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Sur les constantes de structure et le théorème d’existence des algèbres de Lie semi-simples. (French) Zbl 0145.25804

In this note, the author deals with the structure constants of the complex semisimple Lie algebra \(\mathfrak G\) and gives an elementary proof of the existence and uniqueness of the Lie algebra for any given root system. Let \(\mathfrak F\) be a Cartan subalgebra of \(\mathfrak G\), \(\Sigma\) be the system of roots of \(\mathfrak G\) relative to \(\mathfrak F\), \(\mathfrak E_a\) be the eigenspace relative to \(a\in \Sigma\). Then there exists an element \(e_a\) of \(\mathfrak E_a\) for each \(a\in \Sigma\) such that \([e_a, e_b] = t_a\) (resp. \(N_{a,b}e_{a+b}\) or \(0)\), if \(a+b=0\) (resp. \(a + b \in \Sigma\) or \(a + b \notin \Sigma\cup \{0\}\), where \(t_a=-a^*\in\mathfrak F\), \(a^*\) being defined by \(b(a^*) = 2(a,b)/(a, a)\) for all \(b\in \Sigma\). The set \(\{e_a, a\in \Sigma\}\) with a base of \(\mathfrak F\) forms a base of \(\mathfrak G\). Following Chevalley, we can chose the set \(\{e_a, a\in \Sigma\}\) for which \(f(a, b) = \vert N_{a,b}\vert\) is the least positive integer such that \(b-fa\) is not a root.
In the first part, the author determines the sign of \(N_{a,b}\) by a simple relation depending only on the system of roots. Namely, let \(G\) be a simply connected split semisimple algebraic group over the complex number field, \(T\) be a maximal torus of \(G\) such that their Lie algebras are \(\mathfrak G\) and \(\mathfrak F\), respectively, \(N\) be the normalizer of \(T\) in \(G\), \(W = N/T\) be the Weyl group of \(G\), \(G_a\) be the connected subgroup of \(G\) generated by \(e_a\) and \(-e_a\). We set \(N_a = N\cap G_a\), \(T_a = T\cap G_a\) and \(M_a = N_a - T_a\). Then for each \(m\in M_a\), there exists a pair of non-zero elements \(e_{a,m}\) in \(\mathfrak E_a\) and \(e_{-a,m}\) in \(\mathfrak E_{-a}\), such that \([e_{a,m}, e_{-a,m}]= -a^*\). The author gives a description of the system \(\{N, M_a, a\in \Sigma\}\) and deals with the function \(\delta\) on a subset of \(\Sigma\times\Sigma\times\Sigma\times N\times N\times N\) with values in the complex number field such that \([e_{a,m}, e_{-a,m}] = \delta(a, b, c; m, n, p) f(a, b) e_{-c,p}\) where \(a, b, c \in \Sigma\), \(a+b+c=0\), \(m\in M_a\), \(n\in M_b\) and \(p\in M_c\). The restriction of \(\delta\) to a subset of \(\Sigma\times\Sigma\times\Sigma\times N_Z\times N_Z\times N_Z\), where \(N_Z\) is the group of the \(Z\)-rational points of \(N\) (called the extended Weyl group) is dependent only on \(m, n\) and \(p\) and with values \(\pm 1\). Denote it by \(\varepsilon(m, n, p)\), then it is characterized by (1) \(\varepsilon(m, n, p^{-1})= -\varepsilon(m, n, p)\), (2) \(\varepsilon(m,n,p) = \varepsilon(p, m, n)\) and (3) if \(\lambda(a) \ge \lambda(b), \lambda(c)\), then \(\varepsilon(m, n, mnm^{-1}) = (-1)^{f(a,b)+1}\), where \(\lambda(a)\) is the length of \(a\) for any root \(a\in \Sigma\). In particular, this gives the signs of the structure constants of a Chevalley base of \(\mathfrak G\). In the second part, for a given system \(\Sigma\) of roots, the author defines a system \(\{N, M_a, a\in\Sigma\}\) axiomatically and proves the existence of the system and also the semisimple Lie algebra associated to the system. This gives a new proof of the existence theorem.

MSC:

17B20 Simple, semisimple, reductive (super)algebras
17B22 Root systems
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References:

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