Extensions centrales d'algèbres de Lie

Comparaison des homologies du groupe linéaire et de son algèbre de Lie

“…On définit le groupe HCi(A) comme le fe-module engendré par les symboles <a,fc>, a, fceA, soumis aux relations On remarquera que l'on a Hi(st(A)) = H2(6t(A)) = 0 (cf. [19]). …”

Homotopical aspects of Lie algebras

“…for all m, m e M, p, p' e P. We recall from [14] that a crossed module (in the context of Lie algebras) is a Lie homomorphism d: M -> P together with an action of P on ¥ such that (ii) ( …”

“…In [10] a tensor product M ® M was defined for an arbitrary Lie algebra M. It is generated by symbols x ® y for x, y € M subject to the above relations (9), (10), (11) and (14) and the following relations for x,x ,y,y &M:…”

“…In Section 2 we analyse the low dimensional parts of the Moore complex of a simplicial Lie algebra. We recall from [14] that a Moore complex of length one is equivalent to a crossed module. We then capture the structure of a Moore complex of length 2 in the definition of a 2-crossed module, this is the Lie algebraic version of a group theoretic notion denned in [5].…”

Second Homology Groups and Universal Coverings of Steinberg Leibniz Algebras of Small Characteristic

“…For any Leibniz algebra L there is an associated Lie algebra L Lie = L/ [x, x] , where [x, x] is the two-sided ideal generated by all [x, x], x ∈ L. To study the second Leibniz homology group of Lie algebra sl n (R) and Steinberg Lie algebra st n (R), Loday and Pirashvili [LP] introduced also the noncommutative Steinberg Leibniz algebra stl n (R), where R is an associative algebra over a commutative ring K. In [L], Steinberg Leibniz algebra and its superalgebra were dicussed. Steinberg Lie algebras and their universal central extensions have been studied by many authors (e.g., [B1, KL,Ka,G1,G2,GS]). In most situations, the Steinberg Lie algebra st n (R) is the universal central extension of the Lie algebra sl n (R) whose kernel is isomorphic to the first cyclic homology group HC 1 (R) of the associative algebra R and the second Lie algebra homology group H 2 (st n (R)) = 0.…”