In this paper we classify solvable Leibniz algebras whose nilradical is a null-filiform algebra. We extend the obtained classification to the case when the solvable Leibniz algebra is decomposed as a direct sum of its nilradical, which is a direct sum of null-filiform ideals, and a onedimensional complementary subspace. Moreover, in this case we establish that these ideals are ideals of the algebra, as well.2010 Mathematics Subject Classification. 17A32, 17A65, 17B30.

For any category of interest C we define a general category of groups with operations C G , C → C G , and a universal strict general actor USGA(A) of an object A in C, which is an object of C G . The notion of actor is equivalent to the one of split extension classifier defined for an object in more general settings of semi-abelian categories. It is proved that there exists an actor of A in C if and only if the semidirect product USGA(A) A is an object of C and if it is the case, then USGA(A) is an actor of A. We give a construction of a universal strict general actor for any A ∈ C, which helps to detect more properties of this object. The cases of groups, Lie, Leibniz, associative, commutative associative, alternative algebras, crossed and precrossed modules are considered. The examples of algebras are given, for which always exist actors.

We introduce a notion of chain of evolution algebras. The sequence of matrices of the structural constants for this chain of evolution algebras satisfies an analogue of Chapman-Kolmogorov equation. We give several examples (time homogenous, time non-homogenous, periodic, etc.) of such chains. For a periodic chain of evolution algebras we construct a continuum set of non-isomorphic evolution algebras and show that the corresponding discrete time chain of evolution algebras is dense in the set. We obtain a criteria for an evolution algebra to be baric and give a concept of a property transition. For several chains of evolution algebras we describe the behavior of the baric property depending on the time. For a chain of evolution algebras given by the matrix of a two-state evolution we define a baric property controller function and under some conditions on this controller we prove that the chain is not baric almost surely (with respect to Lebesgue measure). We also construct examples of the almost surely baric chains of evolution algebras. We show that there are chains of evolution algebras such that if it has a unique (resp. infinitely many) absolute nilpotent element at a fixed time, then it has unique (resp. infinitely many) absolute nilpotent element any time; also there are chains of evolution algebras which have not such property. For an example of two dimensional chain of evolution algebras we give the full set of idempotent elements and show that for some values of parameters the number of idempotent elements does not depend on time, but for other values of parameters there is a critical time tc such that the chain has only two idempotent elements if time t ≥ tc and it has four idempotent elements if time t < tc.

The structural constants of an evolution algebra are given by a quadratic matrix. In this work we establish an equivalence between nil, right nilpotent evolution algebras and evolution algebras defined by upper triangular matrices. The classification of 2-dimensional complex evolution algebras is obtained. For an evolution algebra with a special form of the matrix, we describe all its isomorphisms and their compositions. We construct an algorithm running under Mathematica which decides if two finite dimensional evolution algebras are isomorphic.

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We consider an evolution algebra which corresponds to a bisexual population with a set of females partitioned into finitely many different types and the males having only one type. We study basic properties of the algebra. This algebra is commutative (and hence flexible), not associative and not necessarily power associative, in general. Moreover it is not unital. A condition is found on the structural constants of the algebra under which the algebra is associative, alternative, power associative, nilpotent, satisfies Jacobi and Jordan identities. In a general case, we describe the full set of idempotent elements and the full set of absolute nilpotent elements. The set of all operators of left (right) multiplications is described. Under some conditions on the structural constants it is proved that the corresponding algebra is centroidal. Moreover the classification of 2-dimensional and some 3-dimensional algebras are obtained.

In this paper we show that the method for describing solvable Lie algebras with given nilradical by means of non-nilpotent outer derivations of the nilradical is also applicable to the case of Leibniz algebras. Using this method we extend the classification of solvable Lie algebras with naturally graded filiform Lie algebra to the case of Leibniz algebras. Namely, the classification of solvable Leibniz algebras whose nilradical is a naturally graded filiform Leibniz algebra is obtained.

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