Ore extensions of Hopf algebras

A. N. Panov.

Abstract In the paper we study Ore extensions in the framework of the theory of Hopf algebras. Our classificaton Theorem provides the method available for classification of all Hopf-Ore extensions of group algebras, U(𝔤), U_{q}(𝔤), quantum ”ax + b”.

1 Main Theorem

The Ore extension construction is the well known general construction available for arbitrary noncommutative rings. As a result we have got a ring which is sometimes called the skew polynomial extension of a given ring. In dozen papers were studed ideals in this extensions, their representations, structure of the center and so on. In recent papers [B],[BDG],[D],[N],[Ne] iterated Ore extensions were used for a construction of new examples of Hopf algebras (usually of finite dimension). The aim of this paper is to study Ore extension of Hopf algebras in general setting. In this section we present a criterion for an Ore extension of a given Hopf algebra to be also a Hopf algebra with the comultiplication (1.3). In the next section we use the above criterion for classification of Hopf-Ore extensions.

Let we have a commutative ring k, a k-algebra A, an endomorphism τ and a τ-derivation δ of A. That is

δ(ab) = δ(a)b + τ(a)δ(b)  (1.1)

An Ore extension R = A[y; τ, δ] of an k-algebra A is the k-algebra R generated by indeterminate y and A with the relation

ya = τ(a)y + δ(a)  (1.2)

for all a ∈ A. Every element in R = A[y; τ, δ] is uniquely expressed in the form ∑ c_{i}y^{i} [MC-R, 2.1]. Definition 1.0 Let A and R = A[y; τ, δ] be Hopf algebras over k. We say that the extension R = A[y; τ, δ] is a Hopf-Ore extension if Δ(y) = y ⊗ r_{1} + r_{2} ⊗ y holds for some r_{1}, r_{2} ∈ A and A is a Hopf subalgebra of R.

In what follows we use common notations for multiplication m, comultiplication Δ, counit ε and antipode S of A and R. We shall use the Sweedler’s notation Δy = a_{1}⊗a_{2} (implying the sum) for comultiplication. We shall write the property of coassociativity of comultiplication as follows a_{11}⊗a_{12}⊗a_{2} = a_{1}⊗a_{21}⊗a_{22}. We shall denote an element defined by the last equality by a_{1} ⊗ a_{2} ⊗ a_{3}.

Note that the elements r_{1}, r_{2} in Definition 1.1 are group-like. Indeed, compairing (1 ⊗ Δ)Δy = y ⊗ Δ(r_{1}) + r_{2} ⊗ y ⊗ r_{1} + r_{2} ⊗ r_{2} ⊗ y and (Δ ⊗ 1)Δy = y ⊗ r_{1} ⊗ r_{1} + r_{2} ⊗ y ⊗ r_{1} + Δ(r_{2}) ⊗ y, we get Δr_{i} = r_{i} ⊗ r_{i}. We change the generator y by y′ = yr_{1}^{−1} and get Δ(y′) = y′ ⊗ 1 + r ⊗ y′. Preserving the previous notations we further require that y in a Hopf-Ore extension obeys

Δ(y) = y ⊗ 1 + r ⊗ y  (1.3)

with group-like r ∈ A. As usual Ad_{r}(a) = raS(r) = rar^{−1}. Lemma 1.1. If R = A[y; τ, δ] is a Hopf-Ore extension then

ε(y) = 0,  (1.4)
S(y) = −r^{−1}y  (1.5)

where r^{−1} = S(r). Proof.

1) Using the main property of counit a = a_{1}ε(a_{2}) we obtain y = ε(y)+ε(r)y = ε(y)+y. This yields (1.4).

2) By the main property of antipode ε(a) = a_{1}S(a_{2}), we get 0 = ε(y) = y + rS(y) and (1.5).□ Lemma 1.2. If Δ(b) = b ⊗ 1 then b ∈ k ⊂ A. Proof. Using the main property of counit a = a_{1}ε(a_{2}) = ε(a_{1})a_{2}, we have b = ε(b) ∈ k. □ Theorem 1.3. R = A[y; τ, δ] is a Hopf-Ore extension if and only if

1) there exists a character χ : A ↦ k such that

τ(a) = χ(a_{1})(a_{2}) = Ad_{r}(a_{1})χ(a_{2})  (1.6)

for all a ∈ A (that is τ is a winding automorphism of A);

