Primitive Deformations of Quantum p-Groups

Abstract

In this paper, working over an algebraically closed field k of prime characteristic p, we introduce a concept, called Primitive Deformation, to provide a structured technique to classify certain finite-dimensional Hopf algebras which are Hopf deformations of restricted universal enveloping algebras. We illustrate this technique for the case when the restricted Lie algebra has dimension 3. Together with our previous classification results, we provide a complete classification of p³-dimensional connected Hopf algebras over k of characteristic p > 2.

Appendices

Appendix A: Classifications of p ³-Dimensional Connected Hopf Algebras

Let H be a p³-dimensional connected Hopf algebra over an algebraically closed field k of characteristic p > 0. Let u(P(H)) be the Hopf subalgebra of H generated by all primitive elements. The isomorphism classes of H will be presented by a quotient of the free algebra k〈x,y,z〉/I. The defining relation I and the comultiplication are provided in terms of the generators x, y and z. We follow the notation in [5] to write the reduced comultiplication of H by ψ, for example, ψ(x) = Δ(x) − x ⊗ 1 − 1 ⊗ x. Throughout, we denote the expressions

$$\begin{array}{@{}rcl@{}} \boldsymbol{\omega}(r)&:=&{\sum}_{1\le i\le p-1}\frac{(p-1)!}{i!\,(p-i)!}\, \left( r^{i}\otimes r^{p-i}\right), \quad \text{ for any } r \in H, \\ \mathcal Z &:=&\boldsymbol{\omega}(x)[y\otimes 1 + 1\otimes y+\boldsymbol{\omega}(x)]^{p-1}+\boldsymbol{\omega}(y), \text{ and} \\ \mathcal Z^{\prime} &:=&\boldsymbol{\omega}(x)(y\otimes 1 + 1\otimes y)^{p-1}+\boldsymbol{\omega}(y). \end{array} $$

By Nichols-Zoeller Theorem, dimension of u(P(H)) divides the dimension of H. Hence, our strategy is to consider all possible dimensions of u(P(H)) when H is p³-dimensional. The classifications in Tables 6, 7, and 8 were detailed in our preceding paper [19], where we classified the types (A), (B), and (C) according to the dimensions p, p² (non-commutative), and p³ of u(P(H)), respectively. For each type, we provide the isomorphism classes, their algebra and coalgebra structures, and indicate whether they are commutative, semisimple, or local. The main results in Section ?? classify those in Table 9, where u(P(H)) is a p²-dimensional commutative Hopf subalgebra of H.

Table 6 Connected p³-dim Hopf algebras, when ${\dim } u(\text {P}(H)) = p$

Table 7 When dimu(P(H)) = p² and u(P(H)) is non-commutative

Table 8 When ${\dim } u(\text {P}(H)) = p^{3}$

Table 9 When p > 2, dimu(P(H)) = p² and u(P(H)) is commutative

For the parametric family A(λ), we can choose λ = 0 when p = 2. When p > 2, A(λ)≅ A(λ^′) if and only if λ = γλ^′ for some $\gamma \in \sqrt [p^{2}+p-1]{1}$.

(B3) only appears in characteristic p > 2. In the algebra structure of (B2), we let

$$f(x)={\sum}_{i = 1}^{p-1} (-1)^{i-1}(p-i)^{-1}x^{i}. $$

(C15) only appears in p > 2 and C(λ,δ) satisfies λ^p− 1 = δ = ± 1. We also have C(λ,δ)≅ C(λ^′,δ^′) if and only if δ = δ^′ and λ = λ^′ or λλ^′ = 1.

Hence, these tables provide the complete classification of p³-dimensional connected Hopf algebras over an algebraically closed field of prime characteristic p.

Appendix B: Some Verifications for Section 5

We provide detailed computations for some cases discussed in Section 5. The remaining cases follow using similar arguments and will be left for the readers.

