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J. Math. Phys. 53, 023502 (2012); http://dx.doi.org/10.1063/1.3681211 (19 pages)
Quasi-periodic functions on the torus and sl(n)-elliptic Lie algebra
Universita degli Studi di Milano-Bicocca, via Roberto Cozzi, 53, 20125, Milano, Italy and Bogoliubov Institute for Theoretical Physics, Metrologichna st.14-b, 03143, Kiev, Ukraine
(Received 17 May 2011; accepted 4 January 2012; published online 1 February 2012)
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(sl(n)). We prove it to be 
quasi-graded Lie algebra which could be viewed as a deformation of a graded loop algebra. We show that it admits the decomposition into the direct sum of two subalgebras: (sl(n)) = (sl(n))++(sl(n))− consistent with the described quasi-grading. We prove that (sl(n))±* = (sl(n))∓, i.e., Lie algebras (sl(n)), (sl(n))+, and (sl(n))− constitute the Manin triple. We explicitly construct a central extension of (sl(n)). We find its algebra of differentiations and its central extension which coincide with the quasi-graded deformation of the Virasoro algebra.© 2012 American Institute of Physics
Article Outline
- INTRODUCTION
- DEFINITIONS AND NOTATIONS
- Quasi-periodic elliptic functions
- Special bases for sl ( n ) algebra
- AUTOMORPHIC ELLIPTIC LIE ALGEBRA
- CENTRAL EXTENSION OF THE ALGEBRA
(sl(n))
- ELLIPTIC VIRASORO ALGEBRA
- Algebra of the differentiations
- Action of the algebra of differentiations on
(sl(n))
- Central extension of the algebra
- RATIONAL DEGENERATION OF
(sl(n))
AND
- Rational degeneration of
(sl(n))
- Rational degeneration of
- Rational degeneration of
- CONCLUSION AND DISCUSSION
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ARTICLE DATA
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Bremner, M., “Degenerated elliptic Krichiver-Novikov algebras,” J. Math. Phys. 31, 2033 (1990)JMAPAQ000031000008002033000001.
Skrypnyk, T., “Quasi-graded Lie algebras on hyperelliptic curves and classical integrable systems,” J. Math. Phys 42(9), 4570–4581 (2001)JMAPAQ000042000009004570000001.
Skrypnyk, T., “Deformations of the loop algebras and hierarchies of integrable equations,” J. Math. Phys 45(12), 4578–4595 (2004)JMAPAQ000045000012004578000001.
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