[Paper Review] A quantum version of the algebra of distributions of $\operatorname{SL}_2$

Iván Angiono|arXiv (Cornell University)|Jul 17, 2016

Algebraic structures and combinatorial models10 references3 citations

TL;DR

This paper introduces a family of finite-dimensional algebras $\mathcal{D}_{\lambda,N}(\mathfrak{sl}_2)$ over $\mathbb{C}$ that generalize the small quantum group $\mathfrak{u}_\lambda(\mathfrak{sl}_2)$, forming a filtration where each algebra is a $\mathfrak{u}_\lambda(\mathfrak{sl}_2)$-cleft extension of the previous. The key result is a Steinberg-type tensor product decomposition: every simple $\mathcal{D}_{\lambda,N}(\mathfrak{sl}_2)$-module is isomorphic to the tensor product of a simple $\mathfrak{u}_\lambda(\mathfrak{sl}_2)$-module and a simple $\mathcal{D}_{\lambda,N-1}(\mathfrak{sl}_2)$-module.

ABSTRACT

Let $λ$ be a primitive root of unity of order $\ell$. We introduce a family of finite-dimensional algebras $\{\mathcal{D}_{λ,N}(\mathfrak{sl}_2)\}_{N\in\mathbb{N}_0}$ over the complex numbers, such that $\mathcal{D}_{λ,N}(\mathfrak{sl}_2)$ is a subalgebra of $\mathcal{D}_{λ,M}(\mathfrak{sl}_2)$ if $N

Motivation & Objective

• To construct a $\mathbb{C}$-algebra whose representation theory closely mirrors that of simply connected algebraic groups over algebraically closed fields of positive characteristic.
• To generalize the small quantum group $\mathfrak{u}_\lambda(\mathfrak{sl}_2)$ into a nested family of finite-dimensional algebras $\mathcal{D}_{\lambda,N}(\mathfrak{sl}_2)$, indexed by $N \in \mathbb{N}_0$.
• To establish a quantum analog of the algebra of distributions of $\mathrm{SL}_2$, filtered by finite-dimensional subalgebras with cleft extension structure.
• To prove a Steinberg-type tensor product decomposition for simple $\mathcal{D}_{\lambda,N}(\mathfrak{sl}_2)$-modules, reflecting the representation-theoretic structure of algebraic groups.

Proposed method

• Define $\mathcal{D}_{\lambda,N}(\mathfrak{sl}_2)$ via generators and relations that generalize those of finite-dimensional subalgebras of the algebra of distributions of $\mathrm{SL}_2$.
• Establish a filtration: $\mathcal{D}_{\lambda,0}(\mathfrak{sl}_2) \simeq \mathfrak{u}_\lambda(\mathfrak{sl}_2)$, with $\mathcal{D}_{\lambda,M}(\mathfrak{sl}_2) \subset \mathcal{D}_{\lambda,N}(\mathfrak{sl}_2)$ for $M < N$.
• Prove that $\mathcal{D}_{\lambda,N-1}(\mathfrak{sl}_2) \subset \mathcal{D}_{\lambda,N}(\mathfrak{sl}_2)$ is a $\mathfrak{u}_\lambda(\mathfrak{sl}_2)$-cleft extension, ensuring a Hopf algebraic structure.
• Use a triangular decomposition of $\mathcal{D}_{\lambda,N}(\mathfrak{sl}_2)$ into positive, zero, and negative parts to show that simple modules are highest weight modules.
• Construct a $\mathfrak{u}_\lambda(\mathfrak{sl}_2)$-comodule algebra structure on $\mathcal{D}_{\lambda,N}(\mathfrak{sl}_2)$, enabling tensor product actions on modules.
• Leverage the comodule structure to prove that each simple $\mathcal{D}_{\lambda,N}(\mathfrak{sl}_2)$-module $\mathcal{L}_N(p)$ decomposes as $\mathcal{L}(p_N) \otimes \mathcal{L}_N(\widehat{p})$, where $p = p_N \ell^N + \widehat{p}$, $0 \leq p_N < \ell$, $0 \leq \widehat{p} < \ell^N$.

Experimental results

Research questions

• RQ1 Can a $\mathbb{C}$-algebra be constructed whose representation category closely emulates that of a simply connected algebraic group over a field of positive characteristic?
• RQ2 How can the algebra of distributions of $\mathrm{SL}_2$ be quantum-deformed to yield a filtration of finite-dimensional algebras with cleft extension structure?
• RQ3 Does the representation theory of such a quantum algebra admit a Steinberg-type tensor product decomposition for simple modules?
• RQ4 Can the simple modules of this algebra be explicitly constructed and classified via a recursive tensor product involving the small quantum group $\mathfrak{u}_\lambda(\mathfrak{sl}_2)$?

Key findings

• The family $\{\mathcal{D}_{\lambda,N}(\mathfrak{sl}_2)\}_{N \in \mathbb{N}_0}$ forms a nested filtration of finite-dimensional $\mathbb{C}$-algebras with $\mathcal{D}_{\lambda,0}(\mathfrak{sl}_2) \simeq \mathfrak{u}_\lambda(\mathfrak{sl}_2)$.
• Each inclusion $\mathcal{D}_{\lambda,N-1}(\mathfrak{sl}_2) \subset \mathcal{D}_{\lambda,N}(\mathfrak{sl}_2)$ is a $\mathfrak{u}_\lambda(\mathfrak{sl}_2)$-cleft extension, generalizing the Hopf algebra structure of distribution algebras.
• Every simple $\mathcal{D}_{\lambda,N}(\mathfrak{sl}_2)$-module $\mathcal{L}_N(p)$ is a highest weight module with highest weight $p = p_N \ell^N + \widehat{p}$, where $0 \leq p_N < \ell$, $0 \leq \widehat{p} < \ell^N$.
• The simple module $\mathcal{L}_N(p)$ admits a tensor product decomposition $\mathcal{L}_N(p) \simeq \mathcal{L}(p_N) \otimes \mathcal{L}_N(\widehat{p})$ as $\mathcal{D}_{\lambda,N}(\mathfrak{sl}_2)$-modules.
• The trivial module $\Bbbk$ is isomorphic to $\mathcal{L}_N(0)$, and $\mathcal{L}_N(p) \simeq \mathcal{L}(p_N) \otimes \Bbb{k}$ when $p = p_N \ell^N$.
• The algebra $\mathcal{D}_{\lambda,N}(\mathfrak{sl}_2)$ is an augmented algebra with counit $\epsilon$ satisfying $\epsilon(E^{[j]}) = \epsilon(F^{[j]}) = 0$, $\epsilon(K^{[j]}) = 1$ for $0 \leq j \leq N$.