## Final Activity Report Summary - ALGEBRAIC LOOPS (Quasigroups, loops and related binary systems on algebraic manifolds)

Regular matrices over a given field form a group with respect to the usual matrix product. Algebraic groups are generalisations of matrix groups; having a beautiful mathematical theory with many applications in physics, chemistry and informatics. By dropping the associative law (xy)z=x(yz) of the product operation but keeping the ability of solving equations of the form ax=b and ya=b, we obtain the concept of algebraic quasigroups. We are interested in the following questions. 1) Which kind of weakened associativity ensures us to obtain structural theorems for algebraic quasigroups which are similar to properties of algebraic groups? 2) Which kind of weakened associativity enables us to apply methods and tools from group theory in order to settle problems of quasigroups?

It was known for a long time that by replacing the associativity with the Moufang identity (xy)(zx)=x((yz)x), both questions have positive answers. In the present research project, we investigated algebraic quasigroups which satisfy the right Bol identity ((xy)z)y=x((yz)y). Quite surprisingly, it turned out that for this class of quasigroups, Question 1 has a negative and Question 2 has a positive answer. For example, local algebraic quasigroups of Bol type need not to be birationally equivalent to global algebraic quasigroups, hence A. Weil's theorem on local algebraic groups does not hold in the Bol context.

The basic tool, which lets us transform non-associative problems to group theory, is the concept of 'loop folders', the terminology is due to M. Aschbacher. Using this concept and deep results from the theory of abstract and algebraic groups, we were able to construct rich classes of simple Bol loops. Simple Bol loops are Bol type quasigroups with a unit, having no proper factor loops. In the finite case, the question of their existence has been considered as one of the main open problems in the theory of loops and quasigroups for more than 30 years.

We obtained our results by combining the theory of abstract and algebraic groups with special non-associative techniques and by using many sophisticated computer calculations. Of these accomplishments we wrote six papers, at the end of this research project five of them are accepted in peer-reviewed scientific journals.

It was known for a long time that by replacing the associativity with the Moufang identity (xy)(zx)=x((yz)x), both questions have positive answers. In the present research project, we investigated algebraic quasigroups which satisfy the right Bol identity ((xy)z)y=x((yz)y). Quite surprisingly, it turned out that for this class of quasigroups, Question 1 has a negative and Question 2 has a positive answer. For example, local algebraic quasigroups of Bol type need not to be birationally equivalent to global algebraic quasigroups, hence A. Weil's theorem on local algebraic groups does not hold in the Bol context.

The basic tool, which lets us transform non-associative problems to group theory, is the concept of 'loop folders', the terminology is due to M. Aschbacher. Using this concept and deep results from the theory of abstract and algebraic groups, we were able to construct rich classes of simple Bol loops. Simple Bol loops are Bol type quasigroups with a unit, having no proper factor loops. In the finite case, the question of their existence has been considered as one of the main open problems in the theory of loops and quasigroups for more than 30 years.

We obtained our results by combining the theory of abstract and algebraic groups with special non-associative techniques and by using many sophisticated computer calculations. Of these accomplishments we wrote six papers, at the end of this research project five of them are accepted in peer-reviewed scientific journals.