Cubic Dirac operator for $U_q(\mathfrak{sl}_2)$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866929686797352960 |
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| author | Krutov, Andrey Pandžić, Pavle |
| author_facet | Krutov, Andrey Pandžić, Pavle |
| contents | We construct the $q$-deformed Clifford algebra of $\mathfrak{sl}_2$ and study its properties. This allows us to define the $q$-deformed noncommutative Weil algebra $\mathcal{W}_q(\mathfrak{sl}_2)$ for $U_q(\mathfrak{sl}_2)$ and the corresponding cubic Dirac operator $D_q$. In the classical case it was done by Alekseev and Meinrenken. We show that the cubic Dirac operator $D_q$ is invariant with respect to the $U_q(\mathfrak{sl}_2)$-action and *-structures on $\mathcal{W}_q(\mathfrak{sl}_2)$, moreover, the square of $D_q$ is central in $\mathcal{W}_q(\mathfrak{sl}_2)$. We compute the spectrum of the cubic element on finite-dimensional and Verma modules of~$U_q(\mathfrak{sl}_2)$ and the corresponding Dirac cohomology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_09591 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Cubic Dirac operator for $U_q(\mathfrak{sl}_2)$ Krutov, Andrey Pandžić, Pavle Representation Theory Mathematical Physics Quantum Algebra 16T20, 81R50, 20G42 We construct the $q$-deformed Clifford algebra of $\mathfrak{sl}_2$ and study its properties. This allows us to define the $q$-deformed noncommutative Weil algebra $\mathcal{W}_q(\mathfrak{sl}_2)$ for $U_q(\mathfrak{sl}_2)$ and the corresponding cubic Dirac operator $D_q$. In the classical case it was done by Alekseev and Meinrenken. We show that the cubic Dirac operator $D_q$ is invariant with respect to the $U_q(\mathfrak{sl}_2)$-action and *-structures on $\mathcal{W}_q(\mathfrak{sl}_2)$, moreover, the square of $D_q$ is central in $\mathcal{W}_q(\mathfrak{sl}_2)$. We compute the spectrum of the cubic element on finite-dimensional and Verma modules of~$U_q(\mathfrak{sl}_2)$ and the corresponding Dirac cohomology. |
| title | Cubic Dirac operator for $U_q(\mathfrak{sl}_2)$ |
| topic | Representation Theory Mathematical Physics Quantum Algebra 16T20, 81R50, 20G42 |
| url | https://arxiv.org/abs/2209.09591 |