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Graded Lie algebras and generalized Jordan triple systems. (English) Zbl 0699.17021
Let $${\mathfrak g}=\sum_{k\in {\mathbb{Z}}}{\mathfrak g}_ k$$ be a graded Lie algebra (GLA) over $${\mathbb{R}}$$ with dim $${\mathfrak g}_ k<\infty$$ for all $$k\in {\mathbb{Z}}$$ and $${\mathfrak g}_{-1}\neq (0)$$. The family of subspaces ($${\mathfrak g}_ k)$$ is called a gradation in $${\mathfrak g}$$. Two gradations ($${\mathfrak g}_ k)$$ and ($${\mathfrak g}_ k')$$ in $${\mathfrak g}$$ are called isomorphic if there exists an automorphism $$a\in Aut {\mathfrak g}$$ such that a($${\mathfrak g}_ k)={\mathfrak g}_ k'$$ for all k. A classification (up to isomorphisms) for gradations in a semisimple Lie algebra is obtained, in such a way that it is possible to construct all gradations in such a Lie algebra.
A GLA $${\mathfrak g}=\sum_{k\in {\mathbb{Z}}}{\mathfrak g}_ k$$ is said to be of the $$\nu$$-th kind $$(\nu >0)$$ if $${\mathfrak g}_{\pm \nu}\neq (0)$$ and $${\mathfrak g}_ k=(0)$$ for $$| k| >\nu$$. A GLA $${\mathfrak g}$$ with dim $${\mathfrak g}\leq \infty$$ is said to be of type $$\alpha_ 0$$ if $${\mathfrak g}_{-k-1}=[{\mathfrak g}_{-k},{\mathfrak g}_{-1}]$$, $${\mathfrak g}_{k+1}=[{\mathfrak g}_ k,{\mathfrak g}_ 1]$$, $$k\geq 1$$. A necessary and sufficient condition is given for a semisimple GLA of the $$\nu$$-th kind to be of type $$\alpha_ 0.$$
A pair $$(U_{-1},B)$$, where $$U_{-1}$$ is a (finite dimensional) vector space and $$B: U_{-1}\times U_{-1}\times U_{-1}\to U_{-1}$$ is a trilinear mapping, is called a triple system. It is called a generalized Jordan triple system (GJTS) if the equality $(uv(xyz))=((uvx)yz)- (x(vuy)z)+(xy(uvz))$ is valid for $$u,v,x,y,z\in U_{-1}$$, where xyz denotes B(x,y,z). Due to I. L. Kantor [Tr. Semin. Vektorn. Tenzorn. Anal. 16, 407-499 (1972; Zbl 0272.17001)] the authors associate to a $$GJTS(U_{-1},B)$$ a graded Lie algebra $${\mathcal L}(B)$$ which is not necessary of finite dimension. The mentioned results for semisimple GLA’s are applied to the classification of compact GJTS’s of the first or the second kind. At the end of the paper, the authors give a method of constructing noncompact simple GJTS’s starting from compact simple GJTS’s.
Reviewer: G.I.Zhitomirskij

##### MSC:
 17B70 Graded Lie (super)algebras 17A40 Ternary compositions 17B20 Simple, semisimple, reductive (super)algebras 17C50 Jordan structures associated with other structures
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