CHAPTER 2 Hopf Algebras, Algebraic, Formal, and Quantum Groups

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5. Quantum Groups

Definition 2.5.1. (Drinfel’d) A quantum group is a noncommutative noncocommutative Hopf algebra.

Remark 2.5.2. We shall consider all Hopf algebras as quantum groups. Observe, however, that the commutative Hopf algebras may be considered as affine algebraic groups and that the cocommutative Hopf algebras may be considered as formal groups. Their property as a quantum space or as a quantum monoid will play some role. But often the (possibly nonexisting) dual Hopf algebra will have the geometrical meaning. The following examples 𝕊𝕃_{q}(2) and 𝔾𝕃_{q}(2) will have a geometrical meaning.

Example 2.5.3. The smallest proper quantum group, i.e. the smallest noncommutative noncocommutative Hopf algebra, is the 4-dimensional algebra

H_{4} := 𝕂⟨g, x⟩/(g^{2} − 1, x^{2}, xg + gx)

which was first described by M. Sweedler. The coalgebra structure is given by

\begin{matrix}Δ(g) = g ⊗ g, & Δ(x) = x ⊗ 1 + g ⊗ x, \\ ε(g) = 1, & ε(x) = 0, \\ S(g) = g^{−1}(= g), & S(x) = −gx. \end{matrix}

Since it is finite dimensional its linear dual H_{4}^{∗} is also a noncommutative noncocommutative Hopf algebra. It is isomorphic as a Hopf algebra to H_{4}. In fact H_{4} is up to isomorphism the only noncommutative noncocommutative Hopf algebra of dimension 4.

Example 2.5.4. The affine algebraic group 𝕊𝕃(n) : 𝕂-cAlg ⟶ Gr defined by 𝕊𝕃(n)(A), the group of n × n-matrices with coefficients in the commutative algebra A and with determinant 1, is represented by the algebra O(𝕊𝕃(n)) = SL(n) = 𝕂[x_{ij}]/(det(x_{ij}) − 1) i.e.

𝕊𝕃(n)(A) ≅ 𝕂-cAlg(𝕂[x_{ij}]/(det(x_{ij}) − 1), A).

Since 𝕊𝕃(n)(A) has a group structure by the multiplication of matrices, the representing commutative algebra has a Hopf algebra structure with the diagonal Δ = ι_{1} ∗ ι_{2} hence

Δ(x_{ik}) = ∑ x_{ij} ⊗ x_{jk},

the counit ε(x_{ij}) = δ_{ij} and the antipode S(x_{ij}) = adj(X)_{ij} where adj(X) is the adjoint matrix of X = (x_{ij}). We leave the verification of these facts to the reader.

We consider 𝕊𝕃(n) ⊆ M_{n} = 𝔸^{n^{2}} as a subspace of the n^{2}-dimensional affine space.

Example 2.5.5. Let M_{q}(2) = 𝕂 ⟨(\begin{matrix}a & b \\ c & d \end{matrix})⟩ /I as in 1.3.6 with I the ideal gen-

erated by

ab − q^{−1}ba, ac − q^{−1}ca, bd − q^{−1}db, cd − q^{−1}dc, (ad − q^{−1}bc) − (da − qcb), bc − cb.

We first define the quantum determinant det_{q} = ad − q^{−1}bc = da − qcb in M_{q}(2). It is a central element. To show this, it suffices to show that det_{q} commutes with the generators a, b, c, d:

\begin{matrix}(ad − q^{−1}bc)a = a(da − qbc), & (ad − q^{−1}bc)b = b(ad − q^{−1}bc), \\ (ad − q^{−1}bc)c = c(ad − q^{−1}bc), & (da − qbc)d = d(ad − q^{−1}bc). \end{matrix}

We can form the quantum determinant of an arbitrary quantum matrix in A by

det_{q} (\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}) := a′d′ − q^{−1}b′c′ = d′a′ − qc′b′ = φ(det_{q})

if φ : M_{q}(2) ⟶ A is the algebra homomorphism associated with the quantum matrix

(\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}).

