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On the h-adic Quantum Vertex Algebras Associated with Hecke Symmetries

Abstract

We study the quantum vertex algebraic framework for the Yangians of RTT-type and the braided Yangians associated with Hecke symmetries, introduced by Gurevich and Saponov. First, we construct several families of modules for the aforementioned Yangian-like algebras which, in the RTT-type case, lead to a certain h-adic quantum vertex algebra \({\mathcal {V}}_c (R)\) via the Etingof–Kazhdan construction, while, in the braided case, they produce (\(\phi \)-coordinated) \({\mathcal {V}}_c (R)\)-modules. Next, we show that the coefficients of suitably defined quantum determinant can be used to obtain central elements of \({\mathcal {V}}_c (R)\), as well as the invariants of such (\(\phi \)-coordinated) \({\mathcal {V}}_c (R)\)-modules. Finally, we investigate a certain algebra which is closely connected with the representation theory of \({\mathcal {V}}_c (R)\).

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Notes

  1. Equivalently, \(\mathop {\underset{\text {LR}}{\cdot }}\) is the standard multiplication in the algebra \({\mathrm {End}}\,{\mathbb {C}}^N \otimes ({\mathrm {End}}\,{\mathbb {C}}^N )^{op}\) and \(\mathop {\underset{\text {RL}}{\cdot }}\) is the product in \(({\mathrm {End}}\,{\mathbb {C}}^N)^{op}\otimes {\mathrm {End}}\,{\mathbb {C}}^N \), where \(A^{op}\) denotes the opposite algebra of A.

  2. Regarding the normalization, the term f(u) in (2.14), as well as \(\overline{f}(x)\) in (2.24) below, is used as it leads to nice properties of quantum determinants, as we demonstrate in Sects. 6.2 and 6.3 below.

  3. At this point, it is worth it to recall that, in contrast, the h-adic quantum vertex algebra structure and the corresponding module structure from Theorem 4.7 are both given in terms of the same R-matrix (2.14) with additive spectral parameter .

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Acknowledgements

The author would like to thank the anonymous referees for careful reading and many valuable comments and suggestions which helped him to improve the manuscript. This work has been supported in part by Croatian Science Foundation under the project UIP-2019-04-8488.

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Kožić, S. On the h-adic Quantum Vertex Algebras Associated with Hecke Symmetries. Commun. Math. Phys. 397, 607–634 (2023). https://doi.org/10.1007/s00220-022-04498-4

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