Notation

We keep the notations from [Reference Andruskiewitsch, Angiono and Heckenberger1]. Let $G$ be a group, let ${ \Bbbk }G$ be its group algebra and let $\widehat G$ be its group of characters. Given $\chi \in \widehat G$, recall that

\[ \operatorname{Der}_{\chi,\chi}({ \Bbbk}G, { \Bbbk}) = \{\eta\in ({ \Bbbk}G)^{*}: \eta(ht) = \chi(h)\eta(t) + \chi(t)\eta(h) \quad\forall h,t \in G\}. \]

A collection $\mathcal {D}=(g,\, \chi,\, \eta )\in Z(G) \times \widehat G \times \operatorname {Der}_{\chi,\chi }({ \Bbbk }G,\, { \Bbbk })$ is a YD-triple for ${ \Bbbk }G$ if ${\eta (g) = 1}$. Then the vector space $\mathcal {V}_g(\chi,\,\eta )$ with a basis $(x_i)_{i\in \mathbb {I}_2}$ belongs to ${{}^{ { \Bbbk }G }_{ { \Bbbk }G }\mathcal {YD}}$, with the coaction $\delta (x_i) = g\otimes x_i$, $i\in \mathbb {I}_2$, and the action given by

\[ h\cdot x_1 = \chi(h) x_1,\quad h\cdot x_2=\chi(h) x_2 + \eta(h)x_{1},\quad h \in { \Bbbk}G. \]

When $\chi (g) = 1$ we say that $\mathcal {D}=(g,\, \chi,\, \eta )$ is a Jordanian YD-triple.

Let $L$ be a Hopf algebra. The $\Delta$, $\varepsilon$ and ${{\mathcal {S}}}$ denote respectively the comultiplication, the counit and the antipode. The group of group-like elements is denoted by $G(L)$. Also the space of $(g,\,h$)-primitive elements is $\mathcal {P}_{g,h}(L) =\{\ell \in L: \Delta (\ell ) = \ell \otimes h + g \otimes \ell \}$, where $g,\,h \in G(L)$, and $\mathcal {P}(L) = \mathcal {P}_{1,1}(L)$ is the space of primitive elements. The adjoint action of $G(L)$ on $L$ is denoted by $g\cdot \ell := g\ell g^{-1}$, $g\in G(L)$, $\ell \in L$.

1.1. The Jordanian enveloping algebra of $s\ell (2)$

Let ${{\widetilde {\mathfrak U}}^{\mathtt {jordan}}}$ be the algebra generated by $a_1,\, a_2,\, g,\, g^{-1}$ with defining relations

(1.1)\begin{equation} g^{{\pm} 1} g^{{\mp} 1} = 1,\quad ga_1 = a_1\,g + (g-g^{2}),\ ga_2 = a_2\,g + a_1\,g. \end{equation}

It is easy to see that ${{\widetilde {\mathfrak U}}^{\mathtt {jordan}}}$ is a Hopf algebra by imposing $g \in G({{\widetilde {\mathfrak U}}^{\mathtt {jordan}}})$ and $a_1,\,a_2 \in \mathcal {P}_{g, 1}({{\widetilde {\mathfrak U}}^{\mathtt {jordan}}})$. We introduce

(1.2)\begin{equation} z= a_1a_2 - a_2a_1 -\dfrac{a_1^{2}}{2} + a_2 +\dfrac{1}{2}a_1 \in {{\widetilde{\mathfrak U}}^{\mathtt{jordan}}} \end{equation}

Proof. We compute

\begin{align*} \Delta (z) & = a_1a_2\otimes 1 + a_1\,g \otimes a_2 + g a_2 \otimes a_1 + g^{2} \otimes a_1a_2 \\ & \quad-a_2a_1\otimes 1 - a_2\,g \otimes a_1 - g a_1 \otimes a_2 - g^{2} \otimes a_2a_1 \\ & \quad-\dfrac{1}{2} a_1^{2}\otimes 1 -\dfrac{1}{2} ( a_1\,g + g a_1) \otimes a_1-\dfrac{1}{2} g^{2} \otimes a_1^{2} \\ & \quad+ a_2\otimes 1 + g \otimes a_2 +\dfrac{1}{2} a_1\otimes 1 + \dfrac{1}{2} g \otimes a_1 \\ & = z \otimes 1 + g^{2} \otimes z + (a_1\,g - g a_1 + g - g^{2} )\otimes a_2 \\ & \quad+ \left(g a_2 - a_2\,g -\dfrac{1}{2} ( a_1\,g + g a_1) + \dfrac{1}{2} g - \dfrac{1}{2} g^{2}\right)\otimes a_1\\ & = z \otimes 1 + g^{2} \otimes z; \end{align*}