2) for Ad_{r}, S, τ : A ↦ A there holds the relation

Ad_{r}S = τSτ;  (1.7)

3) the τ-derivation δ obeys

Δδ(a) = δ(a_{1}) ⊗ a_{2} + ra_{1} ⊗ δ(a_{2}).  (1.8)

Proof. The Proof is divided into three Steps. In Step 1 we shall show that the comultiplication Δ is extended from A to R = A[y; τ, δ] via (1.3) if and only if (1.6) and (1.8) hold. In Step 2 we prove that R (with the above comultiplication) has a counit as an extension of counit of A via (1.4). In Step 3 we show that R (with the above comultiplicaton and counit) has an antipode S extending antipode S|_{A} via (1.5) if and only if (1.7) holds. Step 1. Comultiplication. Suppose that the comultiplication Δ|_{A} can be extended to R = A[y; τ, δ] via (1.3). Then the homomorphism Δ preserves (1.2). That is ΔyΔa = Δτ(a)Δy + Δδ(a). We have

ΔyΔa = (y ⊗ 1 + r ⊗ y)a_{1} ⊗ a_{2} =

(τ(a_{1})y + δ(a_{1})) ⊗ a_{2} + ra_{1} ⊗ (τ(a_{2})y + δ(a_{2})) =
(τ(a_{1}) ⊗ a_{2})(y ⊗ 1) + (ra_{1}r^{−1} ⊗ τ(a_{2}))(r ⊗ y) + δ(a_{1}) ⊗ a_{2} + ra_{1} ⊗ δ(a_{2}),
Δτ(a)Δy + Δδ(a) = Δτ(a)(y ⊗ 1 + r ⊗ y) + Δδ(a) =
Δτ(a)y ⊗ 1 + Δτ(a)r ⊗ y + Δδ(a).

We see that the comultiplication of A extended to a comultiplication R via (1.3) if and only if the following conditions hold

Δτ(a) = τ(a_{1}) ⊗ a_{2},  (1.9),
Δτ(a) = Ad_{r}(a_{1}) ⊗ τ(a_{2}),  (1.10),
Δδ(a) = δ(a_{1}) ⊗ a_{2} + ra_{1} ⊗ δ(a_{2})

for all a ∈ A and Δa = a_{1} ⊗ a_{2}. The last coincides with (1.8).

We are going to prove that (1.9) and (1.10) is equivalent to (1.6). Consider the element χ(a) := τ(a_{1})S(a_{2}) ∈ A. By direct calculations,

Δχ(a) = Δτ(a_{1})ΔS(a_{2}) = (τ(a_{11}) ⊗ (a_{12})(S(a_{22}) ⊗ S(a_{21})) =
τ(a_{11})S(a_{22}) ⊗ (a_{12})S(a_{21})) = τ(a_{1})S(a_{4}) ⊗ (a_{2}S(a_{3})) =
τ(a_{1})S(a_{3}) ⊗ ε(a_{2}) = τ(a_{1}ε(a_{2}))S(a_{3}) ⊗ 1 = τ(a_{1})S(a_{2}) ⊗ 1 = χ(a) ⊗ 1.

By Lemma 1.2, χ(a) ∈ k. One can treat χ as a map χ : A → k. Since τ is an endomorphism, χ is a character:

χ(ab) = τ(a_{1}b_{1})S(a_{2}b_{2}) = τ(a_{1})τ(b_{1})S(b_{2})S(a_{2}) =
τ(a_{1})χ(b)S(a_{2}) = χ(a)χ(b),
χ(a + b) = τ((a + b)_{1})S((a + b)_{2}) = τ(a_{1})S(a_{2}) + τ(b_{1})S(b_{2}) = χ(a) + χ(b).

By χ, one can restore τ:

χ(a_{1})a_{2} = τ(a_{1})S(a_{2})a_{3} = τ(a_{1})ε(a_{2}) = τ(a_{1}ε(a_{2})) = τ(a).

This proves the left equality (1.6). We substitute it in (1.10) and get:

Δ(χ(a_{1})a_{2}) = Ad_{r}(a_{1}) ⊗ χ(a_{2})a_{3},
χ(a_{1})Δ(a_{2}) = Ad_{r}(a_{1})χ(a_{2}) ⊗ a_{3},
(χ(a_{1})a_{2}) ⊗ a_{3} = Ad_{r}(a_{1})χ(a_{2}) ⊗ a_{3}.

