[Paper Review] Categorification of Quantum Generalized Kac-Moody Algebras and Crystal Bases

Seok‐Jin Kang, Se‐jin Oh+1 more|arXiv (Cornell University)|Feb 25, 2011

Algebraic structures and combinatorial models3 citations

TL;DR

This paper constructs Khovanov-Lauda-Rouquier (KLR) algebras and their cyclotomic quotients for quantum generalized Kac-Moody algebras, establishing a categorification via an injective algebra homomorphism from the integral form of the negative part of the quantum generalized Kac-Moody algebra into the Grothendieck group of graded projective modules. It further proves that the crystal structures on the isomorphism classes of irreducible graded modules over these algebras are isomorphic to the classical crystals $ B(ty) $ and $ B(lambda) $, extending categorification to the generalized Kac-Moody setting.

ABSTRACT

We construct and investigate the structure of the Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^\\lambda$ which give a categrification of quantum generalized Kac-Moody algebras. Let $U_\\A(\\g)$ be the integral form of the quantum generalized Kac-Moody algebra associated with a Borcherds-Cartan matrix $A=(a_{ij})_{i,j \\in I}$ and let $K_0(R)$ be the Grothedieck group of finitely generated projective graded $R$-modules. We prove that there exists an injective algebra homomorphism $\\Phi: U_\\A^-(\\g) \ o K_0(R)$ and that $\\Phi$ is an isomorphism if $a_{ii}\ e 0$ for all $i\\in I$. Let $B(\\infty)$ and $B(\\lambda)$ be the crystals of $U_q^-(\\g)$ and $V(\\lambda)$, respectively, where $V(\\lambda)$ is the irreducible highest weight $U_q(\\g)$-module. We denote by $\\mathfrak{B}(\\infty)$ and $\\mathfrak{B}(\\lambda)$ the isomorphism classes of irreducible graded modules over $R$ and $R^\\lambda$, respectively. If $a_{ii}\ e 0$ for all $i\\in I$, we define the $U_q(\\g)$-crystal structures on $\\mathfrak{B}(\\infty)$ and $\\mathfrak{B}(\\lambda)$, and show that there exist crystal isomorphisms $\\mathfrak{B}(\\infty) \\simeq B(\\infty)$ and $\\mathfrak{B}(\\lambda) \\simeq B(\\lambda)$. One of the key ingredients of our approach is the perfect basis theory for generalized Kac-Moody algebras.

Motivation & Objective

• To extend the Khovanov-Lauda-Rouquier categorification framework from symmetrizable Kac-Moody algebras to generalized Kac-Moody algebras.
• To construct KLR algebras $ R $ and their cyclotomic quotients $ R^{lambda} $ for quantum generalized Kac-Moody algebras using Borcherds-Cartan matrices.
• To establish a Grothendieck group isomorphism between $ U_{\mathbb{A}}^{-}(\mathfrak{g}) $ and $ K_0(R) $, and to define crystal structures on the isomorphism classes of irreducible graded modules.
• To prove that the crystal structures on $ \mathfrak{B}(\infty) $ and $ \mathfrak{B}(\blambda) $ are isomorphic to the classical crystals $ B(\infty) $ and $ B(\blambda) $, respectively.

Proposed method

• Define the KLR algebra $ R $ via generators and relations, incorporating homogeneous polynomials $ \mathcal{P}_i(u,v) $ of degree $ 1 - \frac{a_{ii}}{2} $ as twisting factors for commutation and braid relations.
• Introduce a diagrammatic presentation of $ R $ to facilitate structural analysis and computation.
• Construct the cyclotomic quotient $ R^{\blambda} $ for each dominant integral weight $ \blambda $, modeling the irreducible highest weight module $ V(\blambda) $.
• Define a Grothendieck group $ K_0(R) $ of finitely generated graded projective $ R $-modules and establish an injective algebra homomorphism $ \Phi: U_{\mathbb{A}}^{-}(\mathfrak{g}) \to K_0(R) $.
• Define crystal operators $ \tilde{e}_i^\lambda $, $ \tilde{f}_i^\lambda $ on the set of isomorphism classes of irreducible $ R^\lambda $-modules, and show they satisfy the crystal axioms.
• Construct a strict crystal embedding $ \Psi_\lambda: \mathfrak{B}(\lambda) \to \mathfrak{B}(\infty) \otimes T_\lambda \otimes C $, and prove its image is isomorphic to $ B(\lambda) $.

Experimental results

Research questions

• RQ1 Can the Khovanov-Lauda-Rouquier categorification framework be extended to quantum generalized Kac-Moody algebras?
• RQ2 Does the Grothendieck group $ K_0(R) $ of the KLR algebra $ R $ categorify the negative part $ U_{\mathbb{A}}^{-}(\mathfrak{g}) $ of the quantum generalized Kac-Moody algebra?
• RQ3 Do the isomorphism classes of irreducible graded $ R $-modules and $ R^\lambda $-modules carry crystal structures isomorphic to the classical crystals $ B(\infty) $ and $ B(\lambda) $?
• RQ4 How do the twisting factors $ \mathcal{P}_i(u,v) $, depending on $ a_{ii} $, affect the algebraic structure and categorification in the generalized Kac-Moody case?

Key findings

• There exists an injective algebra homomorphism $ \Phi: U_{\mathbb{A}}^{-}(\mathfrak{g}) \to K_0(R) $, which becomes an isomorphism if $ a_{ii} \neq 0 $ for all $ i \in I $.
• The crystal structure on the set $ \mathfrak{B}(\infty) $ of isomorphism classes of irreducible graded $ R $-modules is isomorphic to the classical crystal $ B(\infty) $.
• The crystal structure on the set $ \mathfrak{B}(\lambda) $ of isomorphism classes of irreducible graded $ R^\lambda $-modules is isomorphic to the classical crystal $ B(\lambda) $.
• The map $ \Psi_\lambda: \mathfrak{B}(\lambda) \to \mathfrak{B}(\infty) \otimes T_\lambda \otimes C $ is a strict crystal embedding, and $ \mathfrak{B}(\lambda) \simeq B(\lambda) $ as crystals.
• The construction generalizes the KLR categorification to generalized Kac-Moody algebras by incorporating non-constant twisting factors $ \mathcal{P}_i(u,v) $ for imaginary roots with $ a_{ii} \leq 0 $.
• The proof relies on the perfect basis theory for generalized Kac-Moody algebras and the structure of induced modules over $ R^\lambda $, particularly the properness of certain submodules in the induction process.