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Hopf algebra∗
mhale†
2013-03-21 14:59:11
A Hopf algebra is a bialgebra A over a field K with a K-linear map S :
A → A, called the antipode, such that
m ◦ (S ⊗ id) ◦ ∆ = η ◦ ε = m ◦ (id ⊗ S) ◦ ∆,
(1)
where m : A ⊗ A → A is the multiplication map m(a ⊗ b) = ab and η : K → A
is the unit map η(k) = k1I.
In terms of a commutative diagram:
k A SSSS
kkk
SSS
SS∆S
SSS
SS)
∆ kkkk
A⊗A
S⊗id
k
kkk
ukkk
ε
K
A⊗A
id⊗S
η
A⊗A S
A⊗A
k
SSS
kkk
SSS
kkk
SSS
k
k
m
k m
SSS
SS) ukkkkk
A
The category of commutative Hopf algebras is anti-equivalent to the category
of affine group schemes. The prime spectrum of a commutative Hopf algebra
is an affine group scheme of multiplicative units. And going in the opposite
direction, the algebra of natural transformations from an affine group scheme
to its affine 1-space is a commutative Hopf algebra, with coalgebra structure
given by dualising the group structure of the affine group scheme. Further, a
commutative Hopf algebra is a cogroup object in the category of commutative
algebras.
∗ hHopfAlgebrai
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hDefinitioni h16W30i
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Example 1 (Algebra of functions on a finite group)
Let A = C(G) be the algebra of complex-valued functions on a finite group G and
identify C(G × G) with A ⊗ A. Then, A is a Hopf algebra with comultiplication
(∆(f ))(x, y) = f (xy), counit ε(f ) = f (e), and antipode (S(f ))(x) = f (x−1 ).
Example 2 (Group algebra of a finite group)
Let A = CG be the complex group algebra of a finite group G. Then, A is a
Hopf algebra with comultiplication ∆(g) = g ⊗ g, counit ε(g) = 1, and antipode
S(g) = g −1 .
The above two examples are dual to one another. Define a bilinear form
C(G) ⊗ CG → C by hf, xi = f (x). Then,
hf g, xi = hf ⊗ g, ∆(x)i,
h1, xi = ε(x),
h∆(f ), x ⊗ yi = hf, xyi,
ε(f )
= hf, ei,
hS(f ), xi = hf, S(x)i,
hf ∗ , xi = hf, S(x)∗ i.
Example 3 (Polynomial functions on a Lie group)
Let A = Poly(G) be the algebra of complex-valued polynomial functions on a
complex Lie group G and identify Poly(G × G) with A ⊗ A. Then, A is a Hopf
algebra with comultiplication (∆(f ))(x, y) = f (xy), counit ε(f ) = f (e), and
antipode (S(f ))(x) = f (x−1 ).
Example 4 (Universal enveloping algebra of a Lie algebra)
Let A = U(g) be the universal enveloping algebra of a complex Lie algebra g.
Then, A is a Hopf algebra with comultiplication ∆(X) = X ⊗ 1 + 1 ⊗ X, counit
ε(X) = 0, and antipode S(X) = −X.
The above two examples are dual to one another (if g is the Lie algebra of
.
.
G). Define a bilinear form Poly(G) ⊗ U(g) → C by t. t=0 f (exp(tX))
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