Hence

(χ(a_{1})a_{2})a_{3}S(a_{4}) = Ad_{r}(a_{1})χ(a_{2})a_{3}S(a_{4}),
(χ(a_{1})a_{2})ε(a_{3}) = Ad_{r}(a_{1})χ(a_{2})ε(a_{3}),
χ(a_{1})(a_{2}) = Ad_{r}(a_{1})χ(a_{2}).

This proves the right equality (1.6). We have shown that (1.6) and (1.8) are necessary for an extension of comultiplication.

On the other hand, if (1.6) and (1.8) hold, then Δτ(a) = χ(a_{1})a_{2} ⊗ a_{3} = τ(a_{1}) ⊗ a_{2} and Δτ(a) = Δ(Ad_{r}a_{1}χ(a_{2})) = Ad_{Δr}(a_{1} ⊗ a_{2})χ(a_{3}) = Ad_{r}a_{1} ⊗ Ad_{r}a_{2}χ(a_{3}) = Ad_{r}a_{1} ⊗ τ(a_{2}). This proves that (1.9) and (1.10) is true and the comuliplication Δ|_{A} extends to comultiplication of R via (1.3). Step 2. Counit. Let R admit the above comultiplicaion (i.e (1.6) and (1.8) are true). Let us prove an existence of counit extending ε|_{A} and obeying (1.4) . This data defines a homomorphism ε of R if and only if ε preserves (1.2). That is ε(y)ε(a) = ε(τ(a))ε(y) + ε(δ(a)). Since ε(y) = 0, R admits the counit if and only if

ε(δ(a)) = 0.  (1.11)

Hence, our goal in this Step is to prove (1.11). Acting by m(1 ⊗ ε) on (1.8), we get

m(id ⊗ ε)(Δδ(a)) = m(id ⊗ ε)(δ(a_{1}) ⊗ a_{2} + ra_{1} ⊗ δ(a_{2})),
δ(a) = δ(a_{1})ε(a_{2}) + ra_{1}ε(δa_{2})) = δ(a) + ra_{1}ε(δa_{2})).

It follows a_{1}ε(δ(a_{2})) = 0. Finally,

0 = S(a_{1})a_{2}ε(δ(a_{3})) = ε(a_{1})ε(δa_{2}) = εδ(ε(a_{1})a_{2}) = ε(δ(a)).

This proves (1.11) and an existence of counit. Step 3. Antipode. Let R be as in Step 2. Recall that S is an antihomomorphism of a Hopf algebra and ΔS(a) = S(a_{2}) ⊗ S(a_{1}) [M,1.5.10]. Suppose that R has antipode S which is extended (as an antihomomorphism) from A to R via (1.3). Then S preserves (1.2). That is

S(a)S(y) = S(y)Sτ(a) + Sδ(a).  (1.12)

By Lemma 1.1,

−S(a)r^{−1}y = −r^{−1}ySτ(a) + Sδ(a),
−S(a)r^{−1}y = −r^{−1}τ(Sτ(a))y − r^{−1}δ(Sτ(a)) + Sδ(a).

We get two necessary conditions

S(a)r^{−1} = r^{−1}τSτ(a),  (1.13)
rSδ(a) = δSτ(a).  (1.14)

The first is equivalent to (1.7) (and proves it).

We shall show that (1.14) is a consequence of (1.1) and (1.8). By (1.6), Sτ(a) = S(χ(a_{1})(a_{2})) = χ(a_{1})S(a_{2}). After substitution in (1.14) one can present (1.14) in the equivalent form:

rSδ(a) = χ(a_{1})δS(a_{2}).  (1.15)

Denote L = rSδ(a) and M = χ(a_{1})δS(a_{2}). Acting by m(id ⊗ S) on (1.8), we obtain

m(id ⊗ S)(Δδ(a)) = m(id ⊗ S)(δ(a_{1}) ⊗ a_{2} + ra_{1} ⊗ δ(a_{2})),

0 = εδ(a) = δ(a_{1})S(a_{2}) + ra_{1}Sδ(a_{2}),
a_{1}Sδ(a_{2}) = −r^{−1}δ(a_{1})S(a_{2}).