Given two commuting quantum 2×2-matrices (\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}) , (\begin{matrix}a′′ & b′′ \\ c′′ & d′′ \end{matrix}). The quantum

determinant preserves the product, since

det_{q}((\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}) (\begin{matrix}a′′ & b′′ \\ c′′ & d′′ \end{matrix})) = det_{q} (\begin{matrix}a′a′′ + b′c′′ & a′b′′ + b′d′′ \\ c′a′′ + d′c′′ & c′b′′ + d′d′′ \end{matrix})

= (a′a′′ + b′c′′)(c′b′′ + d′d′′) − q^{−1}(a′b′′ + b′d′′)(c′a′′ + d′c′′) = a′c′a′′b′′ + b′c′c′′b′′ + a′d′a′′d′′ + b′d′c′′d′′

−q^{−1}(a′c′b′′a′′ + b′c′d′′a′′ + a′d′b′′c′′ + b′d′d′′c′′) = b′c′c′′b′′ + a′d′a′′d′′ − q^{−1}b′c′d′′a′′ − q^{−1}a′d′b′′c′′ = b′c′c′′b′′ + a′d′a′′d′′ − q^{−1}b′c′d′′a′′ − q^{−1}a′d′b′′c′′ −q^{−1}b′c′(a′′d′′ − d′′a′′ − q^{−1}b′′c′′ + qc′′b′′) = a′d′a′′d′′ − q^{−1}a′d′b′′c′′ − q^{−1}b′c′(a′′d′′ − q^{−1}b′′c′′) = (a′d′ − q^{−1}b′c′)(a′′d′′ − q^{−1}b′′c′′)

= det_{q} (\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}) det_{q} (\begin{matrix}a′′ & b′′ \\ c′′ & d′′ \end{matrix}) .

(1)

In particular we have Δ(det_{q}) = det_{q} ⊗ det_{q} and ε(det_{q}) = 1. The quantum determinant is a group like element (see 2.1.6).

Now we define an algebra

SL_{q}(2) := M_{q}(2)/(det_{q} − 1).

The algebra SL_{q}(2) represents the functor

𝕊𝕃_{q}(2)(A) = \{(\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}) ∈ M_{q}(2)(A)|det_{q} (\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}) = 1\}.

There is a surjective homomorphism of algebras M_{q}(2) ⟶ SL_{q}(2) and 𝕊𝕃_{q}(2) is a subfunctor of M_{q}(2).

Let X, Y be commuting quantum matrices satisfying det_{q}(X) = 1 = det_{q}(Y). Since det_{q}(X) det_{q}(Y) = det_{q}(XY) for commuting quantum matrices we get det_{q}(XY) = 1, hence 𝕊𝕃_{q}(2) is a quantum submonoid of M_{q}(2) and SL_{q}(2) is a bialgebra with diagonal

Δ (\begin{matrix}a & b \\ c & d \end{matrix}) = (\begin{matrix}a & b \\ c & d \end{matrix}) ⊗ (\begin{matrix}a & b \\ c & d \end{matrix}) ,

and

ε (\begin{matrix}a & b \\ c & d \end{matrix}) = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) .

To show that SL_{q}(2) has an antipode we first define a homomorphism of algebras T : M_{q}(2) ⟶ M_{q}(2)^{op} by

T (\begin{matrix}a & b \\ c & d \end{matrix}) := (\begin{matrix}d & −qb \\ −q^{−1}c & a \end{matrix}) .

We check that T : 𝕂 ⟨(\begin{matrix}a & b \\ c & d \end{matrix})⟩ ⟶ M_{q}(2)^{op} vanishes on the ideal I.

T (ab − q^{−1}ba) = T (b)T (a) − q^{−1}T (a)T (b) = −qbd + q^{−1}qdb = 0.

