Cyclic Representations of $U_q(\hat{\mathfrak{sl}}_2)$ and its Borel Subalgebras at Roots of Unity and Q-operators
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| Format: | Preprint |
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2024
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| _version_ | 1866914400572538880 |
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| author | Weston, Robert |
| author_facet | Weston, Robert |
| contents | We consider the cyclic representations $Ω_{rs}$ of $ U_q(\widehat{\mathfrak{sl}}_2)$ at $q^N=1$ that depend upon two points $r,s$ in the chiral Potts algebraic curve. We show how $Ω_{rs}$ is related to the tensor product $ρ_r\otimes \barρ_s$ of two representations of the upper Borel subalgebra of $U_q(\widehat{\mathfrak{sl}}_2)$. This result is analogous to the factorization property of the Verma module of $U_q(\widehat{\mathfrak{sl}}_2)$ at generic-$q$ in terms of two q-oscillator representation of the Borel subalgebra - a key step in the construction of the Q-operator. We construct short exact sequences of the different representations and use the results to construct Q operators that satisfy TQ relations for $q^N=1$ for both the 6-vertex and $τ_2$ models. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_14811 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cyclic Representations of $U_q(\hat{\mathfrak{sl}}_2)$ and its Borel Subalgebras at Roots of Unity and Q-operators Weston, Robert Mathematical Physics Quantum Algebra 81R10, 81R12, 81R50 (Primary) 82B23, 16T25 (Secondary) We consider the cyclic representations $Ω_{rs}$ of $ U_q(\widehat{\mathfrak{sl}}_2)$ at $q^N=1$ that depend upon two points $r,s$ in the chiral Potts algebraic curve. We show how $Ω_{rs}$ is related to the tensor product $ρ_r\otimes \barρ_s$ of two representations of the upper Borel subalgebra of $U_q(\widehat{\mathfrak{sl}}_2)$. This result is analogous to the factorization property of the Verma module of $U_q(\widehat{\mathfrak{sl}}_2)$ at generic-$q$ in terms of two q-oscillator representation of the Borel subalgebra - a key step in the construction of the Q-operator. We construct short exact sequences of the different representations and use the results to construct Q operators that satisfy TQ relations for $q^N=1$ for both the 6-vertex and $τ_2$ models. |
| title | Cyclic Representations of $U_q(\hat{\mathfrak{sl}}_2)$ and its Borel Subalgebras at Roots of Unity and Q-operators |
| topic | Mathematical Physics Quantum Algebra 81R10, 81R12, 81R50 (Primary) 82B23, 16T25 (Secondary) |
| url | https://arxiv.org/abs/2412.14811 |