2.1 B.1 Verification of Table 3 for T = (T5)

Note that z^[p] = 0, x^[p] = x, y^[p] = 0, and ρ_z(x) = 0, ρ_z(y) = x. Let $P=(a,b,c)\in \mathbb A^{3}\setminus \{0\}$. By Proposition 2.9(iii), a direct computation in $H^{2}({\Omega }\, u(\mathfrak {h}))$ shows that

$$\begin{array}{@{}rcl@{}} [{\Phi}_{z}(\chi_{P})]&=& [\chi_{P}^{p}+\rho_{z}^{p-1}(\chi_{P})] = [(a\,x\otimes y+\boldsymbol{\omega}(b\,x+c\,y))^{p}+\rho_{z}^{p-1}(a\,x\otimes y)\\&&\qquad\qquad\qquad\qquad\quad +\rho_{z}^{p-1}\boldsymbol{\omega}(b\,x+c\,y)]\\ &=& [a^{p} x^{p}\otimes y^{p}+\boldsymbol{\omega}(b^{p}\,x^{p}+c^{p}\,y^{p})+a\rho_{z}^{p-2}(\rho_{z}(x)\otimes y+x\otimes \rho_{z}(y))\\&&\,\,+\rho_{z}^{p-1}\boldsymbol{\omega}(b\,x+c\,y)]\\ &=& [\boldsymbol{\omega}(b^{p}\,x)+a\,\rho_{z}^{p-2}(x\otimes x)]+[\rho_{z}^{p-1}\boldsymbol{\omega}(b\,x+c\,y)]\\ &=& [\boldsymbol{\omega}(b^{p}\,x)]+[\rho_{z}^{p-1}\boldsymbol{\omega}(b\,x+c\,y)] = [\boldsymbol{\omega}(b^{p}\,x)]. \end{array} $$

If $P\in \mathscr{A}^{+}(\text {T})$, one sees that b = 0 since [Φ_z(χ_P)] = 0 by Remark 3.3(i). So $P=(a,0,c)\in \mathbb A^{3}\setminus \{0\}$. Thus, Φ_z(χ_P) can be written as

$$\begin{array}{@{}rcl@{}} {\Phi}_{z}(\chi_{P})&=&\ \rho_{z}^{p-1}\boldsymbol{\omega}(c\, y)=\partial^{1}(-{\sum}_{i_{1}+\cdots+i_{p}=p-2}\, \frac{(p-2)!}{i_{1}!{\cdots} i_{p}!}\, \rho_{z}^{i_{1}}(c\, y){\cdots} \rho_{z}^{i_{p-1}}(c\, y)\rho_{z}^{1+i_{p}}(c\, y))\\ &=&\ \partial^{1}(-{\sum}_{i_{1}+\cdots+i_{p}=p-2}\, \frac{(p-2)!}{i_{1}!{\cdots} i_{p}!}\, c^{p}\, \rho_{z}^{i_{1}}(y){\cdots} \rho_{z}^{i_{p-1}}(y)\rho_{z}^{i_{p}}(x)). \end{array} $$

Since ρ_z(y) = x and ${\rho _{z}^{i}}(y)= 0$ for i ≥ 2, nonzero terms occur in the above summation only if i_p = 0 and, for 1 ≤ k ≤ p − 1, all i_k’s equal to 1 except one of the i_k’s is zero. Hence,

$${\Phi}_{z}(\chi_{P})=\partial^{1}(-(p-1)(p-2)!\, c^{p}\,x^{p-1}y)=\partial^{1}(c^{p}\,x^{p-1}y). $$

Set ${\Theta }=c^{p}(x^{p-1}y-y)\in u(\mathfrak {h})^{+}$. Then Φ_z(χ_P) = ∂¹(Θ) since ∂¹(y) = 0 and ρ_z(Θ) = ρ_z(c^p(x^p− 1y − y)) = c^p(x^p− 1ρ_z(y) − ρ_z(y)) = c^p(x^p − x) = 0. By Proposition 3.2, (Θ,χ_P) is a PD datum and $P\in \mathscr{A}^{+}(\text {T})$ by Remark 4.6. In conclusion, $\mathscr{A}^{+}(\text {T})$ contains all points $P=(a,0,c)\in \mathbb {A}^{3}\setminus \{0\}$.