We leave the check of the other defining relations to the reader. Furthermore T restricts to a homomorphism of algebras S : SL_{q}(2) ⟶ SL_{q}(2)^{op} since T (det_{q}) = T (ad − q^{−1}bc) = T (d)T (a) − q^{−1}T (c)T (b) = ad − q^{−1}(−q^{−1}c)(−qb) = det_{q} hence T (det_{q} −1) = det_{q} −1 = 0 in SL_{q}(2).

One verifies easily that S satisfies ∑ S(x_{(1)})x_{(2)} = ε(x) for all given generators of SL_{q}(2), hence S is a left antipode by 2.1.3. Symmetrically S is a right antipode. Thus the bialgebra SL_{q}(2) is a Hopf algebra or a quantum group.

Example 2.5.6. The affine algebraic group 𝔾𝕃(n) : 𝕂-cAlg ⟶ Gr defined by 𝔾𝕃(n)(A), the group of invertible n×n-matrices with coefficients in the commutative algebra A, is represented by the algebra O(𝔾𝕃(n)) = GL(n) = 𝕂[x_{ij}, t]/(det(x_{ij})t−1) i.e.

𝔾𝕃(n)(A) ≅ 𝕂-cAlg(𝕂[x_{ij}, t]/(det(x_{ij})t − 1), A)).

Since 𝔾𝕃(n)(A) has a group structure by the multiplication of matrices, the representing commutative algebra has a Hopf algebra structure with the diagonal Δ = ι_{1} ∗ ι_{2} hence

Δ(x_{ik}) = ∑ x_{ij} ⊗ x_{jk},

the counit ε(x_{ij}) = δ_{ij} and the antipode S(x_{ij}) = t · adj(X)_{ij} where adj(X) is the adjoint matrix of X = (x_{ij}). We leave the verification of these facts from linear algebra to the reader. The diagonal applied to t gives

Δ(t) = t ⊗ t.

Hence t(= det(X)^{−1}) is a grouplike element in GL(n). This reflects the rule det(AB) = det(A) det(B) hence det(AB)^{−1} = det(A)^{−1} det(B)^{−1}.

Example 2.5.7. Let M_{q}(2) be as in the example 2.5.5. We define

GL_{q}(2) := M_{q}(2)[t]/J

with J generated by the elements t · (ad − q^{−1}bc) − 1. The algebra GL_{q}(2) represents the functor

𝔾𝕃_{q}(2)(A) = \{(\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}) ∈ M_{q}(2)(A)|det_{q} (\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}) invertible in A\}.

In fact there is a canonical homomorphism of algebras M_{q}(2) ⟶ GL_{q}(2). A homomorphism of algebras φ : M_{q}(2) ⟶ A has a unique continuation to GL_{q}(2) iff

det_{q}(φ (\begin{matrix}a & b \\ c & d \end{matrix})) is invertible:

[figure: M_{q}(2); M_{q}(2)[t]; G_{q}(2); A]

with t ↦ det_{q} (\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix})^{−1} . Thus 𝔾𝕃_{q}(2)(A) is a subset of M_{q}(2)(A). Observe that

M_{q}(2) ⟶ GL_{q}(2) is not surjective.

Since the quantum determinant preserves products and the product of invertible elements is again invertible we get 𝔾𝕃_{q}(2) is a quantum submonoid of M_{q}(2), hence

Δ : GL_{q}(2) ⟶ GL_{q}(2)⊗GL_{q}(2) with Δ (\begin{matrix}a & b \\ c & d \end{matrix}) = (\begin{matrix}a & b \\ c & d \end{matrix})⊗(\begin{matrix}a & b \\ c & d \end{matrix}) and Δ(t) = t⊗t.

