1_{(1)})h1_{(2)} = Ad^{r}1(h), h ∈ H —– (13) …….