Skip to main content
SpringerLink
Log in

(H,R)-Lie coalgebras and (H,R)-Lie bialgebras

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

Given an (H,R)-Lie coalgebra Γ, we construct (H,R T )-Lie coalgebra ΓT through a right cocycle T, where (H,R) is a triangular Hopf algebra, and prove that there exists a bijection between the set of (H,R)-Lie coalgebras and the set of ordinary Lie coalgebras. We also show that if (L, [, ], Δ, R) is an (H,R)-Lie bialgebra of an ordinary Lie algebra then (L T, [, ], ΔT, R T ) is an (H,R T )-Lie bialgebra of an ordinary Lie algebra.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Michaelis W., “The Dual Poincaré-Birkhoff-Witt Theorem,” Adv. Math., 57, 93–162 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  2. Nichols W. D., “The structure of the dual Lie algebra of the Witt algebra,” J. Pure Appl. Algebra, 68, No. 3, 359–364 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  3. Drinfel’d V. G., “Quantum groups,” in: Proceedings of the International Congress of Mathematicians, Berkeley, California, 1987, pp. 798–820.

  4. Taft E. J., “Witt and Virasoro algebras as Lie bialgebras,” J. Pure Appl. Algebra, 87, No. 3, 301–312 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  5. Michaelis W., “A class of infinite-dimensional Lie bialgebras containing the Virasoro algebra, ” Adv. Math., 107, No. 3, 365–392 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  6. Bahturin Y., Fischman D., and Montgomery S., “Bicharacters, twistings, and Scheunert’s theorem for Hopf algebras,” J. Algebra, 236, No. 1, 246–276 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  7. Montgomery S., Hopf Algebras and Their Actions on Rings (1993) (Cbms. Lecture Notes 82:1).

  8. Michaelis W., “Lie coalgebras,” Adv. Math., 38, 1–54 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  9. Gurievich D. and Majid S., “Braided groups and Hopf algebras by twisting,” Pacific J. Math., 162, No. 1, 27–43 (1994).

    MathSciNet  Google Scholar 

  10. Ballesteros A., Celeghini E., and Herranz F. J., “Quantum (1+1) extended Galilei algebras: from Lie bialgebras to quantum R-matrices and integrable systems,” J. Phys. A, 33, No. 17, 3431–3444 (2000).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Nanjing Agricultural University; Nanjing University, Nanjing, P. R. China

    Liang-yun Zhang

Authors
  1. Liang-yun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text Copyright © 2006 Zhang Liang-yun

The author was supported by the National Natural Science Foundation of China (Grant 10571153), the Postdoctoral Science Foundation of China (Grant 2005037713), and the Postdoctoral Science Foundation of Jiangsu (Grant 0203003403).

__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 47, No. 4, pp. 932–945, July–August, 2006.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, Ly. (H,R)-Lie coalgebras and (H,R)-Lie bialgebras. Sib Math J 47, 767–778 (2006). https://doi.org/10.1007/s11202-006-0087-5

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11202-006-0087-5

Keywords

Search

Navigation