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Quantum groups of Borcherds-Cartan type and Khovanov-Lauda-Rouquier algebras

Stefano V. Kang, Young Rock Kim, Bolun Tong

TL;DR

The paper develops a framework for quantum groups of Borcherds-Cartan type arising from quivers with loops and provides a categorification via Khovanov-Lauda-Rouquier algebras $R$. It proves that the indecomposable projective modules over $R$ correspond to Lusztig’s canonical basis of $U^-$, and in the Jordan quiver case shows that cyclotomic KLR algebras $R^ ext{\Lambda}$ categorify the irreducible highest-weight module $V(\Lambda)$. The authors connect the algebraic structure to a geometric realization using Steinberg-type algebras and Springer theory, establishing an isomorphism $_{\mathcal{A}}U^- \cong K_0(R)$ and identifying canonical bases with self-dual indecomposable projectives. In the Jordan case they further relate $K_0(R)$ to Specht modules and Schur functions, confirming the Springer correspondence yields a concrete categorification of highest-weight representations, including the cyclotomic cases. Overall, the work extends KLR categorification to quivers with loops, providing both algebraic and geometric perspectives and highlighting the Jordan quiver as a tractable, illuminating example of the general theory.

Abstract

We study a class of quantum groups $U^-$ associated with quivers with loops and provide a categorification by constructing their associated Khovanov-Lauda-Rouquier algebras $R$. We prove that the indecomposable projective module over $R$ corresponds to the canonical basis of $U^-$. In the Jordan quiver case, we show that the cyclotomic Khovanov-Lauda-Rouquier algebras $R^Λ$ categorify the irreducible highest weight $U$-module $V(Λ)$.

Quantum groups of Borcherds-Cartan type and Khovanov-Lauda-Rouquier algebras

TL;DR

The paper develops a framework for quantum groups of Borcherds-Cartan type arising from quivers with loops and provides a categorification via Khovanov-Lauda-Rouquier algebras . It proves that the indecomposable projective modules over correspond to Lusztig’s canonical basis of , and in the Jordan quiver case shows that cyclotomic KLR algebras categorify the irreducible highest-weight module . The authors connect the algebraic structure to a geometric realization using Steinberg-type algebras and Springer theory, establishing an isomorphism and identifying canonical bases with self-dual indecomposable projectives. In the Jordan case they further relate to Specht modules and Schur functions, confirming the Springer correspondence yields a concrete categorification of highest-weight representations, including the cyclotomic cases. Overall, the work extends KLR categorification to quivers with loops, providing both algebraic and geometric perspectives and highlighting the Jordan quiver as a tractable, illuminating example of the general theory.

Abstract

We study a class of quantum groups associated with quivers with loops and provide a categorification by constructing their associated Khovanov-Lauda-Rouquier algebras . We prove that the indecomposable projective module over corresponds to the canonical basis of . In the Jordan quiver case, we show that the cyclotomic Khovanov-Lauda-Rouquier algebras categorify the irreducible highest weight -module .
Paper Structure (16 sections, 19 theorems, 127 equations)

This paper contains 16 sections, 19 theorems, 127 equations.

Table of Contents

  1. A quantum group of Borcherds-Cartan type
  2. Borcherds-Cartan matrices and $U^-$
  3. Geometric setting for $U^-$
  4. Categorification of ${U}^-$
  5. $\mathbb{Z}$-gradings
  6. KLR-algebras $R(\nu)$
  7. Algebras $R(ni)$ and their gr-irreducible modules
  8. Grothendieck groups $K_0(R)$ and $G_0(R)$
  9. Quantum Serre relations
  10. Surjectivity of $\Gamma_{\mathbb{Q}(q)}$ and $\Gamma$
  11. Geometric realization of $R$
  12. Jordan quiver case
  13. Proof of Theorem \ref{['canonical']}
  14. Categorification of irreducible highest weight module in Jordan quiver case
  15. The algebra $U$
  16. ...and 1 more sections

Key Result

Proposition 2.1

If $V_1,\dots,V_{|\mathcal{P}_n|}$ is a complete set of non-isomorphic classes of irreducible $\mathbb{C} S_n$-modules, then $V_1,\dots,V_{|\mathcal{P}_n|}$ is a complete set of non-isomorphic classes of gr-irreducible $R(ni)$-modules. In particular, the gr-Jacobson radical $J^{\text{gr}}(R(ni))=R(n

Theorems & Definitions (31)

  • Definition 1.1
  • Remark 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • Example 2.5
  • Lemma 2.6
  • Theorem 2.7
  • ...and 21 more