Overview
This research investigates skew-commutative subalgebras within Drinfeld-Jimbo quantum groups at a root of unity $\zeta$. The focus is on classifying their centrality and commutativity based on the Lie type and the order of $\zeta$ modulo 8. The study further describes Hopf ideals induced from ideals within these subalgebras, encompassing non-commutative scenarios. A significant aspect involves the construction and analysis of a family of Hopf ideals dependent on a choice of an element in the Weyl group, demonstrating their natural emergence in (partial) $R$-matrix constructions and as vanishing ideals of Bruhat subgroups.
The specialization to the maximal element facilitates a rigorous, choice-independent construction of restricted quantum groups as pre-triangular Hopf algebras. The methodology accommodates even orders of $\zeta$, non-simply laced Lie types, and minimal ground rings. It also extends prior work by De Concini-Kac-Procesi, which primarily focused on odd orders of $\zeta$ and complex ground fields, cases where the subalgebra is strictly central. The research includes identification of these subalgebras for Lie types $\mathsf{A}_n$ and $\mathsf{B}_2$ with coordinate rings of associated algebraic groups in commutative cases, even for even orders of $\zeta$. The descriptions provided are computationally explicit and do not rely on Poisson structures.
Research Context
The work builds upon and extends previous research, specifically referencing the work of De Concini-Kac-Procesi. The extension involves addressing scenarios not covered by their prior studies, such as even orders of $\zeta$, non-simply laced Lie types, and minimal ground rings. The earlier work by De Concini-Kac-Procesi focused on conditions where $\zeta$ had odd orders, leading to strictly central subalgebras, and utilized complex ground fields. This research expands the scope to include more general conditions and maintains computational explicitness without resorting to Poisson structures.
Approach
The approach involves considering skew-commutative subalgebras generated by primitive power elements within Drinfeld-Jimbo quantum groups at a root of unity. A key step is the classification of the centrality and commutativity of these skew-polynomial algebras. This classification depends on the Lie type and the order of $\zeta$ modulo 8. The research then proceeds to describe Hopf ideals induced from ideals in these subalgebras, including those in non-commutative settings.
A family of Hopf ideals is constructed and analyzed, with its dependence on the choice of an element in the Weyl group explicitly observed. Their role in the construction of (partial) $R$-matrices and as vanishing ideals of Bruhat subgroups is established. The rigorous construction of restricted quantum groups as pre-triangular Hopf algebras is achieved through specialization to the maximal element, ensuring independence from specific choices.
Technical preparations undertaken for this research include discussions on PBW bases over minimal rings, considerations of dependencies on choices of convex orderings, and new constructions and relations among automorphisms on quantum groups. Formulas for the Garside element in the Lustzig-Artin group action and the family of Che-transformations are provided as part of these preparations.
Findings
- The centrality and commutativity of skew-polynomial algebras within Drinfeld-Jimbo quantum groups at a root of unity $\zeta$ are classified based on Lie type and the order of $\zeta$ modulo 8.
- Hopf ideals induced from ideals in these subalgebras, including non-commutative cases, have been described.
- A family of Hopf ideals dependent on a choice of an element in the Weyl group was constructed and analyzed. These ideals naturally arise in the construction of (partial) $R$-matrices and as vanishing ideals of Bruhat subgroups.
- Specialization to the maximal element facilitates a rigorous, choice-independent construction of restricted quantum groups as pre-triangular Hopf algebras.
- The treatment includes even orders of $\zeta$, non-simply laced Lie types, and minimal ground rings, extending previous results by De Concini-Kac-Procesi.
- The research identified the subalgebras for Lie types $\mathsf{A}_n$ and $\mathsf{B}_2$ with the coordinate rings of associated algebraic groups in commutative cases, even when $\zeta$ has even order.
- Computational explicitness is maintained, and the descriptions do not utilize Poisson structures.
- Technical discussions include PBW bases over minimal rings, dependencies on choices of convex orderings, and new constructions and relations among automorphisms on quantum groups, such as formulae for the Garside element in the Lustzig-Artin group action and the family of Che-transformations.