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Minimal Generators of Hall Algebras of 1-cyclic Perfect Complexes
- Source :
- International Mathematics Research Notices. 2021:402-425
- Publication Year :
- 2019
- Publisher :
- Oxford University Press (OUP), 2019.
-
Abstract
- Let $A$ be the path algebra of a Dynkin quiver over a finite field, and let $C_1(\mathscr{P})$ be the category of 1-cyclic complexes of projective $A$-modules. In the present paper, we give a PBW-basis and a minimal set of generators for the Hall algebra $\H(C_1(\mathscr{P}))$ of $C_1(\mathscr{P})$. Using this PBW-basis, we firstly prove the degenerate Hall algebra of $C_1(\mathscr{P})$ is the universal enveloping algebra of the Lie algebra spanned by all indecomposable objects. Secondly, we calculate the relations in the generators in $\H(C_1(\mathscr{P}))$, and obtain quantum Serre relations in a quotient of certain twisted version of $\H(C_1(\mathscr{P}))$. Moreover, we establish relations between the degenerate Hall algebra, twisted Hall algebra of $A$ and those of $C_1(\mathscr{P})$, respectively.<br />Comment: 20 pages
- Subjects :
- Pure mathematics
General Mathematics
010102 general mathematics
Degenerate energy levels
Quiver
Mathematics::General Topology
Universal enveloping algebra
Mathematics - Rings and Algebras
01 natural sciences
Finite field
Hall algebra
Rings and Algebras (math.RA)
Mathematics - Quantum Algebra
0103 physical sciences
Lie algebra
FOS: Mathematics
Quantum Algebra (math.QA)
010307 mathematical physics
Representation Theory (math.RT)
0101 mathematics
Mathematics::Representation Theory
Indecomposable module
Mathematics - Representation Theory
Quotient
Mathematics
Subjects
Details
- ISSN :
- 16870247 and 10737928
- Volume :
- 2021
- Database :
- OpenAIRE
- Journal :
- International Mathematics Research Notices
- Accession number :
- edsair.doi.dedup.....befad73d80c38a0bb4e199b61d7c1a7d