Next, we compute the Aut(T)-action on $\mathscr{A}^{+}(\text {T})$. Let ϕ = (γ,G) ∈Aut(T). Since R = e₁₁, M = e₂₁ and λ = δ = 0, by Eq. 4.1.2, G = diag(α,β) and β = αγ, where α,β,γ are nonzero scalars with α^p = α. The ϕ-action on $P=(a,0,c)\in \mathscr{A}^{+}(\text {T})$ is given via (5.0.1) by

$$\phi(a,0,c)=(\gamma^{2}\alpha^{2}a,\, 0,\, \gamma^{\frac{1+p}{p}}\alpha c). $$

A simple calculation shows that the Aut(T)-orbits of $\mathscr{A}^{+}(\text {T})$ contains one single point and one quotient line. The single point can be represented by (1,0,0); and the quotient line is in terms of (ξ,0,1) where two points (ξ,0,1) and (ξ^′,0,1) are in the same orbit if and only if ξ = τξ^′ for some (p² − 1)/2-th root of unity τ. Furthermore, by Theorem 4.10, orbits [(1,0,0)] and [(ξ,0,1)] correspond to [(0,x ⊗ y)] and [(x^p− 1y − y,ξx ⊗ y + ω(y))] in $\mathcal H^{2}(\text {T})$, respectively.

2.2 B.2 Verification of Table 4 for T = (T2)

Note that z^[p] = 0, x^[p] = y^[p] = 0, and ρ_z(x) = y, ρ_z(y) = 0. Let $P=(a,b,c,d,e)\in \mathbb A^{2}\times \mathbb A^{3}$. By Proposition 2.9 (iii),

$$\begin{array}{@{}rcl@{}} {\Phi}_{z}(\chi_{P})&=&\,\rho_{z}^{p-1}\boldsymbol{\omega}(d\, x+e\, y)\\ &=&\ \partial^{1}(-\!\!{\sum}_{i_{1}+\cdots+i_{p}=p-2}\!\ \frac{(p-2)!}{i_{1}!{\cdots} i_{p}!}\, \rho_{z}^{i_{1}}(d\,x\,+\,e\,y){\cdots} \rho_{z}^{i_{p-1}}(d\,x\,+\,e\,y)\rho_{z}^{1+i_{p}}(d\,x\,+\,e\,y))\\ &=&\ \partial^{1}(-\!\!\!{\sum}_{i_{1}+\cdots+i_{p}=p-2}\ \frac{(p-2)!}{i_{1}!{\cdots} i_{p}!}\, \rho_{z}^{i_{1}}(d\,x){\cdots} \rho_{z}^{i_{p-1}}(d\,x)\rho_{z}^{1+i_{p}}(d\,x))\\ &=&\ \partial^{1}(-{\sum}_{i_{1}+\cdots+i_{p}=p-2}\ \frac{(p-2)!}{i_{1}!{\cdots} i_{p}!}\, d^{p}\rho_{z}^{i_{1}}(x){\cdots} \rho_{z}^{i_{p-1}}(x)\rho_{z}^{i_{p}}(y))\\ &=&\ \partial^{1}(-(p-1)(p-2)!\, d^{p}\, xy^{p-1}) = \partial^{1}(d^{p}\, xy^{p-1}). \end{array} $$

Then,

$$\begin{array}{@{}rcl@{}} {\Phi}_{z}(\chi_{P})&=&\ {\chi_{P}^{p}}-\delta\chi_{P}+\rho_{z}^{p-1}(\chi_{P})=\rho_{z}^{p-1}(c\,x\otimes y)+\rho_{z}^{p-1}\boldsymbol{\omega}(d\,x+e\,y)\\ &=&\ c\rho_{z}^{p-2}(\rho_{z}(x)\otimes y+x\otimes \rho_{z}(y))+\partial^{1}(d^{p}\,xy^{p-1})=c\rho_{z}^{p-2}(y\otimes y)\\&&+\partial^{1}(d^{p}\,xy^{p-1})\\ &=&\ \partial^{2}(d^{p}\,xy^{p-1}). \end{array} $$

Suppose $P\in \mathscr{B}^{+}(\text {T})$. By Definition 5.2, set Ψ_P = d^pxy^p− 1. Since ρ_z(d^pxy^p− 1 + ax + by) = d^py^p + ay = ay = 0, one sees that a = 0. Hence, the set $\mathscr{B}^{+}(\text {T})$ contains all points P = (0,b,c,d,e) with Ψ_P = d^pxy^p− 1, where c,d,e are not all zero.

