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Involutive Lie algebras graded by finite root systems and compact forms of IM algebras. (English) Zbl 0884.17012
An intersection-matrix (IM) Lie algebra is defined through an analogue of the Chevalley basis of a simple Lie algebra associated with an intersection matrix $$A= (a_{ij})_{n\times n}$$ $$(a_{ii}=2$$, $$a_{ij}<0 \iff a_{ji}<0$$, $$0<a_{ij} \iff 0<a_{ji}$$). For any IM algebra, there exists an analogue of the Chevalley involution. The author shows that the fixed-point subalgebra of this involution is isomorphic to a certain Steinberg unitary Lie algebra which was introduced by B. N. Allison and J. R. Faulkner [J. Algebra 161, 1-19 (1993; Zbl 0812.17002)]. Moreover, the author introduces an analogue of the elementary unitary Lie algebra of other types and studies involutive Lie algebras graded by finite root systems.

##### MSC:
 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
Zbl 0812.17002
Full Text:
##### References:
 [1] B. N. Allison, J. R. Faulkner, Nonassociative coefficient algebras for Steinberg unitary Lie algebras, J. Algebra161 (1993) 1–19. · Zbl 0812.17002 [2] G. M. Benkart, E. Zelmanov, Lie algebras graded by finite root systems and intersection matrix algebras, Invemt. Math. (to appear) · Zbl 0871.17024 [3] S. Berman, On generators and relations for certain involutory subalgebras of Kac-Moody Lie algebras, Comm. in Alg.17 (1989) 3165–3185. · Zbl 0693.17012 [4] S. Berman, R. V. Moody, Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy, Invent. Math.108 (1992) 323–347. · Zbl 0778.17018 [5] R. Borcherds, Generalized Kac-Moody algebras, J. Alg.115 (1988) 501–512. · Zbl 0644.17010 [6] N. Bourbaki, Groupes et algèbres de Lie, Chap. IV, V, VI. Paris: Hermann 1968 [7] R. W. Carter, Simple groups of Lie type, London: Wiley 1972 · Zbl 0248.20015 [8] Y. Gao, Skew-dihedral homology and involutive Lie algebras graded by finite root systems, Ph D Thesis, University of Saskatchewan 1994 [9] Y. Gao, Steinberg unitary Lie algebras and skew-dihedral homology, J. Algebra179 (1996), 261–304. · Zbl 0837.17011 [10] H. Garland, The arithmetic theory of loop groups, Publ. Math. IHES52 (1980) 5–136. · Zbl 0475.17004 [11] M. Goto, F. D. Grosshans, Semisimple Lie algebras, Lect. Notes. Pure. Appl. Math. Vol38, New York: M. Dekker 1978 · Zbl 0391.17004 [12] J. B. Humphreys, Introduction to Lie algebras and representation theory, Grad. Texts Math, Vol9, Springer 1972 · Zbl 0254.17004 [13] V. G. Kac, Infinite dimensional Lie algebras, third edition, Cambridge Univ. Press 1990 · Zbl 0716.17022 [14] R. V. Moody, A. Pianzona, Lie algebras with triangular decomposition, New York: J. Wiley 1995 [15] E. Neher, Lie algebras graded by 3-graded root systems, Amer. J. Math.118 (1996), 439–491. · Zbl 0857.17019 [16] A. A. Sagle, R. E. Walde, Introduction to Lie groups and Lie algebras, Academic Press, New York and London 1973. · Zbl 0252.22001 [17] G. Seligman, Rational methods in Lie algebras, Lect. Notes Pure Appl. Math, New York: M. Dekker 1976 · Zbl 0334.17002 [18] P. Slodowy, Beyond Kac-Moody algebras and inside, Can. Math. Soc. Conf. Proc.5 (1986) 361–371. · Zbl 0582.17011 [19] P. Slodowy, Singularitäten, Kac-Moody Lie algebren, assoziierte Gruppen und Verallgemeinerungen, Habiliationsschrift, Universitát Bonn (March 1984) [20] R. Steinberg, Lectures on Chevalley groups,(notes by J. Faulkner & R. Wilson), Yale Univ. Lect. Notes 1967 · Zbl 0164.34302
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