Deformation and quantization of the Loday-Quillen-Tsygan isomorphism for Calabi-Yau categories
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908363265146880 |
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| author | Chen, Xiaojun Eshmatov, Farkhod Huang, Maozhou |
| author_facet | Chen, Xiaojun Eshmatov, Farkhod Huang, Maozhou |
| contents | For an associative algebra $A$, the famous theorem of Loday, Quillen and Tsygan says that there is an isomorphism between the graded symmetric product of the cyclic homology of $A$ and the Lie algebra homology of the infinite matrices $\mathfrak{gl}(A)$, as commutative and cocommutative Hopf algebras. This paper aims to study a deformation and quantization of this isomorphism. We show that if $A$ is a Koszul Calabi-Yau algebra, then the primitive part of the Lie algebra homology $\mathrm{H}_\bullet (\mathfrak{gl}(A))$ has a Lie bialgebra structure which is induced from the Poincaré duality of $A$ and deforms $\mathrm{H}_\bullet (\mathfrak{gl}(A))$ to a co-Poisson bialgebra. Moreover, there is a Hopf algebra which quantizes such a co-Poisson bialgebra, and the Loday-Quillen-Tsygan isomorphism lifts to the quantum level, which can be interpreted as a quantization of the tangent map from the tangent complex of $\mathrm{BGL}$ to the tangent complex of K-theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09378 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Deformation and quantization of the Loday-Quillen-Tsygan isomorphism for Calabi-Yau categories Chen, Xiaojun Eshmatov, Farkhod Huang, Maozhou Rings and Algebras 16E40, 19D55 For an associative algebra $A$, the famous theorem of Loday, Quillen and Tsygan says that there is an isomorphism between the graded symmetric product of the cyclic homology of $A$ and the Lie algebra homology of the infinite matrices $\mathfrak{gl}(A)$, as commutative and cocommutative Hopf algebras. This paper aims to study a deformation and quantization of this isomorphism. We show that if $A$ is a Koszul Calabi-Yau algebra, then the primitive part of the Lie algebra homology $\mathrm{H}_\bullet (\mathfrak{gl}(A))$ has a Lie bialgebra structure which is induced from the Poincaré duality of $A$ and deforms $\mathrm{H}_\bullet (\mathfrak{gl}(A))$ to a co-Poisson bialgebra. Moreover, there is a Hopf algebra which quantizes such a co-Poisson bialgebra, and the Loday-Quillen-Tsygan isomorphism lifts to the quantum level, which can be interpreted as a quantization of the tangent map from the tangent complex of $\mathrm{BGL}$ to the tangent complex of K-theory. |
| title | Deformation and quantization of the Loday-Quillen-Tsygan isomorphism for Calabi-Yau categories |
| topic | Rings and Algebras 16E40, 19D55 |
| url | https://arxiv.org/abs/2505.09378 |