Quantum affine vertex algebra at root of unity
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914448775577600 |
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| author | Kong, Fei |
| author_facet | Kong, Fei |
| contents | Let $\mathfrak g$ be a finite simple Lie algebra, and let $r$ denote the ratio of the square length of long roots to that of short roots. Let $\wp>2r$ be an integer and $ζ$ a primitive $\wp$-th root of unity. Denote by $\mathcal U_ζ(\widehat{\mathfrak g})$ the Lusztig big quantum affine algebra at root of unity defined by divided powers. In this paper, we establish a current algebra presentation of $\mathcal U_ζ(\widehat{\mathfrak g})$. Based on this presentation, we construct a $\mathbb Z_\wp$-module quantum vertex algebras $V_{\wp,τ}^\ell(\mathfrak g)$ for each integer $\ell$. Moreover, we establish a fully faithful functor from the category of smooth weighted $\mathcal U_ζ(\widehat{\mathfrak g})$-modules of level $\ell$ to the category of $(\mathbb Z_\wp,χ_ϕ)$-equivariant $ϕ$-coordinated quasi-modules of $V_{\wp,τ}^\ell(\mathfrak g)$, where $χ_ϕ:\mathbb Z_\wp\to\mathbb C^\times$ is the group homomorphism defined by $s\mapsto ζ^s$. We also determine the image of this functor. The structure $V_{\wp,τ}^\ell(\mathfrak g)$ is substantially different from that of affine vertex algebras. We realize $V_{\wp,τ}^\ell(\mathfrak g)$ as a deformation of a simpler quantum vertex algebra $V_{\wp,\varepsilon}^\ell(\mathfrak g)$ by using vertex bialgebras, and decompose $V_{\wp,\varepsilon}^\ell(\mathfrak g)$ into a Heisenberg vertex algebra and a more interesting quantum vertex algebra determined by a quiver. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_04666 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantum affine vertex algebra at root of unity Kong, Fei Quantum Algebra Mathematical Physics Let $\mathfrak g$ be a finite simple Lie algebra, and let $r$ denote the ratio of the square length of long roots to that of short roots. Let $\wp>2r$ be an integer and $ζ$ a primitive $\wp$-th root of unity. Denote by $\mathcal U_ζ(\widehat{\mathfrak g})$ the Lusztig big quantum affine algebra at root of unity defined by divided powers. In this paper, we establish a current algebra presentation of $\mathcal U_ζ(\widehat{\mathfrak g})$. Based on this presentation, we construct a $\mathbb Z_\wp$-module quantum vertex algebras $V_{\wp,τ}^\ell(\mathfrak g)$ for each integer $\ell$. Moreover, we establish a fully faithful functor from the category of smooth weighted $\mathcal U_ζ(\widehat{\mathfrak g})$-modules of level $\ell$ to the category of $(\mathbb Z_\wp,χ_ϕ)$-equivariant $ϕ$-coordinated quasi-modules of $V_{\wp,τ}^\ell(\mathfrak g)$, where $χ_ϕ:\mathbb Z_\wp\to\mathbb C^\times$ is the group homomorphism defined by $s\mapsto ζ^s$. We also determine the image of this functor. The structure $V_{\wp,τ}^\ell(\mathfrak g)$ is substantially different from that of affine vertex algebras. We realize $V_{\wp,τ}^\ell(\mathfrak g)$ as a deformation of a simpler quantum vertex algebra $V_{\wp,\varepsilon}^\ell(\mathfrak g)$ by using vertex bialgebras, and decompose $V_{\wp,\varepsilon}^\ell(\mathfrak g)$ into a Heisenberg vertex algebra and a more interesting quantum vertex algebra determined by a quiver. |
| title | Quantum affine vertex algebra at root of unity |
| topic | Quantum Algebra Mathematical Physics |
| url | https://arxiv.org/abs/2604.04666 |