Skip to main content
SpringerLink
Log in

Odd reflections in the Yangian associated with \(\mathfrak {gl}(m|n)\)

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

The odd reflections are an effective tool in the Lie superalgebra representation theory, as they relate non-conjugate Borel subalgebras. We introduce analogues of the odd reflections for the Yangian \( \mathrm{Y}(\mathfrak {gl}_{m|n})\) and use them to produce a transition rule for the parameters of the highest weight modules corresponding to a change of the parity sequence. This leads to a description of the finite-dimensional irreducible representations of the Yangian associated with an arbitrary parity sequence.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Arnaudon, D., Avan, J., Crampé, N., Frappat, L., Ragoucy, E.: \(R\)-matrix presentation for super-Yangians \(Y({\rm osp}(m\vert 2n))\). J. Math. Phys. 44, 302–308 (2003)

    Article  MathSciNet  Google Scholar 

  2. Brown, J., Brundan, J., Goodwin, S.M.: Principal \(W\)-algebras for \(GL(m|n)\). Algebr. Number Theory 7, 1849–1882 (2013)

    Article  MathSciNet  Google Scholar 

  3. Brundan, J., Kleshchev, A.: Parabolic presentations of the Yangian \({\rm Y}({\mathfrak{gl}}_n)\). Commun. Math. Phys. 254, 191–220 (2005)

    Article  Google Scholar 

  4. Cheng, S.-J., Wang, W.: Dualities and Representations of Lie Superalgebras, Graduate Studies in Mathematics, vol. 144. AMS, Providence (2012)

    Book  Google Scholar 

  5. Drinfeld, V.G.: A new realization of Yangians and quantized affine algebras. Sov. Math. Dokl. 36, 212–216 (1988)

    Google Scholar 

  6. Gow, L.: On the Yangian \({{\rm Y}}({\mathfrak{gl}}_{m|n})\) and its quantum Berezinian. Czech J. Phys. 55, 1415–1420 (2005)

    Article  Google Scholar 

  7. Gow, L.: Gauss decomposition of the Yangian \({\rm Y}({\mathfrak{gl}}_{m|n})\). Commun. Math. Phys. 276, 799–825 (2007)

    Article  Google Scholar 

  8. Leites, D.A., Saveliev, M.V., Serganova, V.V.: Embeddings of \({\mathfrak{osp}}(N/2)\) and the associated nonlinear supersymmetric equations. In: Group Theoretical Methods in Physics, vol. I, VNU Sci. Press, Utrecht, pp. 255–297 (1986)

  9. Lu, K.: A note on odd reflections of super Yangian and Bethe ansatz. arXiv:2111.10655

  10. Molev, A.: Yangians and Classical Lie Algebras, Mathematical Surveys and Monographs, vol. 143. AMS, Providence (2007)

    Book  Google Scholar 

  11. Molev, A.: Representations of the Yangians associated with Lie superalgebras \({\mathfrak{osp}}(1|2n)\). arXiv:2109.02361

  12. Musson, I.M.: Lie Superalgebras and Enveloping Algebras, Graduate Studies in Mathematics, vol. 131. AMS, Providence (2012)

    Book  Google Scholar 

  13. Nazarov, M.L.: Quantum Berezinian and the classical Capelli identity. Lett. Math. Phys. 21, 123–131 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  14. Peng, Y.-N.: Finite \(W\)-superalgebras and truncated super Yangians. Lett. Math. Phys. 104, 89–102 (2014)

    Article  MathSciNet  Google Scholar 

  15. Peng, Y.-N.: Parabolic presentations of the super Yangian \(Y({\mathfrak{gl}}_{M|N})\) associated with arbitrary \(01\)-sequences. Commun. Math. Phys. 346, 313–347 (2016)

    Article  Google Scholar 

  16. Penkov, I., Serganova, V.: Cohomology of \(G/P\) for classical complex Lie supergroups G and characters of some atypical \(G\)-modules. Ann. Inst. Fourier 39, 845–873 (1989)

    Article  MathSciNet  Google Scholar 

  17. Stukopin, V.A.: Yangians of Lie superalgebras of type \(A(m, n)\). Funct. Anal. Appl. 28, 217–219 (1994)

    Article  MathSciNet  Google Scholar 

  18. Stukopin, V.A.: On representations of the Yangian of a Lie superalgebra of type \(A(m, n)\). Izv. Math. 77, 1021–1043 (2013)

    Article  MathSciNet  Google Scholar 

  19. Tarasov, V.O.: Irreducible monodromy matrices for the \(R\)-matrix of the \(XXZ\)-model and lattice local quantum Hamiltonians. Theor. Math. Phys. 63, 440–454 (1985)

    Article  MathSciNet  Google Scholar 

  20. Tsymbaliuk, A.: Shuffle algebra realizations of type \(A\) super Yangians and quantum affine superalgebras for all Cartan data. Lett. Math. Phys. 110, 2083–2111 (2020)

    Article  MathSciNet  Google Scholar 

  21. Yamane, H.: On defining relations of affine Lie superalgebras and affine quantized universal enveloping superalgebras. Publ. Res. Inst. Math. Sci. 35, 321–390 (1999)

    Article  MathSciNet  Google Scholar 

  22. Zhang, R.B.: Representations of super Yangian. J. Math. Phys. 36, 3854–3865 (1995)

    Article  MathSciNet  ADS  Google Scholar 

  23. Zhang, R.B.: The \({\mathfrak{gl}}(M|N)\) super Yangian and its finite-dimensional representations. Lett. Math. Phys. 37, 419–434 (1996)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, University of Sydney, Sydney, NSW, 2006, Australia

    A. I. Molev

Authors
  1. A. I. Molev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. I. Molev.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Molev, A.I. Odd reflections in the Yangian associated with \(\mathfrak {gl}(m|n)\). Lett Math Phys 112, 8 (2022). https://doi.org/10.1007/s11005-021-01501-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11005-021-01501-2

Keywords

Mathematics Subject Classification

Search

Navigation