Introduction to quantum Lie algebras

Gustav Delius

Banach Center Publications (1997)

  • Volume: 40, Issue: 1, page 91-97
  • ISSN: 0137-6934

Abstract

top
Quantum Lie algebras are generalizations of Lie algebras whose structure constants are power series in h. They are derived from the quantized enveloping algebras U h ( g ) . The quantum Lie bracket satisfies a generalization of antisymmetry. Representations of quantum Lie algebras are defined in terms of a generalized commutator. The recent general results about quantum Lie algebras are introduced with the help of the explicit example of ( s l 2 ) h .

How to cite

top

Delius, Gustav. "Introduction to quantum Lie algebras." Banach Center Publications 40.1 (1997): 91-97. <http://eudml.org/doc/252214>.

@article{Delius1997,
abstract = {Quantum Lie algebras are generalizations of Lie algebras whose structure constants are power series in h. They are derived from the quantized enveloping algebras $U_h(g)$. The quantum Lie bracket satisfies a generalization of antisymmetry. Representations of quantum Lie algebras are defined in terms of a generalized commutator. The recent general results about quantum Lie algebras are introduced with the help of the explicit example of $(sl_2)_h$.},
author = {Delius, Gustav},
journal = {Banach Center Publications},
keywords = {quantum Lie algebras; quantized enveloping algebras; quantum Lie bracket},
language = {eng},
number = {1},
pages = {91-97},
title = {Introduction to quantum Lie algebras},
url = {http://eudml.org/doc/252214},
volume = {40},
year = {1997},
}

TY - JOUR
AU - Delius, Gustav
TI - Introduction to quantum Lie algebras
JO - Banach Center Publications
PY - 1997
VL - 40
IS - 1
SP - 91
EP - 97
AB - Quantum Lie algebras are generalizations of Lie algebras whose structure constants are power series in h. They are derived from the quantized enveloping algebras $U_h(g)$. The quantum Lie bracket satisfies a generalization of antisymmetry. Representations of quantum Lie algebras are defined in terms of a generalized commutator. The recent general results about quantum Lie algebras are introduced with the help of the explicit example of $(sl_2)_h$.
LA - eng
KW - quantum Lie algebras; quantized enveloping algebras; quantum Lie bracket
UR - http://eudml.org/doc/252214
ER -

References

top
  1. [1] Dri V. G. Drinfel'd, Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl. 32 (1985) 254. 
  2. [2] Jim M. Jimbo, A q-Difference Analogue of U(g) and the Yang-Baxter Equation, Lett. Math. Phys. 10 (1985) 63. Zbl0587.17004
  3. [3] G. W. Delius, A. Hüffmann, On Quantum Lie Algebras and Quantum Root Systems, q-alg/9506017, J. Phys. A. 29 (1996) 1703. Zbl0916.17011
  4. [4] G. W. Delius, M. D. Gould, Quantum Lie Algebras, their existence, uniqueness and q-antisymmetry, KCL-TH-96-05, q-alg/9605025, Commun. Math. Phys. (in print). 
  5. [5] S. L. Woronowicz, Differential Calculus on Compact Matrix Pseudogroups (Quantum Groups), Comm. Math. Phys. 122 (1989) 125. Zbl0751.58042
  6. [6] P. Aschieri, L. Castellani, An introduction to noncommutative differential geometry on quantum groups, Int. J. Mod. Phys. A8 (1993) 1667 Zbl0802.58007
  7. D. Bernard, Quantum Lie Algebras and Differential Calculus on Quantum Groups, Prog. Theo. Phys. Suppl. 102 (1990) 49 
  8. B. Jurco, Differential Calculus on Quantized Simple Lie Groups, Lett. Math. Phys. 22 (1991) 177 Zbl0753.17020
  9. P. Schupp, P. Watts, B. Zumino, Bicovariant Quantum Algebras and Quantum Lie Algebras, Commun. Math. Phys. 157 (1993) 305 Zbl0808.17010
  10. P. Schupp, Quantum Groups, Non-Commutative Differential Geometry and Applications, hep-th/9312075 (1993) 
  11. K. Schmüdgen, A. Schüler, Classification of Bicovariant Differential Calculi on Quantum Groups of Type A, B, C and D, Comm. Math. Phys. 107 (1995) 635 
  12. A. Sudbery, Quantum Lie Algebras of Type A n , q-alg/9510004. 
  13. [7] Andrew V. Chari, A. Pressley, A Guide to Quantum Groups, Cambridge University Press (1994). Zbl0839.17009
  14. [8] G.W. Delius, M.D. Gould, A. Hüffmann, Y.-Z. Zhang, Quantum Lie algebras associated to U q ( g l n ) and U q ( s l n ) , q-alg/9508013. Zbl0905.17008
  15. [9] V. Lyubashenko and A. Sudbery, Quantum Lie algebras of type A(N), q-alg/9510004. 
  16. [10] World Wide Web, http:/www.mth.kcl.ac.uk/~delius/q-lie.html. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.