here $g a_2 - a_2\,g -\dfrac {1}{2} ( a_1\,g + g a_1) + \dfrac {1}{2} g - \dfrac {1}{2} g^{2} = \dfrac {1}{2} (a_1\,g - g a_1 + (g - g^{2})) = 0$.

It remains to prove that $\gamma (z)=0$, where $\gamma \in \operatorname {End}_{{ \Bbbk }}({{\widetilde {\mathfrak U}}^{\mathtt {jordan}}})$ is given by $\gamma (x) = gxg^{-1}-x$, for all $x\in {{\widetilde {\mathfrak U}}^{\mathtt {jordan}}}$. Note that

\[ \gamma(xy)=\gamma(x)(\gamma(y)+y)+x\gamma(y)\text{ for all } x,y \in {{\widetilde{\mathfrak U}}^{\mathtt{jordan}}}. \]

From (1.1) we have that

(1.3)\begin{equation} \gamma(a_1)=1-g,\quad \gamma(a_2)=a_1. \end{equation}

Therefore,

\begin{align*} \gamma(z)& =\gamma(a_1a_2+\left(a_2+\displaystyle\frac 12 a_1)(1-a_1)\right)\\ & =\gamma(a_1)(\gamma(a_2)+a_2)+a_1\gamma(a_2)\\ & \quad+\gamma\left(a_2+\displaystyle\frac 12a_1\right)(\gamma(1-a_1)+1-a_1) +\left(a_2+\frac 12a_1\right)\gamma(1-a_1). \end{align*}

By using (1.3) we obtain that

\begin{align*} \gamma(z)& =(1-g)(a_2+a_1)+a_1^{2}\\ & \quad +\left(a_1+\frac 12 (1-g)\right)(g-a_1) +\left(a_2+\frac 12a_1\right)(g-1)\\ & =a_2+a_1-(a_2g+2a_1g+g-g^{2})+a_1^{2} +\left(a_2+\frac 12a_1\right)(g-1)\\ & \quad +a_1g-a_1^{2}+\frac 12(g-g^{2})+\frac 12a_1(g-1)+\frac 12(g-g^{2}). \end{align*}

Now it follows easily that $\gamma (z)=0$.

The Jordanian enveloping algebra of $s\ell (2)$ is

(1.4)\begin{align} {{\mathfrak U}^{\text {jordan}}} := {{\widetilde{\mathfrak U}}^{\mathtt{jordan}}} / \langle z\rangle. \end{align}

By Lemma 2, ${{\mathfrak U}^{\text {jordan}}}$ is a Hopf algebra quotient of ${{\widetilde {\mathfrak U}}^{\mathtt {jordan}}}$. By abuse of notation the images of $g,\, a_1,\, a_2$ in ${{\mathfrak U}^{\text {jordan}}}$ are denoted by the same symbols.

Remark 3 For each $\lambda \in { \Bbbk }$ let

(1.5)\begin{equation} {\mathfrak U}^{\text {jordan}}_{\lambda} := {{\widetilde{\mathfrak U}}^{\mathtt{jordan}}} / \langle z-\lambda(1-g^{2})\rangle. \end{equation}

Then ${\mathfrak U}^{\text {jordan}}_{\lambda }$ is a Hopf algebra, since $z-\lambda (1-g^{2})\in \mathcal {P}_{g^{2},1}({{\widetilde {\mathfrak U}}^{\mathtt {jordan}}})$.