Then

L = rSδ(a) = rε(a_{1})Sδ(a_{2}) = rS(a_{1})a_{2}Sδ(a_{3}) =
−rS(a_{1})r^{−1}δ(a_{2})S(a_{3}) = −Ad_{r}S(a_{1})δ(a_{2})S(a_{3}).  (1.16)

On the other hand, acting by δ on the left and right sides of ε(a) = a_{1}Sa_{2}, we get

0 = δ(a_{1})S(a_{2}) + τ(a_{1})δS(a_{2}),
M = χ(a_{1})δS(a_{2}) = χ(a_{1})ε(a_{2})δS(a_{3}) = χ(a_{1})τ(S(a_{2})a_{3})δS(a_{4}) =
= χ(a_{1})τS(a_{2})τ(a_{3})δS(a_{4}) = −χ(a_{1})τS(a_{2})δ(a_{3})S(a_{4}) =
−χ(a_{1})Ad_{r}S(a_{3})χ(S(a_{2}))δ(a_{4})S(a_{(5)}) = −Ad_{r}S(a_{1})δ(a_{2})S(a_{3}).  (1.17)

Compairing (1.17) and (1.16), we get L = M. This proves (1.15).

It is proved that (1.7) is a necessary condition for an existence of antipode in R. The condition (1.7) is also a sufficient condition. If it holds, then (1.13) is true. As above the formula (1.14) is a corollary of properties of δ. This implies (1.12) an existence of antipode in R. □ Corollary 1.4. If R = A[y; τ, δ] is a Hopf-Ore extension and A is cocommutative a Hopf algebra, then r lies in the center of A. Proof. Since τ(a) = χ(a_{1})(a_{2}) = Ad_{r}(a_{1})χ(a_{2}) = χ(a_{1})Ad_{r}a_{2}, then

χ(S(a_{1}))χ(a_{2})a_{3} = χ(S(a_{1}))χ(a_{2})Ad_{r}a_{3},
χ(ε(a_{1}))a_{2} = χ(ε(a_{1}))Ad_{r}a_{2},
a = Ad_{r}(a)

for all a ∈ A. □ Remarks 1.5.

1) The left condition (1.6) states that τ(a) = χ(a_{1})a_{2}. That is τ is a right χ-winding automorphism of R [M,1.6.5]. The automorphism τ′(a) = a_{1}χ(a_{2}) is called a left χwinding automorphism. The right condition (1.6) can be rewritten in an equivalent form τ = Ad_{r}τ′.

2) Let us denote by R_{r} and L_{r} the right and left winding action of r in A^{∗}. By definition, R_{r}λ(a) = λ(ra) and L_{r}λ(a) = λ(ar) for all λ ∈ A^{∗} and a ∈ A. One can prove that the right condition (1.6) has an equivalent form R_{r} = Ad_{χ}L_{r}. Definition 1.6. 1) A map δ, obeying (1.8), will be called a r-coderivaton (notice that is δ is a r-coderivetion, then the map δ^{∗} : A^{∗} → A^{∗} is a R_{r}-derivation);

2) we shall refer δ as < χ, r >-derivation if it is a τ-derivation (see 1.1) and a rcoderivaton. Notation 1.7. The Hopf-Ore extension R = A[y; τ, δ] we shall denote R = A(χ, r, δ) with χ : A → k is a character, r is a grouplike element of A obeying (1.7) and δ is a < χ, r >-derivation. Definition 1.8. We say that δ is an inner < χ, r >-derivation if there exists an element d ∈ A such that δ(a) = τ(a)d − da and Δ(d) = d ⊗ 1 + r ⊗ d. Definition 1.9. We say that Hopf-Ore extensions R and R′ of A and A′ are isomorphic if there exists a Hopf isomorphism Φ : R ↦ R′ passing A to A′. Proposition 1.10. 1) If δ is an inner < χ, r >-derivation, then A(χ, r, δ) is isomorphic to A(χ, r, 0).

2) Two Hopf-Ore extensions R = A(χ, r, δ) and R = A′(χ′, r′, δ′) are isomorphic iff the exists a Hopf isomorphism Φ : A ↦ A′ such that χ′ = χΦ^{−1}, r′ = Φ(r), δ′ = ΦδΦ^{−1}+δ′′ where δ′′ is an inner < χ′, r′ >-derivation of R′. Proof. Changing y by y + d.□ Proposition 1.11. The subHopfalgebra A in R = A(χ, r, δ) is normal (see [M,3.4.1]) if and only if χ = ε and r lies in the center of A. Proof. The left(resp.right) adjoint action on a Hopf algebra H is given by (ad_{l}h)(x) = h_{1}xS(h_{2}) (resp.(ad_{r}h)(x) = S(h_{1})xh_{2}). By definition, a subHopfalgebra is normal if it is stable with respect to the left and the right adjoint action. The subHopfalgebra A in R = A(χ, r, δ) is normal if (ad_{l}y)(a) ∈ A and (ad_{r}y)(a) ∈ A for any a ∈ A. That is ya − rar^{−1}y ∈ A and −r^{−1}ya + r^{−1}ay = r^{−1}(−ya + ay) ∈ A. By (1.2), it follows τ(a) = rar^{−1} and τ(a) = a for any a ∈ A. This proves the claim. □

