# zbMATH — the first resource for mathematics

The closure diagrams for nilpotent orbits of the real forms EVI and EVII of $$\mathbf{E}_7$$. (English) Zbl 1031.17004
Summary: Let $$\mathcal{O}_1$$ and $$\mathcal{O}_2$$ be adjoint nilpotent orbits in a real semisimple Lie algebra. Write $$\mathcal{O}_1\geq\mathcal{O}_2$$ if $$\mathcal{O}_2$$ is contained in the closure of $$\mathcal{O}_1.$$ This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the two noncompact nonsplit real forms of the simple complex Lie algebra $$E_7.$$

##### MSC:
 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B20 Simple, semisimple, reductive (super)algebras 17B05 Structure theory for Lie algebras and superalgebras
Maple; LiE
Full Text:
##### References:
 [1] Dan Barbasch and Mark R. Sepanski, Closure ordering and the Kostant-Sekiguchi correspondence, Proc. Amer. Math. Soc. 126 (1998), no. 1, 311 – 317. · Zbl 0896.22004 [2] W. M. Beynon and N. Spaltenstein, Green functions of finite Chevalley groups of type \?_{\?} (\?=6,7,8), J. Algebra 88 (1984), no. 2, 584 – 614. · Zbl 0539.20025 [3] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 (French). · Zbl 0186.33001 [4] Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. · Zbl 0567.20023 [5] B.W. Char, K.O. Geddes, G.H. Gonnet, B.L. Leong, M.B. Monagan, and S.M. Watt, Maple V Language reference Manual, Springer-Verlag, New York, 1991, xv+267 pp. · Zbl 0758.68038 [6] David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. · Zbl 0972.17008 [7] Dragomir Ž. {\Dj}oković, Classification of nilpotent elements in simple exceptional real Lie algebras of inner type and description of their centralizers, J. Algebra 112 (1988), no. 2, 503 – 524. · Zbl 0639.17005 [8] -, Explicit Cayley triples in real forms of $$E_7,$$ Pacific J. Math. 191 (1999), 1-23. CMP 2000:12 [9] -, The closure diagrams for nilpotent orbits of real forms of $$F_4$$and $$G_2$$, J. Lie Theory 10 (2000), 491-510. CMP 2000:16 [10] -, The closure diagrams for nilpotent orbits of real forms of $$E_6$$, J. Lie Theory (to appear). · Zbl 1049.17006 [11] E.B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik 30 (1952), 349-462. (Amer. Math. Soc. Transl. Ser. 2 6 (1957), 111-245.) · Zbl 0048.01701 [12] Jun-ichi Igusa, A classification of spinors up to dimension twelve, Amer. J. Math. 92 (1970), 997 – 1028. · Zbl 0217.36203 [13] Kenzo Mizuno, The conjugate classes of unipotent elements of the Chevalley groups \?$$_{7}$$ and \?$$_{8}$$, Tokyo J. Math. 3 (1980), no. 2, 391 – 461. · Zbl 0454.20046 [14] M. Sato and T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1 – 155. · Zbl 0321.14030 [15] Mikio Sato, Theory of prehomogeneous vector spaces (algebraic part) — the English translation of Sato’s lecture from Shintani’s note, Nagoya Math. J. 120 (1990), 1 – 34. Notes by Takuro Shintani; Translated from the Japanese by Masakazu Muro. · Zbl 0715.22014 [16] Nicolas Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics, vol. 946, Springer-Verlag, Berlin-New York, 1982 (French). · Zbl 0486.20025 [17] M.A.A. van Leeuwen, A.M. Cohen, and B. Lisser, ”LiE”, a software package for Lie group theoretic computations, Computer Algebra Group of CWI, Amsterdam, The Netherlands.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.