We construct the antipode for GL_{q}(2). We define T : M_{q}(2)[t] ⟶ M_{q}(2)[t]^{op} by

T (\begin{matrix}a & b \\ c & d \end{matrix}) := t (\begin{matrix}d & −qb \\ −q^{−1}c & a \end{matrix})  and  T (t) := det_{q} (\begin{matrix}a & b \\ c & d \end{matrix}) = ad − q^{−1}bc.

As in 2.5.5 T defines a homomorphism of algebras. We obtain an induced homomorphism of algebras S : GL_{q}(2) ⟶ GL_{q}(2)^{op} or a GL_{q}(2)^{op}-point in GL_{q}(2) since S(t(ad − q^{−1}bc) − 1) = (S(d)S(a) − q^{−1}S(c)S(b))S(t) − S(1) = (t^{2}ad − q^{−1}t^{2}cb)(ad − q^{−1}bc) − 1 = t^{2}(ad − q^{−1}bc)^{2} − 1 = 0.

Since S satisfies ∑ S(x_{(1)})x_{(2)} = ε(x) for all given generators, S is a left antipode by 2.1.3. Symmetrically S is a right antipode. Thus the bialgebra GL_{q}(2) is a Hopf algebra or a quantum group.

Example 2.5.8. Let sl(2) be the 3-dimensional vector space generated by the matrices

X = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) ,  Y = (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}) ,  H = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) .

Then sl(2) is a subspace of the algebra M(2) of 2×2-matrices over 𝕂. We easily verify [X, Y] = XY −Y X = H, [H, X] = HX−XH = 2X, and [H, Y] = HY −Y H = −2Y, so that sl(2) becomes a Lie subalgebra of M(2)^{L}, which is the Lie algebra of matrices of trace zero. The universal enveloping algebra U(sl(2)) is a Hopf algebra generated as an algebra by the elements X, Y, H with the relations

[X, Y] = H, [H, X] = 2X, [H, Y] = −2Y.

As a consequence of the Poincaré-Birkhoff-Witt Theorem (that we don’t prove) the Hopf algebra U(sl(2)) has the basis {X^{i}Y^{j}H^{k}|i, j, k ∈ ℕ}. Furthermore one can prove that all finite dimensional U(sl(2))-modules are semisimple.

Example 2.5.9. We define the so-called q-deformed version U_{q}(sl(2)) of U(sl(2)) for any q ∈ 𝕂, q ≠ 1, −1 and q invertible. Let U_{q}(sl(2)) be the algebra generated by the elements E, F, K, K′ with the relations

KK′ = K′K = 1,

KEK′ = q^{2}E, KF K′ = q^{−2}F,

EF − F E = \frac{K − K′}{q − q^{−1}} .

Since K′ is the inverse of K in U_{q}(sl(2)) we write K^{−1} = K′. The representation theory of this algebra is fundamentally different depending on whether q is a root of unity or not.

We show that U_{q}(sl(2)) is a Hopf algebra or quantum group. We define

Δ(E) = 1 ⊗ E + E ⊗ K, Δ(F) = K^{−1} ⊗ F + F ⊗ 1,
Δ(K) = K ⊗ K,
ε(E) = ε(F) = 0, ε(K) = 1,
S(E) = −EK^{−1}, S(F) = −KF, S(K) = K^{−1}.

.

First we show that Δ can be expanded in a unique way to an algebra homomorphism Δ : U_{q}(sl(2)) ⟶ U_{q}(sl(2)) ⊗ U_{q}(sl(2)). Write U_{q}(sl(2)) as the residue class algebra 𝕂⟨E, F, K, K^{−1}⟩/I where I is generated by

KK^{−1} − 1, K^{−1}K − 1,

KEK^{−1} − q^{2}E, KF K^{−1} − q^{−2}F,

EF − F E − \frac{K − K^{−1}}{q − q^{−1}} .

