Abstract. In 1988, Brown and Ellis published [3] a generalised Hopf formula for the higher homology of a group. Although substantially correct, their result lacks one necessary condition. We give here a counterexample to the result without that condition. The main aim of this paper is, however, to generalise this corrected result to derive formulae of Hopf type for the n-foldČech derived functors of the lower central series functors Z k . The paper ends with an application to algebraic K-theory.

Abstract. Non-abelian homology of Lie algebras with coefficients in Lie algebras is constructed and studied, generalising the classical Chevalley-Eilenberg homology of Lie algebras. The relationship between cyclic homology and Milnor cyclic homology of non-commutative associative algebras is established in terms of the long exact nonabelian homology sequence of Lie algebras. Some explicit formulae for the second and the third non-abelian homology of Lie algebras are obtained.2000 Mathematics Subject Classification. 17B40, 17B56, 18G10, 18G50. 0. Introduction. The non-abelian homology of groups with coefficients in groups was constructed and investigated in [16,17], using the non-abelian tensor product of groups of Brown and Loday [4, 5] and its non-abelian left derived functors. It generalises the classical Eilenberg-MacLane homology of groups and extends Guin's low dimensional non-abelian homology of groups with coefficients in crossed modules [9], having an interesting application to the algebraic K-theory of non-commutative local rings [9,17].The purpose of this paper is to set up a similar non-abelian homology theory for Lie algebras and is mainly dedicated to state and prove several desirable properties of this homology theory.In [8] Ellis introduced and studied the non-abelian tensor product of Lie algebras which is a Lie structural and purely algebraic analogue of the non-abelian tensor product of groups of Brown and Loday [4,5], arising in applications to homotopy theory of a generalised Van Kampen theorem.Applying this tensor product of Lie algebras, Guin defined the low-dimensional non-abelian homology of Lie algebras with coefficients in crossed modules [10].We construct a non-abelian homology H * (M, N) of a Lie algebra M with coefficients in a Lie algebra N as the non-abelian left derived functors of the tensor product of Lie algebras, generalising the classical Chevalley-Eilenberg homology of Lie algebras and extending Guin's non-abelian homology of Lie algebras [10]. We give an application of our long exact homology sequence to cyclic homology of associative algebras, correcting the result of [10]. In fact, for a unital associative (non-commutative) algebra A we obtain a long exact non-abelian homology available at https://www.cambridge.org/core/terms. https://doi

Abstract. The Hochschild and (cotriple) cyclic homologies of crossed modules of (notnecessarily-unital) associative algebras are investigated. Wodzicki's excision theorem is extended for inclusion crossed modules in the category of crossed modules of algebras. The cyclic and cotriple cyclic homologies of crossed modules are compared in terms of long exact homology sequence, generalising the relative cyclic homology exact sequence.

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We construct a pair of adjoint functors between the categories of crossed modules of groups and associative algebras and establish an equivalence of categories between module structures over a crossed module of groups and its respective crossed module of associative algebras.

Abstract. We give interpretations of some known key agreement protocols in the framework of category theory and in this way we give a method of constructing of many new key agreement protocols.

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