2 Classifying Hopf-Ore extensions

In this section we classify Hopf-Ore extensions for some classes of Hopf algebras. Point 1. Cocommmutative Hopf algebras. We present here a method of constructing < χ, r >-derivations of cocommutative Hopf algebras.

Let A be a cocommutative Hopf algebra. Let χ be a A-charater and r be a grouplike element in the center of A. Let α be a linear form α : A → k obeying

α(uv) = α(u)ε(v) + χ(a)α(b).  (2.1)

Proposition 2.1.The map

δ(a) := α(a_{1})(1 − r)a_{2}  (2.2)

is a < χ, r >-derivation. This provides an example A(χ, τ, δ) of a Hopf-Ore extenson of A. Proof. The map δ is a τ-derivation:

δ(ab) = α((ab)_{(1)})(1 − r)(ab)_{(2)} = (α(a_{1})ε(b_{1}) + χ(a_{1})α(b_{1}))(1 − r)a_{2}b_{2} =
α(a_{1})(1 − r)a_{2}ε(b_{1})b_{2} + χ(a_{1})a_{2}α(b_{1})(1 − r)b_{2} = δ(a)b + τ(a)δ(b).

The map δ is r-coderivation (see 1.8):

Δδ(a) = α(a_{1})Δ(1 − r)a_{2} = α(a_{1})(1 ⊗ 1 − r ⊗ r)a_{2} ⊗ a_{3} =
α(a_{1})(1 − r)a_{2} ⊗ a_{3} + α(a_{1})ra_{2} ⊗ (1 − r)a_{3} =
α(a_{1})(1 − r)a_{2} ⊗ a_{3} + ra_{1} ⊗ α(a_{2})(1 − r)a_{3} = δ(a_{1}) ⊗ a_{2} + ra_{1} ⊗ δ(a_{2}).

□ Remark. In the case A = kG the linear form α obeying is determined by α(g),g ∈ G and obeys α(gh) = α(g) + χ(g)α(h). That is α is a 1-cocycle (α ∈ Z_{χ}^{1}(G)). Point 2. Group algebras. Propositon 2.2 Any Hopf-Ore extension of A = kG has the form A(χ, r, δ) where χ is a group character, r in the center of G and δ have the form (2.2) for some α ∈ Z_{χ}^{1}(G). Proof. By Theorem 1.3, all Hopf-Ore extesions have the form A(χ, r, δ) where χ is a kG-character (i.e. χ is a G-character), r is a group-like element of kG (one can prove that all group-like elements of kG are elements of G), and δ is a < χ, r >-derivation. We have proved that r lies in the center of G (Corollary 1.4). The condition (1.8) have the form

Δδ(g) = δ(g) ⊗ g + rg ⊗ δ(g)  (2.3)

for all g ∈ G. The element c := δ(g)g^{−1} satisfies

Δ(c) = c ⊗ 1 + r ⊗ c.  (2.4)

For any μ ∈ (kG)^{∗} we have

μ^{2}(c) = μ ⊗ μ(Δc) = μ(c)μ(1) + μ(r)μ(c).  (2.5)

The element c is the sum c = ∑ γ(g)g. Let g_{0} be an element of G which is different from 1 and r. Take μ_{0} ∈ (kG)^{∗} in (2.5) such that μ_{0}(g_{0}) = 1 and μ_{0}(g) = 0 for all other elements of G. We put μ = μ_{0} in (2.5) and get

γ^{2}(g_{0}) = ∑ γ(g)μ_{0}^{2}(g) = μ_{0}^{2}(∑ γ(g)g) = μ_{0}^{2}(c) = μ_{0}(c)μ_{0}(1) + μ_{0}(r)μ_{0}(c) = 0.

