Hall algebra of morphism category

Chen, QingHua ; Zhang, Liwang

Czechoslovak Mathematical Journal, Tome 74 (2024) no. 4, pp. 1145-1164

Abstract (VO)

This paper investigates a universal PBW-basis and a minimal set of generators for the Hall algebra $\mathcal {H}(C_2(\mathcal {P}))$, where $C_2(\mathcal {P})$ is the category of morphisms between projective objects in a finitary hereditary exact category $\mathcal A$. When $\mathcal A$ is the representation category of a Dynkin quiver, we develop multiplication formulas for the degenerate Hall Lie algebra $\mathcal {L}$, which is spanned by isoclasses of indecomposable objects in $C_2(\mathcal {P})$. As applications, we demonstrate that $\mathcal {L}$ contains a Lie subalgebra isomorphic to the central extension of the Heisenberg Lie algebra and construct the Borel subalgebra of the simple Lie algebra associated with $\mathcal A$ as a Lie subquotient algebra of $\mathcal {L}$.

Détail

DOI : 10.21136/CMJ.2024.0103-24

Classification : 16G20, 17B20, 17B30, 18G05
Keywords: Hall algebra; morphism category; Heisenberg Lie algebra; simple Lie algebra

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