Let us now fix $\lambda,\,\mu \in { \Bbbk }$. Let $U$ be the algebra

\[ U ={ \Bbbk} \langle g,g^{{-}1},a_1,a_2\rangle /\langle gg^{{-}1}-1, g^{{-}1}g-1\rangle. \]

Then $U$ has a unique Hopf algebra structure such that $g,\,g^{-1}\in G(U)$ and $a_1,\,a_2\in \mathcal {P}_{g,1}(U)$. Moreover, there exists a well-defined Hopf algebra map

\[ \varphi_{\lambda,\mu}:U\to {\mathfrak U}^{\text {jordan}}_{\lambda}, \quad g\mapsto g,\, a_1\mapsto a_1,\, a_2\mapsto a_2+\mu(1-g). \]

It is easily checked that

\[ ga_1-a_1g-g+g^{2},\quad ga_2-(a_2+a_1)g\in \ker \varphi_{\lambda,\mu}. \]

Moreover, for $z\in U$ defined as in (1.2) we obtain that

\[ \varphi_{\lambda,\mu}(z)-z=a_1\mu(1-g)-\mu(1-g)a_1+\mu(1-g)=\mu(1-g^{2}). \]

Since $z=\lambda (1-g^{2})\in {\mathfrak U}^{\text {jordan}}_{\lambda }$, we conclude that $\varphi _{\lambda,\mu }$ induces a surjective Hopf algebra map

\[ \varphi_{\lambda,\mu}:{\mathfrak U}^{\text {jordan}}_{\lambda+\mu}\to {\mathfrak U}^{\text {jordan}}_\lambda. \]

It follows that $\varphi _{0,\lambda }:{\mathfrak U}^{\text {jordan}}_{\lambda }\to {{\mathfrak U}^{\text {jordan}}}$ is a Hopf algebra isomorphism.

Remark 4 For any ${\unicode{x2644}} \in { \Bbbk }$, the Hopf algebra $U_{{\unicode{x2644}} }$ was introduced by Christian Ohn in [Reference Ohn3]; this is the algebra generated over ${ \Bbbk }$ by $K,\,Y,\,T^{\pm 1}$ with relations:

(1.6)\begin{align} TT^{{-}1} = T^{{-}1}T = 1, \quad [K,T] & =T^{2}-1, \quad [Y,T] ={-}\displaystyle\frac{{\unicode{x2644}}}{2}(KT+TK), \end{align}

(1.7)\begin{align} [K,Y] & ={-}\frac 12(YT+TY+YT^{{-}1}+T^{{-}1}Y), \end{align}

with the Hopf algebra structure of $U_{{\unicode{x2644}} }$ determined by $T\in G(U_{{\unicode{x2644}} })$ and $X,\, Y \in \mathcal {P}_{T^{-1},\, T}(U_{{\unicode{x2644}} })$. It is easy to see that the the Hopf algebras $U_{{\unicode{x2644}} }$ with ${\unicode{x2644}} \neq 0$ are all isomorphic so we fix one of them. The appellative Jordanian was introduced by Alev and Dumas to the best of our knowledge. We claim that ${\mathfrak U}^{\text {jordan}}_{\lambda }$ is isomorphic to the Hopf subalgebra $\mathfrak U$ of $U_{{\unicode{x2644}} }$ generated by

(1.8)\begin{equation} x= KT^{{-}1},\quad y = YT^{{-}1}, \ g =T^{{-}2};\end{equation}

we choose these variables to have $x,\, y \in \mathcal {P}_{g, 1}(\mathfrak U)$. Now (1.6) implies

(1.9)\begin{equation} g \cdot x = x+ 2 (1-g), \quad g \cdot y = y - 2{\unicode{x2644}} (x + (1- g)). \end{equation}

We perform a new change of variables:

\[ a_1= \displaystyle\frac{1}{2}x,\quad a_2={-}\frac{1}{4{\unicode{x2644}}}y - \frac{1}{4}x; \]

these new variables satisfy (1.1). Now (1.7) translates succesively into

\[ xy-yx ={-}2y-{{\unicode{x2644}}}x^{2}+\displaystyle\frac{{\unicode{x2644}}}4(1-g^{2}) \]

and then into

\[ z={-}\displaystyle\frac{1}{32}(1-g^{2}). \]

That is, $\mathfrak U \simeq {\mathfrak U}^{\text {jordan}}_{-\frac {1}{32}}$.