This yields γ(g_{0}) = 0 and c = γ(1) + γ(r)r. Substituting in (2.4), we obtain c = γ(1 − r) for some γ ∈ k. Hence δ(g) = α(g)(1 − r)g. Since δ(g) is τ-derivation, then δ(gh) = δ(g)h + χ(g)δ(h). After subsitution δ(g) = α(g)(1 − r)g we obtain (1 − r)(α(gh) − α(g) − χ(g)α(h))gh = 0. Finally, α(g) ∈ Zχ^{1}(G). □ Point 3. A = U(𝔤) where 𝔤 is a Lie algebra over a field of zero characteristic. By [D,2.7.2], one can prove that a group-like element in U(𝔤) equals to 1 and any primitive element lies in 𝔤. We have r = 1. A U(𝔤) character χ is determined by 𝔤character, and τ(x) = x+χ(x) for all x ∈ 𝔤. For x ∈ 𝔤 we also have Δ(x) = x⊗1+1⊗x and (1.8) has the form Δδ(x) = δ(x) ⊗ 1 + 1 ⊗ δ(x). Therefore, δ(x) ∈ 𝔤, δ : 𝔤 → 𝔤 and any Hopf-Ore extension R of U(𝔤) is generated by y and U(𝔤) with the relation yx = (x + χ(x))y + δ(x). In the case χ = 0 the algebra R is universal enveloping algebra U(𝔤′) of the Lie algebra 𝔤′ = ky + 𝔤. Point 4. Let A be U_{q}(𝔤) in the Drilfeld-Jimbo sense, and q be an indeterminate. It as generated by K_{i}, e_{i}, f_{i}, i = 1, …, n related as usual and ΔK_{i} = K_{i} ⊗ K_{i}, Δe_{i} = e_{i} ⊗ 1 + K_{i} ⊗ e_{i} and Δf_{i} = f_{i} ⊗ K_{i}^{−1} + 1 ⊗ e_{i}. Proposition 2.3 Any Hopf-Ore extension of A = U(𝔤) is isomorphic to A(1, 1, δ) with δ(K_{i}) = 0, δ(e_{i}) = ξ_{i}e_{i} and δ(f_{i}) = −ξ_{i}f_{i}. Proof Any character of A has the form χ(K_{i}) = ε_{i} = ±1, χ(e_{i}) = χ(f_{i}) = 0. Then τ(K_{i}) = ε_{i}K_{i}, τ(e_{i}) = ε_{i}e_{i}, τ(f_{i}) = f_{i}. The generator y in a Hopf-Ore extension of U_{q}(𝔤) obeys (1.3) with r = K_{1}^{γ_{1}} ⋯ K_{n}^{γ_{n}} = K^{γ}. For any simple root α_{i} we have

Ad_{r}S(e_{i}) = Ad_{r}(−K_{i}^{−1}e_{i}) = −K_{i}^{−1}q^{(α_{i},γ)}e_{i}.

On the other hand,

τSτ(e_{i}) = τS(ε_{i}e_{i}) = −τ(ε_{i}K_{i}^{−1}e_{i}) = −ε_{i}K_{i}^{−1}e_{i}.

By (1.7), γ = 0 and ε_{i} = 1. Hence, r = 1 and τ = id. We obtain, in particular, that y is primitive. By (1.8), Δδ(K) = δ(K)⊗K +K ⊗δ(K). Whence δ(k)K^{−1} is a primitive element. By PBW Theorem, one can prove that U_{q}(𝔤) has no primitive elements. This implies δ(K) = 0. We get ye_{i} = e_{i}y + δ(e_{i}), yf_{i} = f_{i}y + δ(f_{i}), yK_{i} = K_{i}y. By (1.8), we have Δδ(e_{i}) = δ(e_{i}) ⊗ 1 + K_{i} ⊗ δ(e_{i}) and Δδ(f_{i}) = δ(f_{i}) ⊗ K_{i}^{−1} + 1 ⊗ δ(f_{i}). It follows δ(e_{i}) = ξ_{i}e_{i}, δ(f_{i}) = −ξ_{i}f_{i} for some numbers ξ_{i}.□ Point 5. Hopf-Ore extensions of quantum ”ax + b”. Let q be an indeterminate. Quantum ”ax + b” is generated over the ring C = k[q, q^{−1}] by K, x with the relation Kx = q^{2}xK and Δ(K) = K ⊗ K, Δ(x) = x ⊗ 1 + K ⊗ x, ε(K) = 1, ε(x) = 0, S(K) = K^{−1} and S(x) = −K^{−1}x. Proposition 2.4. There are exist exactly four Hopf-Ore extensions of quantum ”ax+ b”: yK = Ky, yx = xy + γx with γ = 0, 1, y is primitive and Ky = q^{−2}yK, yx = xy + γ(K − K^{−1}) with γ = 0, 1, Δy = y ⊗ K^{−1} + 1 ⊗ y. Among them quantum sl_{q}(2). Proof. An character χ of A has the form χ(K) = α = C^{∗} and χ(x) = 0. Then τ(K) = χ(K)K = αK and τ(x) = χ(K)x = αx. The element r is group-like. Hence, r = K^{m} for some m ∈ ℤ and