Since K^{−1} must be mapped to the inverse of Δ(K) we must have Δ(K^{−1}) = K^{−1} ⊗ K^{−1}. Now Δ can be expanded in a unique way to the free algebra Δ : K⟨E, F, K, K^{−1}⟩ ⟶ U_{q}(sl(2)) ⊗ U_{q}(sl(2)). We have Δ(KK^{−1}) = Δ(K)Δ(K^{−1}) = 1 and similarly Δ(K^{−1}K) = 1. Furthermore we have Δ(KEK^{−1}) = Δ(K)Δ(E)Δ(K^{−1}) = (K ⊗ K) (1 ⊗ E + E ⊗ K)(K^{−1} ⊗ K^{−1}) = KK^{−1} ⊗ KEK^{−1} + KEK^{−1} ⊗ K^{2}K^{−1} = q^{2}(1 ⊗ E + E ⊗ K) = q^{2}Δ(E) = Δ(q^{2}E) and similarly Δ(KF K^{−1}) = Δ(q^{−2}F). Finally we have

Δ(EF − F E) = (1 ⊗ E + E ⊗ K)(K′ ⊗ F + F ⊗ 1)
−(K′ ⊗ F + F ⊗ 1)(1 ⊗ E + E ⊗ K)

= K′ ⊗ EF + F ⊗ E + EK′ ⊗ KF + EF ⊗ K

−K′ ⊗ F E − K′E ⊗ F K − F ⊗ E − F E ⊗ K

= K′ ⊗ (EF − F E) + (EF − F E) ⊗ K

= \frac{K′ ⊗ (K − K′) + (K − K′) ⊗ K}{q − q^{−1}}

= Δ (\frac{K − K′}{q − q^{−1}} )

hence Δ vanishes on I and can be factorized through a unique algebra homomorphism

Δ : U_{q}(sl(2)) ⟶ U_{q}(sl(2)) ⊗ U_{q}(sl(2)).

In a similar way, actually much simpler, one gets an algebra homomorphism

ε : U_{q}(sl(2)) ⟶ 𝕂.

To check that Δ is coassociative it suffices to check this for the generators of the algebra. We have (Δ ⊗ 1)Δ(E) = (Δ ⊗ 1)(1 ⊗ E + E ⊗ K) = 1 ⊗ 1 ⊗ E + 1 ⊗ E ⊗ K + E ⊗ K ⊗ K = (1 ⊗ Δ)(1 ⊗ E + E ⊗ K) = (1 ⊗ Δ)Δ(E). Similarly we get (Δ ⊗ 1)Δ(F) = (1 ⊗ Δ)Δ(F). For K the claim is obvious. The counit axiom is easily checked on the generators.

Now we show that S is an antipode for U_{q}(sl(2)). First define S : 𝕂⟨E, F, K, K^{−1}⟩ ⟶ U_{q}(sl(2))^{op} by the definition of S on the generators. We have

S(KK^{−1}) = 1 = S(K^{−1}K),

S(KEK^{−1}) = −KEK^{−1}K^{−1} = −q^{2}EK^{−1} = S(q^{2}E),
S(KF K^{−1}) = −KKF K^{−1} = −q^{−2}KF = S(q^{−2}F),
S(EF − F E) = KF EK^{−1} − EK^{−1}KF = KF K^{−1}KEK − EF

= \frac{K^{−1} − K}{q − q^{−1}} = S (\frac{K − K^{−1}}{q − q^{−1}} ) .

So S defines a homomorphism of algebras S : U_{q}(sl(2)) ⟶ U_{q}(sl(2)). Since S satisfies ∑ S(x_{(1)})x_{(2)} = ε(x) for all given generators, S is a left antipode by 2.1.3. Symmetrically S is a right antipode. Thus the bialgebra U_{q}(sl(2)) is a Hopf algebra or a quantum group.

This quantum group is of central interest in theoretical physics. Its representation theory is well understood. If q is not a root of unity then the finite dimensional U_{q}(sl(2))-modules are semisimple. Many more properties can be found in [Kassel: Quantum Groups].