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We discuss a first simple application.
Corollary 3.1.7.
Let H be a finite-dimensional Hopf algebra. If I ⊂ H is a right ideal and right coideal, then
I = H or I = (0).
Proof.
As a right ideal, I is a right submodule of H. Similarly, as a right coideal, it is a right H-
subcomodule. The condition of a Hopf module is inherited, so I is a Hopf submodule. The
fundamental theorem implies
I ∼ = I coH ⊗ K H .
Taking dimensions, we find
which only leaves I = (0) or I = H.
dim K H ∙ dim K I coH = dim K I ≤ dim K H
✷
Definition 3.1.8
1. Let H be a Hopf algebra. The K-linear subspace
I l (H) := {x ∈ H | h ∙ x = ɛ(h)x for all h ∈ H}
is called the space of left integrals of the Hopf algebra H. Similarly,
I r (H) := {x ∈ H | x ∙ h = ɛ(h)x for all h ∈ H}
is called the space of right integrals of H.
2. Similarly, the subspace of H ∗
CI l (H) := {φ ∈ H ∗ | (id H ⊗ φ) ◦ Δ H (h) = 1 H φ(h) for all h ∈ H}
is called the space of left cointegrals. Right cointegrals are defined analogously.
3. A Hopf algebra is called unimodular, if I l (H) = I r (H).
Remarks 3.1.9.
1. The space of left integrals is the space of left invariants for the left action of H on itself
by multiplication. Alternatively, h ∈ I l (H) ⊂ H ∼ = Hom K (K, H) is a morphism of left
H-modules. A similar statement holds for right integrals.
2. Even if a Hopf algebra H is cocommutative, it might not be unimodular. For an example,
see Montgomery, p. 17.
3. Let H be finite-dimensional. Then φ ∈ H ∗ is a left integral for the dual Hopf algebra H ∗ ,
if and only if
μ ∗ (β, φ) = ɛ ∗ (β)φ for all β ∈ H ∗ .
Using the definition of the bialgebra structure on H ∗ , this amounts to the equality
β(h (1) ) ∙ φ(h (2) ) = β(1 H ) ∙ φ(h) for all β ∈ H ∗ and h ∈ H .
Thus φ is an integral of H ∗ , if and only if
i.e. if φ is a left cointegral.
h (1) 〈φ, h (2) 〉 = 〈φ, h〉1 H for all h ∈ H ,
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