Remark 5 The algebra $U_{{\unicode{x2644}} }$ can be described as an iterated Ore extension:

(1.10)\begin{equation} U_{{\unicode{x2644}}}={ \Bbbk}[T^{{\pm}}][x\,;\delta][y\,;\,\sigma,D]\end{equation}

with $\delta$ a derivation of ${ \Bbbk }[T^{\pm }]$, $\sigma$ an automorphism of ${ \Bbbk }[T^{\pm }][x\,;\delta ]$ and $D$ a $\sigma$-derivation of ${ \Bbbk }[T^{\pm }][x\,;\delta ]$ defined by:

(1.11)\begin{align} \textstyle xT& =Tx+\underbrace{(T-T^{{-}1})}_{=\delta(T)} \end{align}

(1.12)\begin{align} yT& =\underbrace{T}_{=\sigma(T)}y+\underbrace{(-{\unicode{x2644}} Tx-\displaystyle\frac {\unicode{x2644}} 2 (T-T^{{-}1}))}_{{=}D(T)} \end{align}

(1.13)\begin{align} yx& =\underbrace{(x+2)}_{=\sigma(x)}y+\underbrace{{\unicode{x2644}} x^{2}-\frac {\unicode{x2644}} 4(1-T^{{-}4})}_{{=}D(x)}. \end{align}

Proof. We leave the verification of the first claim to the reader as a long but straightforward exercise: the derivations $\delta _1:R\to R$, $\delta _2:S\to S$ satisfy

\[ \delta_1(g) = g^{2}-g,\quad \delta_2(g)={-}a_1\,g, \delta_2(a_1)= \displaystyle\frac{1}{2}a_1(1-a_1), \]

and $\sigma$ is given by $\sigma (g)=g$, $\sigma (a_1)=a_1+1$. The rest is standard.

1.2. The gap and how to fix it

We fix a group $G$. Let $H$ be a pointed Hopf algebra with coradical filtration $(H_n)_{n\in \mathbb {N}_0}$ such that $G(H) \simeq G$. Then $H_1/ H_0 \simeq V \# { \Bbbk }G$, where $V \in {{}^{ { \Bbbk }G }_{ { \Bbbk }G }\mathcal {YD}}$ is the infinitesimal braiding of $H$. For $g\in G$, the space of $(g,\,1)$ skew-primitives $\mathcal {P}_{g,1}(H)$ satisfies

\[ \mathcal{P}_{g,1}(H) \cap H_0 = { \Bbbk} (1-g) \text{ and } \mathcal{P}_{g,1}(H) / \left(\mathcal{P}_{g,1}(H) \cap H_0 \right) \simeq V_g. \]

Now assume that $V \simeq \mathcal {V}_g(\chi,\,\eta )$ for a YD-triple $\mathcal {D} = (g,\, \chi,\, \eta )$ over ${ \Bbbk }G$. Thus $V = V_g$ and we have an exact sequence of $G$-modules

Since $g\in Z(G)$, one has ${ \Bbbk } (1-g) \subset \mathcal {P}_{g,1}(H)^{\varepsilon }$. Hence $\chi \neq \varepsilon$ implies that

\[ \mathcal{P}_{g,1}(H) \simeq { \Bbbk} (1-g) \oplus \mathcal{V}_g(\chi,\eta) \]

and we have a morphism of Hopf algebras $\pi : \mathcal {T}(\mathcal {V}_g(\chi,\,\eta )) \to H$, where $\mathcal {T}(\mathcal {V}_g(\chi,\,\eta )) = T(\mathcal {V}_g(\chi,\,\eta )) \# { \Bbbk }G$. In particular the proof of [Reference Andruskiewitsch, Angiono and Heckenberger1, Prop. 4.3] goes over without changes.

We assume for the rest of this Section that the infinitesimal braiding $V$ of $H$ is isomorphic to $\mathcal {V}_g(\varepsilon,\,\eta )$ for a YD-triple $\mathcal {D} = (g,\, \varepsilon,\, \eta )$ as Yetter-Drinfeld module over ${ \Bbbk }G$. Under this assumption, $\mathcal {P}_{g,1}(H)$ might be indecomposable.

