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Introduction to Hopf Algebras
Rob Ray
February 26, 2004
The last twenty years has seen a number of advances in
the area of Hopf algebras. Among these are the intro-
duction of quantum groups, and the unification of group
actions, actions of Lie Algebras, and graded algebras .
As an introduction to Hopf algebras, this talk will cover
basic definitions and some examples. We first briefly dis-
cuss the idea of a tensor product of vector spaces and
then move on to algebras, coalgebras, Hopf algebras and
finally quantum groups. In order to cover the amount of
material, many of the details will be omitted and there
will be no proofs, i.e., in order to present a view of the
forest, there will be no pictures of pine needles. How-
ever, those interested can find most of the material in
Montgomery[Mon91] and Kassel [Kas95]. If time per-
mits, some quantum groups will also be discussed.
1
Tensor Product
↓Algebras and Coalgebras
↓Bialgebras and Hopf Algebras
↓Quantum Groups
↓Examples
2
Tensor Products
Let V and W be any two vector spaces over a field F.
V ⊗ W can be thought of as the space of objects that
look like
v1 ⊗ w1 + v2 ⊗ w2 + · · · + vk ⊗ wk
where
vi ∈ V, wi ∈ W
and with the following bilinear relations
a1(v1 ⊗ w) + a2(v2 ⊗ w) = ((a1v1 + a2v2)⊗ w) (1)
and
a1(v ⊗ w1) + a2(v ⊗ w2) = (v ⊗ (a1w1 + a2w2)) (2)
where ai ∈ F, v, vi ∈ V and w, wi ∈ W .
3
Remark 0.1 If BV is a basis for V and BW is a basis
for W , then
BV⊗W = {e⊗ f |e ∈ BV and f ∈ BW}
is a basis for V ⊗W
4
Example 0.1 Consider the polynomials of one vari-
able over CC[x]
These polynomials are a vector space with basis
B = {1, x, x2, x3, x4, . . .}
So examples of objects in C[x]⊗ C[x] are
4x2 ⊗ 3x
x⊗ 3
(x2 − 4)⊗ (x3 − 2x2 + x)
x2 ⊗ x2 + x2 ⊗ 4x3 + (x− 2x2)⊗ x3
Note: We may use the bilinear relations 1 and 2 to do
the following:
x2 ⊗ x2 + x2 ⊗ 4x3 + (x− 2x2)⊗ x3)
= x2 ⊗ x2 + 4x2 ⊗ x3
+(x− 2x2)⊗ x3
= x2 ⊗ x2 + (4x2 + x− 2x2)⊗ x3
= x2 ⊗ x2 + (2x2 + x)⊗ x3
5
Algebras and Coalgebras
Definition 0.1 An algebra over a field k is a vector
space, A, together with two linear maps, a multiplica-
tion µ : A⊗ A → A, and a unit map η : k → A such
that the following diagrams commute:
A⊗ A A-µ
A⊗ A⊗ A A⊗ A-µ⊗ id
?
id⊗ µ
?
µ
and
A
@@
@@
@@
@R
k ⊗ A A⊗ A-η ⊗ id
A⊗ k�id⊗ η
?
µ
��
��
��
�
where the lower left and right maps are simply scalar
multiplication.
6
Example 0.2 Considering the example of C[x]⊗C[x],
we define µ as the standard polynomial multiplication.
E.g.
µ(x⊗ 3) = 3x
µ(4x2 ⊗ 3x) = 12x3
µ((x2 − 4)⊗ (x3 − 2x2 + x)
)= x5 − 2x4 − 3x3 + 8x2 − 4x
µ(x2 ⊗ x2 + x2 ⊗ 4x3 + (x− 2x2)⊗ x3
)= x4 + 4x5 + x4 − 2x5
= 2x4 + 2x5
7
Example 0.3 (A(X)) Now consider the polynomial
algebra
A(X) = C[x1,1, x1,2, x2,1, x2,2] (3)
As a vector space, it’s basis is
{xi1,1x
j1,2x
k2,1x
l2,2 : i, j, k, l ≥ 0 ∈ Z}
and examples of elements of A(X)
x1,2x2,2 − 3x1,1x2,2 + x2,2
x1,1x2,2 − x1,2x2,1
If we think of
X =
[x1,1 x1,2
x2,1 x2,2
]then we can think of the polynomials of A(X) as func-
tions from Mat(2, C) to C. They are in fact often
called the regular functions of Mat(2, C). Let f =
x1,1x2,2 − x1,2x2,1 then
f
([1 1
0 1
])= 1 · 1− 1 · 0 = 1
8
Definition 0.2 If A and B are algebras with respec-
tive multiplications µA and µB then
A linear map g : A → B is an algebra mor-
phism if g ◦ µA = µB ◦ (g ⊗ g) i.e. the following
diagram commutes.