Δy = y ⊗ 1 + K^{m} ⊗ y  (2.6).

The right condition (1.6) has the form:

τ(K) = Ad_{K^{m}}(K)χ(K) = αK,
τ(x) = Ad_{K^{m}}(x)χ(1) = q^{2m}x.

It implies

α = q^{2m}.  (2.7)

and

Ad_{r}S(x) = −q^{2m}K^{−1}x = ατ(−K^{−1}x) = τSτ(x).

This proves that (1.7) holds.

The equation (1.8) has the form Δδ(K) = δ(K) ⊗ K + K^{m+1} ⊗ δ(K). The element b := δ(K)K^{−1} obeys

Δb = b ⊗ 1 + K^{m} ⊗ b.  (2.8)

One can prove that any solution of (2.8) has the form: b = γx + γ′(1 − K) (if m = 1) or b = γ(1 − K^{m}). We shall show that changing the generator y one can consider that δ(K) = 0.

1) Let b = γx + γ′(1 − K) and m = 1. Then δ(K) = γxK + γ′(1 − K)K and α = q^{2}. Consider the inner derivations δ_{x}(a) = τ(a)x−xa and δ_{1−K}(a) = τ(a)(1−K)−(1−K)a (see Definition 1.7). Then δ_{x}(K) = τ(K)x − xK = xK(αq^{2} − 1) = xK(q^{4} − 1) and δ_{1−K}(K) = (q^{2}−1)(1−K)K. Changing y by y′ = y+ξx+ξ′(1−K) with ξ = γ(q^{4}−1)^{−1} and ξ′ = γ′(q^{2} − 1)^{−1}, we get the new generator y′ obeying (2.6) with δ′(K) = 0.

2) Let b = γ(1 − K^{m}). Then δ(K) = γ(1 − K^{m})K. The element b satifies (2.6). Consider the inner derivation δ_{b}. Then δ_{b}(K) = τ(K)b − bK = γ(α − 1)(1 − K^{m})K. If α ≠ 1 we change y by y′ = y + ξb with ξ = (α − 1)^{−1} and get δ′(K) = 0. If α = 1, the equality (2.7) yields m = 0 and δ(K) = 0.

We have proved that, changing y, one can consider that δ(K) = 0. Hence yK = αKy. Our next goal is to determine δ(x). We act by δ on the relation Kx = q^{2}xK and get q^{2m}Kδ(x) = q^{2}δ(x)K and

Kδ(x) = q^{2(1−m)}δ(x)K.  (2.9)

The equality (1.8) yields:

Δδ(x) = δ(x) ⊗ 1 + rK ⊗ δ(x) = δ(x) ⊗ 1 + K^{m+1} ⊗ δ(x).

Again two solutions

δ(x) = \{ \begin{matrix}γx + γ′(1 − K) & for & m = 0, \\ γ(1 − K^{m+1}) & for & m ≠ 0. \end{matrix}  (2.11)

By (2.9) and (2.11),

δ(x) = \{ \begin{matrix}γx & for & m = 0, \\ γ(1 − K^{2}) & for & m = 1. \end{matrix}  (2.12)

In the first case a Hopf-Ore extension R is generated by y and A with yK = Ky, yx = x(y + γ) and y is primitive. One may consider that γ = 0 or γ = 1.

In the second case δ(x) = γ(1 − K^{2}), m = 1 and α = q^{2}. We obtain yK = q^{2}Ky, yx = q^{2}xy + γ(1 − K^{2}).

After changing y′ = yK^{−1} we get Δy′ = y′ ⊗ K^{−1} + 1 ⊗ y′ and y′x = xy′ + γq^{−2}(K^{−1} − K). We may assume that −γq^{−2} is equal to 0 or 1. □

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