2.3 B.3 Verification of Table 5 for T = (T2) and (T9)

Case T=(T2):

The representative points $P\in \mathscr{B}^{+}(\text {T})$ have one of the following forms:

$$\begin{array}{@{}rcl@{}} &&(0,b,1,0,0),\quad (0,b,0,1,0),\quad (0,b,0,0,1),\quad (0,b,1,1,0),\quad (0,b,1,0,1),\\ &&\text{for some}\ b\in \textbf{k}. \end{array} $$

If P is one of the first three cases (0,b,1,0,0), (0,b,0,1,0), (0,b,0,0,1), it is easy to find some $\phi \in \widetilde {\text {Aut}}(\text {T})$ such that the parameter b can be further taken as δ = 0,1. Hence, we obtain the first six isomorphism classes. By Lemma 5.6, for any $\phi =(\gamma ,G)\in \widetilde {\text {Aut}}(\text {T})$, we can write

$$G=\left( \begin{array}{ll} \alpha\gamma & \beta\\ 0 & \alpha \end{array} \right) $$

for some α≠ 0. Suppose P = (0,b,1,1,0). Then by the group action (5.6.1),

$$\phi(0,b,1,1,0)=(0,\, \gamma^{p}\alpha b,\, (\alpha \gamma)^{2},\, \alpha\gamma^{\frac{1+p}{p}},\, 0). $$

Since $(\alpha \gamma )^{2}=\alpha \gamma ^{\frac {1+p}{p}}= 1$, we have αγ = ± 1. Thus γ^pα = ± 1. So the parameter b is parametrized by k/μ₂. Suppose P = (0,b,1,0,1). Then,

$$\phi(0,b,1,0,1)=(0,\, \gamma^{p}\alpha b,\, (\alpha \gamma)^{2},\, 0,\, \alpha\gamma^{\frac{1}{p}}). $$

By similar reason that $(\alpha \gamma )^{2}=\alpha \gamma ^{\frac {1}{p}}= 1$, then γ^pα = 1. So b is parametrized by k.

Case T=(T9)

The representative points P have one of the following forms:

$$(a,0,1,0,0),\quad (a,0,0,0,1),\quad (a,0,1,0,1),\ \text{for some}\ b\in \textbf{k}. $$

Let P = (a,0,1,0,0). If a = 0, then P = (0,0,1,0,0). If a≠ 0, choose $\phi =(\gamma ,G)\in \widetilde {\text {Aut}}(\text {T})$ such that

$$\gamma=a^{-\frac{p + 1}{p^{2}+p-1}},\quad G=\left( \begin{array}{ll} a^{\frac{1}{p^{2}+p-1}} & 0\\ 0& a^{\frac{p}{p^{2}+p-1}} \end{array}\right). $$

Then by the group action (5.6.1), we have ϕ(P) = (1,0,1,0,0).

Let P = (a,0,0,0,1). If a = 0, then P = (0,0,0,0,1). If a≠ 0, choose $\phi =(\gamma ,G)\in \widetilde {\text {Aut}}(\text {T})$ such that

$$\gamma=a^{-\frac{p^{2}}{p^{3}-1}},\quad G=\left( \begin{array}{ll} a^{\frac{1}{p^{3}-1}} & 0\\ 0& a^{\frac{p}{p^{3}-1}} \end{array}\right). $$

Similarly, we have ϕ(P) = (1,0,0,0,1).

Let P = (a,0,1,0,1). For any $\phi =(\gamma ,G)\in \widetilde {\text {Aut}}(\text {T})$, we can write G = diag(α,α^p). Thus,

$$\phi(P)=(\gamma^{p}\alpha a,\, 0,\, \gamma\alpha^{p + 1},\, 0,\, \gamma^{\frac{1}{p}}\alpha^{p}). $$

Since $\gamma \alpha ^{p + 1}=\gamma ^{\frac {1}{p}}\alpha ^{p}= 1$, so γ = α^−p− 1 and $\alpha ^{p^{2}-p-1}= 1$. Hence $\gamma ^{p}\alpha a=\alpha ^{-p^{2}-p + 1}a=\alpha ^{-2p}a$. Moreover, since 2p and p² − p − 1 are coprime, the orbits containing P is parametrized by $\textbf {k}/\mu _{(p^{2}-p-1)}$.