B ⊗B B-µB
A⊗ A A-µA
?
g ⊗ g?
g
I.e. if a, b ∈ A then
g(ab) = g(a)g(b)
9
Definition 0.3 A coalgebra is a vector space C to-
gether with two linear maps, comultiplication ∆ : C →C ⊗ C and counit ε : C → k, such that the following
two diagrams commute.
C ⊗ C C ⊗ C ⊗ C-
id⊗∆
C C ⊗ C-∆
?
∆?
∆⊗ id
and
C ⊗ C
ε⊗ id
@@
@@
@@
@I
k ⊗ C C� 1⊗C ⊗ k-⊗1
?
∆ id⊗ ε
��
��
��
��
10
Example 0.4 (A(X)) We see that if we define the
comultiplication and counit maps on A(X) in the fol-
lowing manner, then A(X) is a coalgebra.
∆(xi,j) = xi,1 ⊗ x1,j + xi,2 ⊗ x2,j (4)
ε(xi,j) = δi,j =
{1, i = j
0, i 6= j
}(5)
We extend the action of ∆ to the rest of A(X) by
defining it to be an algebra morphism. I.e.
∆(xi,jxk,l) = ∆(xi,j)∆(xk,l)
E.g.
∆(x1,1) = x1,1 ⊗ x1,1 + x1,2 ⊗ x2,1
ε(x1,1) = 1
ε(x1,2) = 0
11
Example 0.5 (Coassociativity of ∆ on A(X))
(id⊗∆) ◦∆(x1,2) = (id⊗∆)(x1,1 ⊗ x1,2 + x1,2 ⊗ x2,2)
= (x1,1 ⊗ (x1,1 ⊗ x1,2 + x1,2 ⊗ x2,2)
+x1,2 ⊗ (x2,1 ⊗ x1,2 + x2,2 ⊗ x2,2))
= x1,1 ⊗ x1,1 ⊗ x1,2 + x1,1 ⊗ x1,2 ⊗ x2,2
+x1,2 ⊗ x2,1 ⊗ x1,2 + x1,2 ⊗ x2,2 ⊗ x2,2
and
(∆⊗ id) ◦∆(x1,2) = (∆⊗ id)(x1,1 ⊗ x1,2 + x1,2 ⊗ x2,2)
= (x1,1 ⊗ x1,1 + x1,2 ⊗ x2,1)⊗ x1,2
+(x1,1 ⊗ x1,2 + x1,2 ⊗ x2,2)⊗ x2,2
= x1,1 ⊗ x1,1 ⊗ x1,2 + x1,2 ⊗ x2,1 ⊗ x1,2
x1,1 ⊗ x1,2 ⊗ x2,2 + x1,2 ⊗ x2,2 ⊗ x2,2
Example 0.6 (Counit , ε, of A(X))
x1,1 ⊗ x1,2 + x1,2 ⊗ x2,2
ε⊗ id
@@
@@
@@
@I
1⊗ x1,2 x1,2�1⊗ x1,2 ⊗ 1-⊗1
?
∆ id⊗ ε
��
��
��
��
12
Definition 0.4 Let C be any coalgebra, and let c ∈C. c is group-like if ∆(c) = c⊗ c and ε(c) = 1. The
set of group-like elements in C is denoted by G(C).
Example 0.7 In A(X) there is a special element
det = x1,1x2,2 − x1,2x2,1
It can be shown that det is group-like.