Back to our situation, let us pick $a_1,\, a_2 \in \mathcal {P}_{g,1}(H)$ such that $\varpi (a_j) = x_j$, $j= 1,\,2$ and set $a_0 = 1-g$. Then there are $\zeta \in \operatorname {Der}_{\varepsilon,\varepsilon }({ \Bbbk }G,\, { \Bbbk })$ and a linear map $\xi : { \Bbbk }G \to { \Bbbk }$ such that the action of $h\in G$ on $\mathcal {P}_{g,1}(H)$ is given in the basis $(a_0,\, a_1,\, a_2)$ by

(1.14)\begin{equation} \rVert h\rVert = \begin{pmatrix} 1 & \zeta(h) & \xi(h) \\ 0 & 1 & \eta(h) \\ 0 & 0 & 1 \end{pmatrix}. \end{equation}

Notice that $\operatorname {Der}_{\varepsilon,\varepsilon }({ \Bbbk }G,\, { \Bbbk }) = \operatorname {Hom}_{\text {gps}}(G,\, ({ \Bbbk },\, +))$ and that $\xi$ is a kind of differential operator of degree 2, meaning that

(1.15)\begin{equation} \xi (hk)= \xi(h)+\zeta(h)\eta(k) +\xi(k) \text{ for all }h,k\in G. \end{equation}

Thus if $\zeta \neq 0$, then the claim [Reference Andruskiewitsch, Angiono and Heckenberger1, Prop. 4.2, page p. 669, line 8] is not true. To correct this we consider the subalgebra $A$ generated by $g$ and $\mathcal {P}_{g,1}(H)$, a Hopf subalgebra of $H$. The action of $g$ on $\mathcal {P}_{g,1}(H) = \mathcal {P}_{g,1}(A)$ in the basis $(a_0,\, a_1,\, a_2)$ is given by

(1.16)\begin{equation} \rVert g\rVert =\begin{pmatrix} 1 & \zeta(g) & \xi(g) \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix}. \end{equation}

As $g\in Z(G)$, we have that $\xi (gh) = \xi (hg)$ for all $h\in G$, so (1.15) says that

(1.17)\begin{equation} \zeta(h) = \eta(h)\zeta(g)\text{ for all }h\in G. \end{equation}

We consider two cases:

1. (A) $\zeta (g) = 0$. Then $\zeta =0$ by (1.17) and $\xi \in \operatorname {Der}_{\varepsilon,\varepsilon }({ \Bbbk }G,\, { \Bbbk })$ by (1.15).

2. (B) $t:= \zeta (g) \neq 0$, the Jordanian case. In the basis $(a_0,\, t^{-1}a_1,\, t^{-1}a_2 - t^{-2}\xi (g) a_1)$, the action of $g$ is given by $\small \begin {pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end {pmatrix}$. We still denote the new basis by $(a_0,\, a_1,\, a_2)$; that is, we may assume that $\zeta (g)=1$, $\xi (g)=0$. By (1.17), $\zeta =\eta$, and by (1.15), $\xi (hk)= \xi (h)+\eta (h)\eta (k) +\xi (k)$ for all $h,\,k\in G$.

We shall see that the following Hopf algebras exhaust the case (A).

Definition 9 Let $\mathcal {D} = (g,\, \varepsilon,\, \eta )$ be a YD-triple, $\xi \in \operatorname {Der}_{\varepsilon,\varepsilon }({ \Bbbk }G,\, { \Bbbk })$ and $\lambda \in \Bbbk$. We define $\mathfrak U_{\xi }(\mathcal {D},\,\lambda )$ as the algebra generated by $h\in G$, $a_1$, $a_2$ with defining relations being those of $G$ and

(1.18)\begin{align} & ha_1 - a_1\,h, \quad h\in G; \end{align}

(1.19)\begin{align} & ha_2 - (a_2+\eta(h)a_1+\xi(h)(1-g))h, \quad h\in G; \end{align}

(1.20)\begin{align} & a_1a_2 - a_2a_1 -\dfrac{a_1^{2}}{2} -\lambda(1-g^{2}). \end{align}

As we said already, $\mathfrak U_{0}(\mathcal {D},\,\lambda )\simeq \mathfrak U(\mathcal {D},\,\lambda )$, introduced in [Reference Andruskiewitsch, Angiono and Heckenberger1, §4.1].

