In thiswork, we give the notion of braiding for categorical Lie algebrasand crossed modules of lie algebras and we give an equivalence between them. | Turk J Math 31 (2007) , 239 – 255. ¨ ITAK ˙ c TUB Braiding for Categorical and Crossed Lie Algebras and Simplicial Lie Algebras E. Ulualan Abstract In this work, we give the notion of braiding for categorical Lie algebras and crossed modules of Lie algebras and we give an equivalence between them. Key Words: Simplicial Lie Algebra, Categorical Lie Algebra Crossed Module. 1. Introduction Crossed modules of groups were introduced by Whitehead in [19]. The commutative algebra analogue of crossed modules was given by Porter in [18]. Kassel and Loday [15] introduced crossed modules of Lie algebras as computational algebraic objects equivalent to simplicial Lie algebras with Moore complex of length 1. Conduch´e [6] defined 2crossed module of groups and he gave a link between 2-crossed modules and simplicial groups. Ellis [11] captured the algebraic structure of a Moore complex of length 2 in his definition of a 2-crossed module of Lie algebras. Ak¸ca and Arvasi [1] have defined higher dimensional Peiffer elements for Lie algebras in the image of the Moore complex of a simplicial Lie algebra, and they then gave a functor from simplicial Lie algebras to 2-crossed Lie algebras in terms of hypercrossed complex pairings. Joyal and Street [13, 14] have defined the notion of braiding for a monoidal category and they have showed that braided monoidal categories are equivalent to crossed semimodules with bracket operations. Brown and Gilbert introduced in [4] the notion of AMS Mathematics Subject Classification: 18G50, 18G55 239 ULUALAN braided, regular crossed module as an algebraic model for homotopy 3-type equivalent to Conduch´e’s 2-crossed module and simplicial groups with Moore complex of length 2. The reduced case of braided regular crossed module is called a braided crossed module of groups. Of course, braided crossed modules of groups are equivalent to reduced 2-crossed modules and braided categorical groups (cf. [13, 14]) and reduced simplicial groups with Moore .