∆(det) = det⊗ det
13
∆(det) = ∆(x1,1x2,2 − x1,2x2,1)
= ∆(x1,1)∆(x2,2)−∆(x1,2)∆(x2,1)
= (x1,1 ⊗ x1,1 + x1,2 ⊗ x2,1)(x2,1 ⊗ x1,2 + x2,2 ⊗ x2,2)
−(x1,1 ⊗ x1,2 + x1,2 ⊗ x2,2)(x2,1 ⊗ x1,1 + x2,2 ⊗ x2,1)
= (x1,1 ⊗ x1,1)(x2,1 ⊗ x1,2) + (x1,1 ⊗ x1,1)(x2,2 ⊗ x2,2)
+(x1,2 ⊗ x2,1)(x2,1 ⊗ x1,2) + (x1,2 ⊗ x2,1)(x2,2 ⊗ x2,2)
−(x1,1 ⊗ x1,2)(x2,1 ⊗ x1,1)− (x1,1 ⊗ x1,2)(x2,2 ⊗ x2,1)
−(x1,2 ⊗ x2,2)(x2,1 ⊗ x1,1)− (x1,2 ⊗ x2,2)(x2,2 ⊗ x2,1)
= (x1,1x2,1 ⊗ x1,1x1,2) + (x1,1x2,2 ⊗ x1,1x2,2)
+(x1,2x2,1 ⊗ x2,1x1,2) + (x1,2x2,2 ⊗ x2,1x2,2)
−(x1,1x2,1 ⊗ x1,2x1,1)− (x1,1x2,2 ⊗ x1,2x2,1)
−(x1,2x2,1 ⊗ x2,2x1,1)− (x1,2x2,2 ⊗ x2,2x2,1)
= (x1,1x2,2 ⊗ x1,1x2,2)− (x1,1x2,2 ⊗ x1,2x2,1)
+(x1,2x2,1 ⊗ x2,1x1,2)− (x1,2x2,1 ⊗ x2,2x1,1)
= x1,1x2,2 ⊗ (x1,1x2,2 − x1,2x2,1)
+x1,2x2,1 ⊗ (x2,1x1,2 − x2,2x1,1)
= x1,1x2,2 ⊗ (x1,1x2,2 − x1,2x2,1)
−x1,2x2,1 ⊗ (x2,2x1,1 − x2,1x1,2)
= (x1,1x2,2 − x1,2x2,1)⊗ (x1,1x2,2 − x1,2x2,1)
= det⊗ det
14
Bialgebras and Hopf Algebras
Definition 0.5 Given a space B, B is a bialgebra
if (B, ∆, ε) is a coalgebra, (B, µ, η) is an algebra and
either of the following equivalent conditions is true:
1. ∆ and ε are algebra morphisms
2. µ and η are coalgebra morphisms
This bialgebra structure is often denoted by (B, µ, η, ∆, ε).
Remark 0.2 A(X) is a bialgebra.
15
Definition 0.6 (Convolution) Given an algebra (A, µ, η),
a coalgebra (C, ∆, ε) and two linear maps f, g : C →A then the convolution of f and g is the linear map
f ? g : C → A defined by
(f ? g)(c) = µ ◦ (f ⊗ g) ◦∆(c), c ∈ C
Definition 0.7 (Antipode and Hopf Algebra) Let
(H, µ, η, ∆, ε) be a bialgebra. An endomorphism S of
H is called an antipode for the bialgebra H if
idH ? S = S ? idH = η ◦ ε
A Hopf algebra is a bialgebra with an antipode.
16
Example 0.8 (A(G)) Unfortunately, A(X) has no an-
tipode, so we construct a new space.
A(G) = C[x1,1, x1,2, x2,1, x2,2, det−1
](6)
The comultiplication and counit are defined as with
A(X)
∆(xi,j) =
2∑k=1
xi,k ⊗ xk,j (7)
ε(xi,j) = δi,j (8)
Then
S(x1,1) = x2,2det−1
S(x1,2) = −x1,2det−1
S(x2,1) = −x2,1det−1
S(x2,2) = x1,1det−1
defines the antipode for A(G) and makes A(G) a Hopf
algebra.
17
Quantum Groups
Example 0.9 Let q ∈ C such that it is not a root of
unity, then define
Aq(X) = C[x1,1, x1,2, x2,1, x2,2] (9)
with added relations:
x1,1x1,2 = qx1,2x1,1
x2,1x2,2 = qx2,2x2,1
x1,1x2,1 = qx2,1x1,1
x1,2x2,2 = qx2,2x1,2
x1,2x2,1 = x2,1x1,2
x1,1x2,2 = x2,2x1,1 +
(q − 1
q
)x1,2x2,1
Now if we define ∆ and ε as we did for A(X), Aq(X)
becomes a bialgebra.
