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Bolun Tong

Publications and source records attributed to Bolun Tong.

8 recordsLinked to original sources
arXiv2024

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

We categorify a class of quantum groups associated with quivers, possibly with loops, by constructing the corresponding Khovanov-Lauda-Rouquier algebras (KLR) algebras $R$. We prove that the indecomposable projective $R$-modules realize the canonical basis of the negative part $U^-$ of the quantum group. Moreover, for $\Lambda \in P^+$, the cyclotomic KLR algebra $R^\Lambda$ provide a categorification of the irreducible highest weight $U$-module $V(\Lambda)$.

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arXiv2024

Quiver Hecke algebras for Borcherds-Cartan datum II

We give the crystal structure of the Grothendieck group $G_0(R)$ of irreducible modules over the quiver Hecke algebra $R$ constructed in \cite{TW2023}. This leads to the categorification of the crystal $B(\infty)$ of the quantum Borcherds algebra $U_q(\mathscr g)$ and its irreducible highest weight crystal $B(λ)$ for arbitrary Borcherds-Cartan data. Additionally, we study the cyclotomic categorification of irreducible highest weight $U_q(\mathscr g)$-modules.

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arXiv2021

Braid group actions of quantum Borcherds-Bozec algebras

In this paper, we construct the Lusztig symmetries for quantum Borcherds-Bozec algebra $U_q(\mathscr g)$ and its weight module $M\in \mathcal O$, on which the generators with real indices of $U_q(\mathscr g)$ act nilpotently. We show that these symmetries satisfy the defining relations of the braid group, associated to the Weyl group $W$ of $U_q(\mathscr g)$, which gives a braid group action.

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arXiv2020

Abstract crystals for quantum Borcherds-Bozec algebras

In this paper, we develop the theory of abstract crystals for quantum Borcherds-Bozec algebras. Our construction is different from the one given by Bozec. We further prove the crystal embedding theorem and provide a characterization of ${B}(\infty)$ and ${B}(λ)$ as its application, where ${B}(\infty)$ and ${B}(λ)$ are the crystals of the negative half part of the quantum Borcherds-Bozec algebra $U_q(\mathfrak g)$ and its irreducible highest weight module $V(λ)$, respectively.

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arXiv2020

Classical limit of quantum Borcherds-Bozec algebras

Let $\mathfrak{g}$ be a Borcherds-Bozec algebra, $U(\mathfrak{g})$ be its universal enveloping algebra and $U_{q}(\mathfrak{g})$ be the corresponding quantum Borcherds-Bozec algebra. We show that the classical limit of $U_{q}(\mathfrak{g})$ is isomorphic to $U(\mathfrak{g})$ as Hopf algebras. Thus $U_{q}(\mathfrak{g})$ can be regarded as a quantum deformation of $U(\mathfrak{g})$. We also give explicit formulas for the commutation relations among the generators of $U_{q}(\mathfrak{g})$.

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