## 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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