18
Definition 0.8 We define the quantum determinant
by
detq = x1,1x2,2 − qx1,2x2,1 ∈ Aq(X) (10)
19
Example 0.10 (Aq(Gl(n)), a quantum Hopf algebra)
Aq(Gl(n)) = C[x1,1, x1,2, x2,1, x2,2, detq
−1]
(11)
The comultiplication and counit are defined as with
A(X) Then
S(x1,1) = det−1q x2,2
S(x1,2) = −qdet−1q x1,2
S(x2,1) = −1
qdet−1
q x2,1
S(x2,2) = det−1q x1,1
defines an antipode.
20
More Examples
Example 0.11 (Divided Powers) If we let C = C[t]
be the polynomials of one variable over C, then we can
define a comultiplication and counit by
∆(tn) =∑
p+q=n
tp ⊗ tq
ε(tn) = δn0
We can demonstrate the coassociativity (for n = 2)
with the following calculations:
(id⊗∆) ◦∆(t2) = (id⊗∆)(t2 ⊗ 1 + t⊗ t + 1⊗ t2)
= t2 ⊗ 1⊗ 1 + t⊗ 1⊗ t + t⊗ t⊗ 1
+1⊗ t2 ⊗ 1 + 1⊗ t⊗ t + 1⊗ 1⊗ t2
= (∆⊗ id)(t2 ⊗ 1 + t⊗ t + 1⊗ t2)
= (∆⊗ id) ◦∆(t2)
To show that ε(tn) = δn0 defines a counit map we
check that (id⊗ ε) ◦∆(tn) = tn ⊗ 1 and that (ε⊗ id) ◦∆(tn) = 1⊗ tn
(id⊗ ε) ◦∆(tn) = (id⊗ ε)
( ∑p+q=n
tp ⊗ tq
)= tn ⊗ 1
Similarly, (ε⊗ id) ◦∆(tn) = 1⊗ tn.
21
Example 0.12 If we let G be a group then B = CG,
the associated group algebra, becomes a bialgebra with
the following defined maps
∆(g) = g ⊗ g, ∀g ∈ G
ε(g) = 1, ∀g ∈ G
and a Hopf algeba with the antipode defined by
S(g) = g−1
Remark 0.3 The set of group-like elements of this
Hopf algebra is the original group G.
22
Example 0.13 (U(sl(2))) Consider the universal en-
veloping algebra of sl(2), U(sl(2)). One can think of
U(sl(2)) as the polynomial algebra of three generators
e, f , and h, with the added relations
[e, f ] = h, [h, e] = 2x, [h, f ] = −2y (12)
Also, note that the set {eif jhk : i, j, k ≥ 0 ∈ Z+} is a
basis of U(sl(2)) as a result of the Poincare-Birkhoff-
Witt theorem [Kas95]. If we define the comultipli-
cation and counit maps on U(sl(2)) in the following
manner, then it has a bialgebra structure.
∆(x) = x⊗ 1 + 1⊗ x, ε(x) = 0, ∀x ∈ sl(2) (13)
S(x) = −x (14)
Infact, the enveloping algebra of any Lie algebra is a Hopf
algebra with the above definitions (13 and 14).
23
Example 0.14 (Quantum Plane) Choose q to be
an invertible element in C and define
Cq [x, y] = C {x, y|xy = qyx} (15)
With comultiplication and counits defined as
∆(x) = x⊗ x, ∆(y) = y ⊗ 1 + 1⊗ y (16)
and
ε(x) = 1, ε(y) = 0 (17)
Cq [x, y] is a bialgebra.
24
Example 0.15 A four dimensional noncommutative,
noncocomutative Hopf algebra [Mon91].
H4 ={1, g, x, gx|g2 = 1, x2 = 0, xg = −gx
}(18)
with structure maps
∆g = g ⊗ g
∆x = 1⊗ x + x⊗ 1
ε(g) = 1
ε(x) = 0
S(g) = g = g−1
S(x) = −gx
25
References
[Kas95] Christian Kassel. Quantum Groups. Springer-
Verlag New York, Inc., 1995.
[Mon91] Susan Montgomery. Hopf algebras and their ac-
tions on rings. In CBMS Regional Conference
Series in Mathematics, volume 82. Conference
Board of the Mathematical Sciences, American
Mathematical Society, 1991.
26
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