The R-matrix of the affine Yangian
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915939641982976 |
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| author | Appel, Andrea Gautam, Sachin Wendlandt, Curtis |
| author_facet | Appel, Andrea Gautam, Sachin Wendlandt, Curtis |
| contents | Let g be an affine Lie algebra with associated Yangian Y_hg. We prove the existence of two meromorphic R-matrices associated to any pair of representations of Y_hg in the category O. They are related by a unitary constraint and constructed as products of the form R(s)=R^+(s)R^0(s)R^-(s), where R^+(s) = R^-_{21}(-s)^{-1}. The factor R^0(s) is a meromorphic, abelian R-matrix, and R^-(s) is a rational twist. Our proof relies on two novel ingredients. The first is an irregular, abelian, additive difference equation whose difference operator is given in terms of the q-Cartan matrix of g. The regularization of this difference equation gives rise to R^0(s) as the exponentials of the two canonical fundamental solutions. The second key ingredient is a higher order analogue of the adjoint action of the affine Cartan subalgebra of g on Y_hg. This action has no classical counterpart, and produces a system of linear equations from which R^-(s) is recovered as the unique solution. Moreover, we show that both operators give rise to the same rational R-matrix on the tensor product of any two highest-weight representations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_02377 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The R-matrix of the affine Yangian Appel, Andrea Gautam, Sachin Wendlandt, Curtis Representation Theory Mathematical Physics Quantum Algebra Primary: 81R10. Secondary: 17B37, 39A45 Let g be an affine Lie algebra with associated Yangian Y_hg. We prove the existence of two meromorphic R-matrices associated to any pair of representations of Y_hg in the category O. They are related by a unitary constraint and constructed as products of the form R(s)=R^+(s)R^0(s)R^-(s), where R^+(s) = R^-_{21}(-s)^{-1}. The factor R^0(s) is a meromorphic, abelian R-matrix, and R^-(s) is a rational twist. Our proof relies on two novel ingredients. The first is an irregular, abelian, additive difference equation whose difference operator is given in terms of the q-Cartan matrix of g. The regularization of this difference equation gives rise to R^0(s) as the exponentials of the two canonical fundamental solutions. The second key ingredient is a higher order analogue of the adjoint action of the affine Cartan subalgebra of g on Y_hg. This action has no classical counterpart, and produces a system of linear equations from which R^-(s) is recovered as the unique solution. Moreover, we show that both operators give rise to the same rational R-matrix on the tensor product of any two highest-weight representations. |
| title | The R-matrix of the affine Yangian |
| topic | Representation Theory Mathematical Physics Quantum Algebra Primary: 81R10. Secondary: 17B37, 39A45 |
| url | https://arxiv.org/abs/2309.02377 |