Simplicial and crossed Lie algebras
The authors examine higher order Peiffer elements in simplicial Lie algebras and apply them to the Lie 2-crossed module and Lie crosssed squares introduced by \textit{G. J. Ellis} [J. Aust. Math. Soc., Ser. A 54, No.~3, 393-419 (1993; Zbl 0824.17022)] The notion of simplicial algebras used here is that considered by \textit{E. B. Curtis} [``Simplicial homotopy theory'', Adv. Math. 6, 107-209 (1971; Zbl 0225.55002)]. Let \(L\) be a simplicial Lie algebra with Moore complex \(NL\). Let \(\vartheta\) denote the boundary homomorphism of the Moore complex. For \(n > 1\) let \(D_n\) be the ideal in \(L_n\) generated by the degenerate elements. In the beginning it is proved that if \(L_n = D_n\), then \(NL_n = I_n\), where \(I_n\) is an ideal in \(L_n\) generated by certain shuffles. Then, the authors use this equality to prove the following theorem (in which the face homomorphisms of the simplicial Lie algebra \(L\) are denoted by \(d_i\), and for \(I\subseteq \{1,\dots,n\}\) the intersection \(\bigcap_{i\in I}\text{Ker }d_i\) is denoted by \(K_I\)). Theorem 1. Let \(L\) be a simplicial Lie algebra. (i) \(L_2=D_2\) then \(\vartheta_2(NL_2) = [\text{Ker} d_0, \text{Ker} d_1]\); (ii) \(L_3=D_3\) then \(\vartheta_3(NL_3) = [K_{\{0,1\}},K_{\{0,2\}}]+[K_{\{0,2\}}, K_{\{1,2\}}] + [K_{\{0,1\}}, K_{\{1,2\}}]+ \sum_{I,J}[K_I,K_J]\), where \(I\cup J=\{0,1,2\}\), \(I\cap J=\emptyset\). (iii) If \(L_n=D_n\) then \(\vartheta_n(NL_2)\supseteq \sum_{I,J}[K_I,K_J]\), where \(I\), \(J\) are nonempty subsets of the set \(\{0,\dots,n-1\}\) with \(I\cup J = \{0,\dots,n-1\}\). In the last section of the paper theorem 1 is used to construct a functor from simplicial Lie algebras to 2-crossed modules, and the authors also explain how theorem 1 yields a functor from simplicial Lie algebras to those crossed \(n\)-cubes of Lie algebras introduced by \textit{G. J. Ellis} [J. Pure Appl. Algebra 52, No.~3, 277-282 (1988; Zbl 0658.17010)].
scientific article; zbMATH DE number 1748379