Lemma 10 $\mathfrak U_{\xi }(\mathcal {D},\,\lambda )$ is a Hopf algebra with comultiplication determined by

\[ G(\mathfrak U_{\xi}(\mathcal{D},\lambda)) = G \text{ and } a_1,a_2 \in \mathcal{P}_{g,1}(\mathfrak U_{\xi}(\mathcal{D},\lambda)). \]

Thus $\mathfrak U_{\xi }(\mathcal {D},\,\lambda )$ is pointed. The set $\{a_1^{m} a_2^{n} h \, | \, m,\,n\in \mathbb {N}_0,\, \, h\in G \}$ is a basis of $\mathfrak U_{\xi }(\mathcal {D},\,\lambda )$; $\operatorname {gr} \mathfrak U_{\xi }(\mathcal {D},\,\lambda )\simeq \mathscr {B}(\mathcal {V}(1,\,2))\# { \Bbbk }G$ and

\[ \operatorname{GKdim} \mathfrak U_{\xi}(\mathcal{D},\lambda) = \operatorname{GKdim} \Bbbk G +2. \]

In particular, if $G$ is nilpotent-by-finite, then $\operatorname {GKdim} \mathfrak U_{\xi }(\mathcal {D},\,\lambda ) < \infty$.

We shall see that the following Hopf algebras exhaust the case (B).

Definition 11 Let $\mathcal {D} = (g,\, \varepsilon,\, \eta )$ be a YD-triple and define $\xi \in ({ \Bbbk }G)^{*}$ by $\xi (h) =\tfrac {1}{2}(\eta (h)^{2}-\eta (h))$, $h \in G$. We introduce ${{\mathfrak U}^{\text {jordan}}}(\mathcal {D})$ as the algebra generated by $h\in G$, $a_1$, $a_2$ with defining relations those of $G$, (1.19) and

(1.21)\begin{align} & ha_1 - (a_1 +\eta(h)(1-g) ) h, \quad h\in G. \end{align}

(1.22)\begin{align} & a_1a_2 - a_2a_1 -\dfrac{a_1^{2}}{2} + a_2 +\dfrac{1}{2}a_1. \end{align}

Observe that $\xi$, needed in (1.19), satisfies (1.15) with $\zeta = \eta$. The proof of the following Lemma is also standard.

Lemma 12 ${{\mathfrak U}^{\text {jordan}}}(\mathcal {D})$ is a Hopf algebra with structure determined by

\[ G({{\mathfrak U}^{\text {jordan}}}(\mathcal{D})) = G \text{ and } a_1,a_2 \in \mathcal{P}_{g,1}({{\mathfrak U}^{\text {jordan}}}(\mathcal{D})). \]

Thus ${{\mathfrak U}^{\text {jordan}}}(\mathcal {D})$ is pointed. The set $\{a_1^{m} a_2^{n} h \, | \, m,\,n\in \mathbb {N}_0,\, \, h\in G \}$ is a basis of ${{\mathfrak U}^{\text {jordan}}}(\mathcal {D})$; $\operatorname {gr} {{\mathfrak U}^{\text {jordan}}}(\mathcal {D})\simeq \mathscr {B}(\mathcal {V}(1,\,2))\# { \Bbbk }G$ and

\[ \operatorname{GKdim} {{\mathfrak U}^{\text {jordan}}}(\mathcal{D}) = \operatorname{GKdim} \Bbbk G +2. \]

In particular, if $G$ is nilpotent-by-finite, then $\operatorname {GKdim} {{\mathfrak U}^{\text {jordan}}}(\mathcal {D}) < \infty$.

1.3. Proof of proposition 1

Let $G$ be a nilpotent-by-finite group and let $H$ be a pointed Hopf algebra with finite $\operatorname {GKdim}$ such that $G(H) \simeq G$ and the infinitesimal braiding $V$ of $H$ is isomorphic to $\mathcal {V}(1,\, 2)$. By [Reference Andruskiewitsch, Angiono and Heckenberger1, Lemma 2.3], there exists a unique YD-triple $\mathcal {D} = (g,\, \chi,\, \eta )$ such that $V \simeq \mathcal {V}_g(\chi,\,\eta )$ in ${{}^{ { \Bbbk }G }_{ { \Bbbk }G }\mathcal {YD}}$. By [Reference Andruskiewitsch, Angiono and Heckenberger1, Lemma 3.7], $\operatorname {gr} H \simeq \mathscr {B}(\mathcal {V}(1,\,2)) \# { \Bbbk }G$, hence $H$ is generated by $\mathcal {P}_{g,1}(H)$ and $G$ as algebra.

