Abstract
For the Kac-Moody superalgebra associated with the loop superalgebra with values in a finite-dimensional Lie superalgebra g, we show what its quadratic Casimir element is equal to if the Casimir element for g is known (if g has an even invariant supersymmetric bilinear form). The main tool is the Wick normal form of the even quadratic Casimir operator for the Kac-Moody superalgebra associated with g; this Wick normal form is independently interesting. If g has an odd invariant supersymmetric bilinear form, then we compute the cubic Casimir element. In addition to the simple Lie superalgebras g = g(A) with a Cartan matrix A for which the Shapovalov determinant was known, we consider the Poisson Lie superalgebra poi(0|n) and the related Kac-Moody superalgebra.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Springer, New York (1997).
A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Nucl. Phys. B, 241, 333–380 (1984).
V. S. Dotsenko and V. A. Fateev, Nucl. Phys. B, 240, 312–348 (1984).
V. Kac, “Some problems on infinite-dimensional Lie algebras and their representations,” in: Lie Algebras and Related Topics (Lect. Notes Math., Vol. 933, D. Winter, ed.), Springer, Berlin (1982), pp. 117–126.
P. Goddard and D. Olive, Nucl. Phys. B, 257, 226–252 (1985).
V. G. Knizhnik and A. B. Zamolodchikov, Nucl. Phys. B, 247, 83–103 (1984).
V. Kac and M. Wakimoto, Adv. Math., 185, 400–458 (2004); 193, 453–455 (2005).
P. Grozman, A. Lebedev, and D. Leites, “The Shapovalov determinant for stringy Lie superalgebras,” (in preparation).
V. G. Kac and D. A. Kazhdan, Adv. Math., 34, 97–108 (1979).
P. Grozman and D. Leites, J. Nonlinear Math. Phys., 8, 220–228 (2001); arXiv:math.QA/0104287v1 (2001).
M. Gorelik, “On a generic Verma module at the critical level over affine Lie superalgebras,” arXiv: math.RT/0504253v2 (2005).
B. B. Shoikhet, J. Math. Sci., 92, 3764–3806 (1998); arXiv:q-alg/9703029v1 (1997).
M. Gorelik and V. Serganova, “Shapovalov forms for Poisson Lie superalgebras,” in: Noncommutative Geometry and Representation Theory in Mathematical Physics (Contemp. Math., Vol. 391, J. Fuchs et al., ed.), Amer. Math. Soc., Providence, R. I. (2005), pp. 111–121.
P. Grozman and D. Leites, “Lie superalgebras of supermatrices of complex size: Their generalizations and related integrable systems,” in: Complex Analysis and Related Topics (Oper. Theory Adv. Appl., Vol. 114, E. Ramírez de Arellano, M. Shapiro, L. Tovar, and N. Vasilevski, eds.), Birkhäuser, Basel (2000), pp. 73–105; arXiv:math.RT/0202177v1 (2002).
S. E. Konstein, Resenhas, 6, 249–255 (2004); arXiv:math-ph/0112063v3 (2001); S. E. Konstein and M. A. Vasiliev, J. Math. Phys., 37, 2872–2891 (1996); arXiv:math-ph/9904032v1 (1999).
M. Gorelik, “Shapovalov determinants of Q-type Lie superalgebras,” arXiv:math.RT/0511623v1 (2005).
P. Grozman, D. Leites, and I. Shchepochkina, Acta Math. Vietnam., 26, 27–63 (2001); arXiv:hep-th/9702120v1 (1997).
M. Gorelik and V. Serganova, “On representations of affine superalgebras q(n)(2),” Mosc. Math. J., 8, No. 1 (2008).
P. Deligne et al., eds., Quantum Fields and Strings: A Course for Mathematicians, Vols. 1 and 2, Amer. Math. Soc., Providence, R. I. (1999).
V. A. Bunegina, T. N. Naumova, and A. L. Onishchik, “Cartan subalgebras in Lie superalgebras [in Russian],” in: Questions of Group Theory and Homological Algebra (A. L. Onishchik, ed.), Yaroslavl State Univ., Yaroslavl (1989), pp. 99–104.
I. Penkov and V. Serganova, Internat. J. Math., 5, 389–419 (1994).
V. G. Kac, Infinite-Dimensional Lie Algebras (3rd ed.), Cambridge Univ. Press, Cambridge (1994).
B. L. Feigin, D. A. Leites, and V. V. Serganova, Group Theoretical Methods in Physics, Vol. 1, Nauka, Moscow (1983); English transl., Harwood Academic, Chur (1985).
V. V. Serganova, Math. USSR Izv., 24, 539–551 (1985).
I. Musson, Represent. Theory, 1, 405–423 (1997).
Author information
Authors and Affiliations
Additional information
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 3, pp. 378–397, September, 2008.
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Lebedev, A.V., Leites, D.A. Shapovalov determinant for loop superalgebras. Theor Math Phys 156, 1292–1307 (2008). https://doi.org/10.1007/s11232-008-0107-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11232-008-0107-7