×

Nilpotent orbits of Kac-Moody algebras and their parameterization for \(\mathfrak{sl}_n^{(1)}(\mathbb{C})\). (English) Zbl 1490.17028

The orbit of a nilpotent element in a complex semisimple Lie algebra under the adjoint action of its adjoint algebraic group is called a nilpotent orbit. They play important roles in representation theory of Lie algebras and groups. The authors generalize the notion of nilpotent orbits to an untwisted affine Kac-Moody algebra. The discussion is based on their loop realization.
Let \(\dot{\mathfrak{g}}\) be a finite dimensional simple complex Lie algebra. The authors consider the following extended version of an affine Lie algebra \[\hat{\mathfrak{g}}:=\mathbb{C}[[t]][t^{-1}]\otimes\dot{\mathfrak{g}}\oplus\mathbb{C}c\oplus\mathbb{C}d\] where \(\mathbb{C}[[t]]{t^{-1}}\) is the ring of formal Laurent series whose powers are bounded from below, \(c\) is central and \(d\) is a derivation. The authors show that an element \(X\in\hat{\mathfrak{g}}\) is actually nilpotent if it is locally nilpotent, which guarantees the well-defined notion of a nilpotent orbit of an nilpotent element in \(\hat{\mathfrak{g}}\) under the action of the associated affine Kac-Moody group.
In the case of \(\mathfrak{sl}_n^{(1)}(\mathbb{C})\), the authors obtain a parameterization of all nilpotent orbits by finding a distinguished element in each nilpotent orbit with the use of denominated quasi-Jordan matrices.

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B08 Coadjoint orbits; nilpotent varieties
PDFBibTeX XMLCite
Full Text: arXiv Link

References:

[1] P. Bala, R. Carter:Classes of unipotent elements in simple algebraic groups I, Math. Proc. Camb. Phil. Soc. 79 (1976) 401-425. · Zbl 0364.22006
[2] A. Borel:Linear Algebraic Groups, Graduate Texts in Mathematics 126, Springer, New York (1991). · Zbl 0726.20030
[3] N. Bourbaki:Groupes et Algèbres de Lie, Chapitres 1-9, Masson, Paris (1990).
[4] D. Collingwood, W. McGovern:Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold, New York (1993). · Zbl 0972.17008
[5] D. Djokovic:Classification of nilpotent elements in simple exceptional real Lie algebras of inner type and description of their centralizers, J. Algebra 112 (1988), 503-524. · Zbl 0639.17005
[6] D. Djokovic:Classification of nilpotent elements in the simple real Lie algebras E6(6) and E6... and description of their centralizers, J. Algebra 116 (1988) 196-207. · Zbl 0653.17004
[7] J. Fuchs:Affine Lie Algebras and Quantum Groups, Cambridge University Press, Cambridge (1992). · Zbl 0925.17031
[8] H. Garland:The arithmetic theory of loop groups, Publ. Math. IHES 52 (1980) 5-136. · Zbl 0475.17004
[9] L. Gohberg, P. Lancaster, L. Rodman:Matrix Polynomials, Society for Industrial and Applied Mathematics, Philadelphia (2009). · Zbl 1170.15300
[10] J. Humphreys:Introduction to Lie Algebras and Representations Theory, Graduate Texts in Mathematics 9, Springer, New York (1972). · Zbl 0254.17004
[11] J. Humphreys:Conjugacy Classes in Semisimple Algebraic Groups, Mathematical Surveys and Monographs 43, American Mathematical Society, Providence (1995). · Zbl 0834.20048
[12] T. Hungerford:Algebra, Graduate Texts in Mathematics 73, Springer, New York (1974).
[13] J. C. Jantzen, K. Neeb:Lie Theory, Lie Algebras and Representations, Progress in Mathematics 228, Birkhäuser, Basel (2004). · Zbl 1054.17001
[14] V. Kac:Infinite Dimensional Lie Algebras, 3rd ed., Cambridge University Press, Cambridge (1990). · Zbl 0716.17022
[15] W. Knapp:Lie Groups Beyond an Introduction, Progress in Mathematics 140, Birkhäuser, Boston (1996). · Zbl 0862.22006
[16] S. Kumar:Kac Moody Groups, Their Flag Varieties and Representation Theory, Birkhäuser, Basel (2002). · Zbl 1026.17030
[17] A. Noel:Nilpotent orbits and theta-stable parabolic subalgebras, Representation Theory 2 (1998) 1-32. · Zbl 0891.17006
[18] D. Peterson, V. Kac:Infinite flag varieties and conjugacy theorem, Proc. Natl. Acad. Sci. USA 80 (1983) 1778-1782. · Zbl 0512.17008
[19] A. Pressley, G. Segal:Loop Groups, Clarendon Press, Oxford (1986). · Zbl 0618.22011
[20] T. Springer:Linear Algebraic Groups, 2nd ed., Progress in Mathematics 9, Birkhäuser, Basel (1998). · Zbl 0927.20024
[21] L. Valencia:Órbitas Nilpotentes en Algebras de Kac-Moody Afines, Tesis Doctoral, FAMAF, Universidad Nacional, Córdoba (2018).
[22] M. Wakimoto:Infinite-Dimensional Lie Algebras, Translations of Mathematical Monographs 195, American Mathematical Society, Providence (1999)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.