If $\chi \neq \varepsilon$, then the proof of [Reference Andruskiewitsch, Angiono and Heckenberger1, Prop. 4.1] implies that $H$ is isomorphic either to $\mathfrak U(\mathcal {D},\,0)$ or $\mathfrak U(\mathcal {D},\,1)$, the Hopf algebras introduced in [Reference Andruskiewitsch, Angiono and Heckenberger1, §4.1].

Assume that $\chi = \varepsilon$. Pick a basis $(a_0 = 1-g,\, a_1,\, a_2)$ such that any $h\in G$ acts on $\mathcal {P}_{g,1}(H)$ by (1.14) where $\zeta \in \operatorname {Der}_{\varepsilon,\varepsilon }({ \Bbbk }G,\, { \Bbbk })$ and $\xi \in ({ \Bbbk }G )^{*}$ satisfies (1.15). Let $A$ be the subalgebra generated by $\mathcal {P}_{g,1}(H)$. As explained above we consider two cases.

Case (A): $\zeta (g) = 0$, thus $\zeta = 0$. Even if [Reference Andruskiewitsch, Angiono and Heckenberger1, Proposition 4.2] does not apply in general since we may have $\xi \ne 0$, it does apply to $A$ up to changing the base to $(a_0,\, a_1,\, \widetilde {a}_2)$ where $\widetilde {a}_2 := a_2 - \xi (g) a_1$, see (1.14). Call the new basis again $(a_0,\, a_1,\, a_2)$ by abuse of notation. Hence $A \simeq \mathfrak U(\mathcal {D}',\,\lambda )$ where $\mathcal {D}' = (g,\, \chi _{\vert \langle g \rangle },\, \eta _{\vert \langle g \rangle })$ is a YD-triple over the subgroup $\langle g \rangle$ of $G$ and $\lambda \in \{0,\,1\}$. In particular the following equality holds in $H$:

\[ a_2a_1=a_1a_2-\tfrac{1}{2}a_1^{2}+\lambda(1-g^{2}) . \]

We first claim that $A$ is stable under the action of $G$. Indeed let $G$ act on the free algebra generated by $g^{\pm 1}$, $a_1$, $a_2$, where $G$ acts trivially on $g$, and by (1.14) on $a_1$, $a_2$. As $g$ is central, the action of each $h\in G$ preserves the defining ideal of $A$, so $G$ acts on $A$.

We next claim that $H \simeq A\rtimes { \Bbbk }G /I$, where $I$ is the ideal that identifies the two copies of $g$ where $\rtimes$ stands for smash product. Indeed, the inclusions $A\hookrightarrow H$, $\Bbbk G\hookrightarrow H$ induce a Hopf algebra map $\psi :A\rtimes \Bbbk G/ I \to H$. As $\operatorname {gr} H\simeq \mathscr {B}(V)\#\Bbbk G$, $H$ is generated by $a_1$, $a_2$ and $G$, so $\psi$ is surjective. On the other hand, $(A\rtimes \Bbbk G/ I )_1$ is spanned by the set $\{1\otimes h,\, a_1\otimes h,\, a_2\otimes h | h\in G\}$. The image of this set under $\psi$ is linearly independent, which implies that $\psi _{\vert (A\rtimes \Bbbk G/ I )_1}$ is injective. By [Reference Montgomery2, 5.3.1], $\psi$ is injective, and the claim follows. As a consequence, the set $\{a_1^{m} a_2^{n} h \, | \, m,\,n\in \mathbb {N}_0,\, \, h\in G \}$ is a basis of $H$.

Finally, we see that there is a Hopf algebra map $\mathfrak U_{\xi }(\mathcal {D},\,\lambda ) \to H$; since this map sends a basis to a basis, we conclude that $H\simeq \mathfrak U_{\xi }(\mathcal {D},\,\lambda )$.

Case (B). $\zeta (g) \neq 0$. As discussed above, we may assume that $\zeta = \eta$. Recall that we are assuming that $\operatorname {GKdim} H < \infty$. We claim that

1. (i) There exists a Hopf algebra isomorphism $A \simeq {{\mathfrak U}^{\text {jordan}}}$, cf. (1.4).

