Lie algebras arising from two-periodic projective complex and derived categories
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866909323553144832 |
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| author | Fang, Jiepeng Lan, Yixin Xiao, Jie |
| author_facet | Fang, Jiepeng Lan, Yixin Xiao, Jie |
| contents | Let $A$ be a finite-dimensional $\mathbb{C}$-algebra of finite global dimension and $\mathcal{A}$ be the category of finitely generated right $A$-modules. By using of the category of two-periodic projective complexes $\mathcal{C}_2(\mathcal{P})$, we construct the motivic Bridgeland's Hall algebra for $\mathcal{A}$, where structure constants are given by Poincaré polynomials in $t$, then construct a $\mathbb{C}$-Lie subalgebra $\mathfrak{g}=\mathfrak{n}\oplus \mathfrak{h}$ at $t=-1$, where $\mathfrak{n}$ is constructed by stack functions about indecomposable radical complexes, and $\mathfrak{h}$ is by contractible complexes. For the stable category $\mathcal{K}_2(\mathcal{P})$ of $\mathcal{C}_2(\mathcal{P})$, we construct its moduli spaces and a $\mathbb{C}$-Lie algebra $\tilde{\mathfrak{g}}=\tilde{\mathfrak{n}}\oplus \tilde{\mathfrak{h}}$, where $\tilde{\mathfrak{n}}$ is constructed by support-indecomposable constructible functions, and $\tilde{\mathfrak{h}}$ is by the Grothendieck group of $\mathcal{K}_2(\mathcal{P})$. We prove that the natural functor $\mathcal{C}_2(\mathcal{P})\rightarrow \mathcal{K}_2(\mathcal{P})$ together with the natural isomorphism between Grothendieck groups of $\mathcal{A}$ and $\mathcal{K}_2(\mathcal{P})$ induces a Lie algebra isomorphism $\mathfrak{g}\cong\tilde{\mathfrak{g}}$. This makes clear that the structure constants at $t=-1$ provided by Bridgeland in [5] in terms of exact structure of $\mathcal{C}_2(\mathcal{P})$ precisely equal to that given in [30] in terms of triangulated category structure of $\mathcal{K}_2(\mathcal{P})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_06664 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lie algebras arising from two-periodic projective complex and derived categories Fang, Jiepeng Lan, Yixin Xiao, Jie Representation Theory Quantum Algebra Rings and Algebras 16G20, 17B37, 17B65, 18G80 Let $A$ be a finite-dimensional $\mathbb{C}$-algebra of finite global dimension and $\mathcal{A}$ be the category of finitely generated right $A$-modules. By using of the category of two-periodic projective complexes $\mathcal{C}_2(\mathcal{P})$, we construct the motivic Bridgeland's Hall algebra for $\mathcal{A}$, where structure constants are given by Poincaré polynomials in $t$, then construct a $\mathbb{C}$-Lie subalgebra $\mathfrak{g}=\mathfrak{n}\oplus \mathfrak{h}$ at $t=-1$, where $\mathfrak{n}$ is constructed by stack functions about indecomposable radical complexes, and $\mathfrak{h}$ is by contractible complexes. For the stable category $\mathcal{K}_2(\mathcal{P})$ of $\mathcal{C}_2(\mathcal{P})$, we construct its moduli spaces and a $\mathbb{C}$-Lie algebra $\tilde{\mathfrak{g}}=\tilde{\mathfrak{n}}\oplus \tilde{\mathfrak{h}}$, where $\tilde{\mathfrak{n}}$ is constructed by support-indecomposable constructible functions, and $\tilde{\mathfrak{h}}$ is by the Grothendieck group of $\mathcal{K}_2(\mathcal{P})$. We prove that the natural functor $\mathcal{C}_2(\mathcal{P})\rightarrow \mathcal{K}_2(\mathcal{P})$ together with the natural isomorphism between Grothendieck groups of $\mathcal{A}$ and $\mathcal{K}_2(\mathcal{P})$ induces a Lie algebra isomorphism $\mathfrak{g}\cong\tilde{\mathfrak{g}}$. This makes clear that the structure constants at $t=-1$ provided by Bridgeland in [5] in terms of exact structure of $\mathcal{C}_2(\mathcal{P})$ precisely equal to that given in [30] in terms of triangulated category structure of $\mathcal{K}_2(\mathcal{P})$. |
| title | Lie algebras arising from two-periodic projective complex and derived categories |
| topic | Representation Theory Quantum Algebra Rings and Algebras 16G20, 17B37, 17B65, 18G80 |
| url | https://arxiv.org/abs/2305.06664 |