# A Dold-Kan theorem for simplicial Lie algebras

## P. Carrasco and A.M. Cegarra

We introduce and study hypercrossed complexes of Lie algebras, that is, non-negatively graded chain complexes of Lie algebras $L=(L_n,\partial_n)$ endowed with an additional structure by means of a suitable set of bilinear maps $L_r\times L_s\rightarrow L_n$. The Moore complex of any simplicial Lie algebra acquires such a structure and, in this way, we prove a Dold-Kan type equivalence between the category of simplicial Lie algebras and the category of hypercrossed complexes of Lie algebras. Several consequences of examining particular classes of hypercrossed complexes of Lie algebras are presented.

Keywords: Dold-Kan theorem, simplicial Lie algebra, chain complex, Moore complex, hypercrossed complex

2010 MSC: 55U10, 18G30, 18G50

Theory and Applications of Categories, Vol. 32, 2017, No. 34, pp 1165-1212.

Published 2017-09-05.

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