Coactions of cocommutative Hopf algebras on skew polynomial rings

Technical Summary: Coactions of Cocommutative Hopf Algebras on Skew Polynomial Rings

Problem Statement
The paper addresses the classification of cocommutative Hopf algebras that coact inner-faithfully on one-parameter skew polynomial rings, denoted $A_q(n) = k \langle x_1, \dots, x_n \rangle / (x_j x_i - q x_i x_j \mid i < j)$, for $n=2$ and $n=3$. This problem is framed within the broader context of understanding quantum group symmetries of Artin–Schelter (AS) regular algebras. Specifically, the authors aim to generalize Crawford's classification of group gradings on two-dimensional AS regular algebras to higher dimensions and to determine the extent to which non-abelian group gradings or non-commutative cocommutative Hopf coactions exist on these rings.

Methodology
The authors employ Manin's universal coacting Hopf algebra construction, denoted $\text{aut}(A_q(n))$, as the central tool. The strategy relies on the observation that a Hopf algebra is a group algebra if and only if it is cocommutative and spanned by its group-like elements. Consequently, the classification of group gradings (and more generally, cocommutative coactions) is reduced to classifying the cocommutative quotients of $\text{aut}(A_q(n))$.

The methodology proceeds in the following steps:

1. Explicit Presentation of $\text{aut}(A_q(n))$: The authors first derive an explicit presentation of the universal right coacting Hopf algebra $\text{aut}(A_q(n))$ for arbitrary $q \in k^*$ and $n \in \mathbb{N}$. This is achieved by constructing the left coacting Hopf algebra of the quadratic dual $A_q^!$ (a quantum Grassmann algebra) and utilizing the isomorphism $\text{aut}_r(A_q) \cong \text{aut}_\ell(A_q^!)$.
2. Involutization: Since cocommutative Hopf algebras are involutive (i.e., $S^2 = \text{id}$), the authors first compute the "involutization" of $\text{aut}(A_q(n))$, denoted $H_q(n) = \text{aut}(A_q(n))_{\text{inv}}$, by quotienting by the ideal generated by $S^2(x) - x$.
3. Classification of Cocommutative Quotients: The core of the work involves analyzing the maximal cocommutative quotients of $H_q(n)$. This is done by imposing the cocommutativity condition ($\Delta = \Delta^{\text{op}}$) on the generators of $H_q(n)$ and solving the resulting system of relations.
4. Dimensional Analysis:
   • For $n=2$, the authors analyze the relations in $H_q(2)$ to determine the structure of its cocommutative quotients.
   • For $n=3$, they utilize a reduction strategy. By analyzing the consequences of cocommutativity, they show that if a cocommutative quotient exists, certain off-diagonal generators must vanish, allowing the problem to be reduced to the $n=2$ case via specific subalgebras.

Key Contributions and Results

1. Explicit Presentation of $\text{aut}(A_q(n))$:
   The paper provides a complete presentation of $\text{aut}(A_q(n))$ for arbitrary $q$ and $n$. The algebra is generated by elements $x_{ij}$ and $D^{-1}$ (where $D$ is the quantum determinant), subject to relations involving $q$-commutation of rows, quantum minor relations, and specific formulas for the coproduct, counit, and antipode involving quantum minors. This fills a gap in the literature where explicit presentations for arbitrary $q$ and $n$ were previously unavailable.

2. Classification for $n=2$ (Theorem 1.2):
   For the two-variable skew polynomial ring $A_q(2)$, the authors classify the maximal cocommutative quotients of $\text{aut}(A_q(2))$:

   • If $q \neq -1$, the only inner-faithful cocommutative coactions are by Hopf quotients of the group algebra $k\mathbb{Z}_2$ and the Hopf algebras $A(0, q^{\pm 1})$ (defined as $k\langle x^{\pm 1}, y \rangle / (yx - qxy)$ with $x$ group-like and $y$ $x^n$-skew primitive).
   • If $q = -1$, an additional coaction exists by Hopf quotients of $k\Gamma$, where $\Gamma = \langle f, g \mid f^2 = g^2 \rangle$.
   • Corollary: This recovers Crawford's result that $A_q(2)$ admits a faithful grading by a non-abelian group if and only if $q = -1$.

3. Classification for $n=3$ (Theorem 1.3):
   For the three-variable ring $A_q(3)$ with $q \neq \pm 1$, the authors prove that the only inner-faithful cocommutative coactions are by Hopf quotients of:

   • The group algebra $k\mathbb{Z}_3$.
   • The Hopf algebras $B_{q^{\pm 1}}$ and $C_{q^{\pm 1}}$, which are specific extensions of $A(0, q^{\pm 1})$ involving three generators.
   • Corollary: $A_q(3)$ does not admit a faithful grading by a non-abelian group for $q \neq \pm 1$. The paper notes that for $q = -1$, non-abelian gradings do exist (via the Ore extension structure $A_{-1}(3) \cong A_{-1}(2)[z; \sigma]$).

4. Applications to GK-Dimension and Cosemisimplicity:

   • The authors address a question by Chan–Walton–Zhang regarding non-commutative Hopf algebras of GK-dimension 1 coacting on $A_q(2)$. They show that if $q$ is not a root of unity, no such non-commutative, cocommutative Hopf algebras exist.
   • Regarding Question 8.4 (Bichon) on cosemisimple Hopf coactions on commutative domains, the authors demonstrate that the universal involutive Hopf algebra $H_1(3)$ coacting on $k[x, y, z]$ is non-commutative, non-Noetherian, and has infinite GK-dimension. This contrasts with the $n=2$ case where the universal involutive algebra is commutative.

Significance
The paper claims significance in three main areas:

1. Generalization of Crawford's Result: It successfully extends the classification of group gradings from dimension 2 to dimension 3, establishing that non-abelian group gradings on skew polynomial rings are highly restricted (essentially limited to the $q=-1$ case).
2. Structural Insight: By providing an explicit presentation of $\text{aut}(A_q(n))$, the authors enable further study of quantum symmetries for these algebras. The construction of the involutization $H_q(n)$ serves as a necessary intermediate step for analyzing cocommutative symmetries.
3. Distinction between Dimensions: The results highlight a sharp distinction between low-dimensional ($n=2$) and higher-dimensional ($n \ge 3$) AS regular algebras regarding their symmetry structures. Specifically, the paper shows that while $n=2$ allows for non-abelian group gradings at $q=-1$, $n=3$ does not (for generic $q$), and the universal involutive coacting algebra for $n \ge 3$ exhibits significantly more complex behavior (non-commutative, infinite GK-dimension) compared to the commutative case at $n=2$.

The authors explicitly state that their methods do not provide information on coactions by non-cocommutative Hopf algebras, nor do they fully classify group gradings for all $q$ in dimension 3 (specifically excluding $q = \pm 1$ due to computational complexity, though they provide partial results for $q=-1$). Future work on more general regular algebras is noted as a separate endeavor involving superpotentials.