Hall algebra of morphism category

QingHua Chen; Liwang Zhang

Czechoslovak Mathematical Journal (2024)

  • Volume: 74, Issue: 4, page 1145-1164
  • ISSN: 0011-4642

Abstract

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This paper investigates a universal PBW-basis and a minimal set of generators for the Hall algebra ℋ ( C 2 ( 𝒫 ) ) , where C 2 ( 𝒫 ) is the category of morphisms between projective objects in a finitary hereditary exact category 𝒜 . When 𝒜 is the representation category of a Dynkin quiver, we develop multiplication formulas for the degenerate Hall Lie algebra ℒ , which is spanned by isoclasses of indecomposable objects in C 2 ( 𝒫 ) . As applications, we demonstrate that 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 𝒜 as a Lie subquotient algebra of ℒ .

How to cite

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Chen, QingHua, and Zhang, Liwang. "Hall algebra of morphism category." Czechoslovak Mathematical Journal 74.4 (2024): 1145-1164. <http://eudml.org/doc/299607>.

@article{Chen2024,
abstract = {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 $\mathcal \{L\}$.},
author = {Chen, QingHua, Zhang, Liwang},
journal = {Czechoslovak Mathematical Journal},
keywords = {Hall algebra; morphism category; Heisenberg Lie algebra; simple Lie algebra},
language = {eng},
number = {4},
pages = {1145-1164},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Hall algebra of morphism category},
url = {http://eudml.org/doc/299607},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Chen, QingHua
AU - Zhang, Liwang
TI - Hall algebra of morphism category
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 4
SP - 1145
EP - 1164
AB - 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 $\mathcal {L}$.
LA - eng
KW - Hall algebra; morphism category; Heisenberg Lie algebra; simple Lie algebra
UR - http://eudml.org/doc/299607
ER -

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