doc. RNDr. Natalie Žukovec, Ph.D.

A base of the free alternative nil-superalgebra of index n, for n3, on one odd generator is constructed. In particular, its index of solvability is computed. As a corollary over a field of characteristic zero an infinite family of solvable alternative algebras which are not associative of arbitrary big solvability index with explicit multiplication table is obtained.

Some results on central and nuclear skew-symmetric elements in alternative algebras are presented.

As mentioned in previous articles, we try to obtain some general results in problematic of finding bases and the indices of solvability of alternative nil-algebras of different indices using the superalgebra method. This way, we constructed the bases of the free alternative nil-superalgebras on one odd generator of different indices, more precisely for the nil index 2 and 3 respectively (published in [1,3]). Finding of the base of the alternative nil-superalgebra of nil index 4 and 5, is in process.

A base of the free alternative nil-superalgebra of index 3 on one odd generator is constructed. In particular, its index of solvability is computed. We consider also the corresponding Grassmann algebra and show that the well-known Dorofeev's example of solvable non-nilpotent alternative algebra is its homomorphic image

It is a short survey of obtained results on alternative superalgebras with one odd generator.

Quadratic alternative superalgebras are introduced and their super- identities and central functions on one odd generator are described. As a corollary, all multilinear skew-symmetric identities and central polynomials of octonions are classied.

A base of the free alternative superalgebra on one odd generator is constructed. As a corollary, a base of the alternative Grassmann algebra is given. We also find a new element of degree 5 from the radical of the free alternative algebra of countable rank.

A base of the free alternative superalgebra on one odd generator is constructed. As a corollary, a base of the alternative Grassmann algebra is given. We also find a new element of degree 5 from the radical of the free alternative algebra of countable rank.

A base of the universal multiplicative envelope of free Malcev superalgebra M on one odd generator is constructed. Some corollaries for skew-symmetric functions and central elements in free Malcev and free alternative algebras are obtained. Moreover, a base of the Poisson Malcev superalgebra of M is constructed. As a corollary, a set of elements that spans the free alternative superalgebra on one odd generator is obtained.

It is proved that every Malcev superalgebra generated by an odd element is special, that is, isomorphic to a subsuperalgebra of the commutator Malcev superalgebra $A^-$ for a certain alternative superalgebra $A$. As a corollary, it is shown that the kernel of the natural homomorphism of the free Malcev algebra $Malc[X]$ of countable rank into the commutator Malcev algebra $Alt[X]^-$ of the corresponding free alternative algebra $Alt[X]$, does not contain skew-symmetric multilinear elements. In other words, there are no skew-symmetric Malcev $s$-identities. Another corollary is speciality of the Malcev Grassmann algebra.

A base of the Malcev Poisson superalgebra related with the free Malcev superalgebra on one odd generator is constructed. As a corollary, a pre-base, that is, a set of elements that spans the free alternative superalgebra on one odd generator is obtained.

A base of the universal multiplicative envelope for the free Malcev superalgebra on one odd generator is constructed. Some corollaries for central skew-symmetric elements in free Malcev and free alternative algebras are obtained. In particular, the Malcev Grassmann algebra is constructed.

We examine two particular constructions that derive from a \mbox{$2$-group} $G=G(\cdot)$ another $2$-group $G(*)$ for the case when $G(\cdot)$ is one of $D_{2^n}$, $SD_{2^n}$, $Q_{2^n}$. The constructions (the cyclic one and the dihedral one) have the property that $x*y=x\cdot y$ for exactly $3/4$ of all pairs $(x,y)\in G\times G$.

We study Kuzmin's conjecture on the index of nilpotency for the variety $\M_5$ of associative nil-algebras of degree 5. Due to Vaughan-Lee~\cite{vau2} the problem is reduced to that for $k$-generator $\M_5$-superalgeb\-ras, where $k\leq 5$. We confirm Kuzmin's conjecture for 2-generator superalgebras proving that they are nilpotent of degree 15

We establish a one-to-one correspondence between the irreducible finite degree partial representations of a group $G$ and the (usual) irreducible representations of certain ideals of a groupoid algebra constructed from $G$. We derive a structural result about the irreducible partial representations on finite dimensional vector spaces and give the description ``up to usual representations'' of the irreducible partial representations of abelian groups of degrees $\leq4$. We treat simultaneously irreducible and indecomposable partial representations