Mathematics

First, we construct some families of nonsolvable anticommutative algebras, solvable Lie algebras and even nilpotent Lie algebras, that can be endowed with the structure of simple Hom-Lie algebras. This situation shows that a classification of simple Hom-Lie algebras would be unrealistic without any further restrictions. Therefore, we introduce the class of \emph{strongly simple Hom-Lie algebras}, which is the class of anticommutative algebras that are simple Hom-Lie with respect to all their twisting maps. We show some of its properties, provide a characterisation and explore some of its subclasses. Then, we classify completely regular simple Hom-Lie algebras over any arbitrary field. Furthermore, we establish that every simple anticommutative algebra of dimension 3 turns out to be a simple Lie algebra where its Lie bracket is deformed by a bijective linear map, and also we determine all the simple Hom-Lie algebras in dimension 2 , that were wrongly claimed to be nonexistent in \cite{CH}.

We prove that the Bowen-Franks group classifies the Leavitt path algebras of purely infinite simple finite graphs over a regular supercoherent commutative ring with involution where 2 is invertible, equipped with their standard involutions, up to matricial stabilization and involution preserving homotopy equivalence. We also consider a twisting of the standard involution on Leavitt path algebras and obtain partial results in the same direction for purely infinite simple graphs. Our tools are K -theoretic, and we prove several results about (Hermitian, bivariant) K -theory of Leavitt path algebras.

Let R⊆E be two Lie conformal algebras and Q be a given complement of R in E . Classifying complements problem asks for describing and classifying all complements of R in E up to an isomorphism. It is known that E is isomorphic to a bicrossed product of R and Q . We show that any complement of R in E is isomorphic to a deformation of Q associated to the bicrossed product. A classifying object is constructed to parameterize all R -complements of E . Several explicit examples are provided. Similarly, we also develop a classifying complements theory of associative conformal algebras.

We investigate finitary functions from Z pq to Z pq for two distinct prime numbers p and q . We show that the lattice of all clones on the set Z pq which contain the addition of Z pq is finite. We provide an upper bound for the cardinality of this lattice through an injective function to the direct product of the lattice of all ( Z p , Z q ) -linearly closed clonoids to the p+1 power and the lattice of all ( Z q , Z p ) -linearly closed clonoids to the q+1 power. These lattices are studied in arXiv:1910.11759 and there we can find the exact cardinality of them. Furthermore, we prove that these clones can be generated by a set of functions of arity at most max({p,q}) .

We investigate the lattice I(n) of clones on the ring Z_n between the clone of polynomial functions and the clone of congruence preserving functions. The crucial case is when n is a prime power. For a prime p, the lattice I(p) is trivial and I(p^2) is known to be a 2-element lattice. We provide a description of I(p^3). To achieve this result, we prove a reduction theorem, which says that I(p^k) is isomorphic to a certain interval in the lattice of clones on Z_p^(k-1).

We investigate the finitary functions from a finite product of finite fields ∏ m j=1 F q j =K to a finite product of finite fields ∏ n i=1 F p i =F , where |K| and |F| are coprime. An (F,K) -linearly closed clonoid is a subset of these functions which is closed under composition from the right and from the left with linear mappings. We give a characterization of these subsets of functions through the F p [ K × ] -submodules of F K p , where K × is the multiplicative monoid of K= ∏ m i=1 F q i . Furthermore we prove that each of these subsets of functions is generated by a set of unary functions and we provide an upper bound for the number of distinct (F,K) -linearly closed clonoids.

The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of characteristic 2, Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic. Also, the author constructed a family of 2-generated restricted Lie algebras of slow polynomial growth with a nil p -mapping. Now, we construct a family of so called clover 3-generated restricted Lie algebras T(Ξ) , where a field of positive characteristic is arbitrary and Ξ an infinite tuple of positive integers. We prove that 1≤GKdimT(Ξ)≤3 , moreover, the set of Gelfand-Kirillov dimensions of clover Lie algebras with constant tuples is dense on [1,3] . We construct a subfamily of non-isomorphic nil restricted Lie algebras T( Ξ q,κ ) , where q∈N , κ∈ R + , with extremely slow quasi-linear growth of type: γ T( Ξ q,κ ) (m)=m( ln (q) m ) κ+o(1) , as m→∞ . The present research is motivated by a construction by Kassabov and Pak of groups of oscillating growth. As an analogue, we construct nil restricted Lie algebras of intermediate oscillating growth in another paper. We call them "Phoenix algebras" because, for infinitely many periods of time, the algebra is "almost dying" by having a "quasi-linear" growth as above, for infinitely many n the growth function behaves like exp(n/(lnn ) λ ) , for such periods the algebra is "resuscitating". The present construction of 3-generated nil restricted Lie algebras of quasi-linear growth is an important part of that result, responsible for the lower quasi-linear growth in that construction.

As an algebraic study of differential equations, differential algebras have been studied for a century and and become an important area of mathematics. In recent years the area has been expended to the noncommutative associative and Lie algebra contexts and to the case when the operator identity has a weight in order to include difference operators and difference algebras. This paper provides a cohomology theory for differential algebras of any weights. This gives a uniform approach to both the zero weight case which is similar to the earlier study of differential Lie algebras, and the non-zero weight case which poses new challenges. As applications, abelian extensions of a differential algebra are classified by the second cohomology group. Furthermore, formal deformations of differential algebras are obtained and the rigidity of a differential algebra is characterized by the vanishing of the second cohomology group.

In this article, we introduce a deformation cohomology of Leibniz superalgebras. Also, we introduce formal deformation theory of Leibniz superalgebras. Using deformation cohomology we study the formal deformation theory of Leibniz superalgebras.

We introduce a notion of left-symmetric Rinehart algebras, which is a generalization of a left-symmetric algebras. The left multiplication gives rise to a representation of the corresponding sub-adjacent Lie-Rinehart algebra. We construct left-symmetric Rinehart algebra from O -operators on Lie-Rinehart algebra. We extensively investigate representations of a left-symmetric Rinehart algebras. Moreover, we study deformations of left-symmetric Rinehart algebras, which is controlled by the second cohomology class in the deformation cohomology. We also give the relationships between O -operators and Nijenhuis operators on left-symmetric Rinehart algebras.