2. (ii) $\xi (h)=\tfrac {1}{2}(\eta (h)^{2}-\eta (h))$ for all $h\in G$.

3. (iii) $A$ is stable under the adjoint action of $G$ and $H \simeq A\rtimes { \Bbbk }G /I$, where $I$ is the ideal that identifies the two copies of $g$.

4. (iv) The set $\{a_1^{m} a_2^{n} h \, | \, m,\,n\in \mathbb {N}_0,\, \, h\in G \}$ is a basis of $H$ and $H\simeq {{\mathfrak U}^{\text {jordan}}}(\mathcal {D})$.

(i): It is easy to see that there exists a Hopf algebra surjective map $\widetilde {\pi }:{{\widetilde {\mathfrak U}}^{\mathtt {jordan}}}\to A$, which applies $g$, $a_1$, $a_2$ to the corresponding elements of $A$. Hence $\widetilde {\pi }(z)\in \mathcal {P}_{g^{2},1}(A)$, by Lemma 2. Now, as $g\ne g^{2}$ and $\operatorname {gr} H\simeq \mathscr {B}(V)\# \Bbbk G$, we have that $\mathcal {P}_{g^{2},1}(H)=\mathcal {P}_{g^{2},1}(H) \cap H_0 = { \Bbbk } (1-g^{2})$; thus there exists $\lambda \in \Bbbk$ such that $\widetilde {\pi }(z)=\lambda (1-g^{2})$, which implies that $\widetilde {\pi }$ factors through a map $\pi : {\mathfrak U}^{\text {jordan}}_{\lambda } \twoheadrightarrow A$. The set $\{g^{k},\,a_1g^{k},\, a_2g^{k}: k\in \mathbb {Z}\}$ is linearly independent in $H$, so $\pi _{\vert ({\mathfrak U}^{\text {jordan}}_{\lambda })_1}$ is injective. By [Reference Montgomery2, 5.3.1], $\pi$ is an isomorphism. Up to composing with $\varphi _{0,\lambda }$, see Remark 3, we may assume that $\lambda =0$.

(ii): Given $h\in G$, let $\gamma _h\in \operatorname {End}_{{ \Bbbk }} H$ be given by

\[ \gamma_h (x)=h x h^{{-}1} -x \text{ for all } x\in H. \]

Note that $\gamma _h(xy)=\gamma _h(x)(\gamma _h(y)+y)+x\gamma _h(y)$ for all $x,\,y\in H$. From (1.14),

(1.23)\begin{equation} \gamma_h(a_1)=\eta(h)(1-g),\quad \gamma_h(a_2)=\eta(h)a_1+\xi(h)(1-g). \end{equation}

Therefore,

\begin{align*} \gamma_h(z) & =\eta(h)(1-g)(\eta(h)a_1+\xi(h)(1-g)+a_2)+a_1(\eta(h)a_1+\xi(h)(1-g))\\ & \quad+\left(\eta(h)a_1+\xi(h)(1-g)+\frac 12 \eta(h)(1-g)\right)(-\eta(h)(1-g)+1-a_1) \\ & \quad -\left(a_2+\frac{1}{2}a_1\right)\eta(h)(1-g) =\left(\frac{1}{2} \eta(h) -\frac{1}{2}\eta(h)^{2}+\xi(h)\right)(1-g^{2} ). \end{align*}

By (i), $z=0$, so $\gamma _h(z)=0$. Thus, $\xi (h)=\tfrac 12( \eta (h)^{2}- \eta (h))$.

(iii): Let $G$ act on the free algebra generated by $g^{\pm 1}$, $a_1$, $a_2$, where $G$ acts trivially on $g$, and by (1.14) on $a_1$, $a_2$. Each $h\in G$ fixes the defining relations $gg^{-1} - 1$, $g^{-1}g - 1$, $ga_1 - a_1\,g - g+g^{2}$, $z$, and

\[ h\cdot ( ga_2 - a_2\,g - a_1\,g ) = ga_2- a_2g- a_1g+\eta(h) (ga_1-a_1\,g -(1-g)g ), \]

so the action descends to $A$. The proof of (iv) is as in Case (A).