## Mid-Term Report Summary - COIMBRA (Combinatorial methods in noncommutative ring theory.)

The main objectives of this project are to:

Study basic open questions on infinite dimensional associative noncommutative algebras; tackle problems from a number of related areas of mathematics using noncommutative ring theory; develop new ways to approach existing problems that could benefit future researchers. In particular, the project members aim to create conceptual tools to help with the study of a variety of fundamental algebraic structures, for example tools related to the Diamond Lemma and the Golod-Shafarevich theorem.

Project Achievements: We have developed several new conceptual tools for the study of a variety of fundamental algebraic structures, some of these directed toward the Diamond Lemma and the Golod-Shafarevich theorem. We have then used these methods to solve several longstanding open questions in noncommutative ring theory. We have also used algebraic methods to investigate topics from other research areas; for example mathematical physics, group theory and numerical analysis.

A brief summary of new techniques which we have developed to-date now follows, along with results obtained by us in consequence.

1. For working with noncommutative rational identities; this facilitated proof of a longstanding Kontsevich conjecture on noncommutative birational transformations. Our result shows that matrix rings over quotients of free algebras satisfy some quite surprising and strong properties.

2. For working with differential polynomial rings; by applying these methods we have shown that Amitsur's theorem on polynomial rings cannot be generalised to differential polynomial rings. This answers an open question which goes back to the 1980's.

3. For the study of adjoint groups of Jacobson radical rings, and adjoint groups of nil rings and nilpotent rings; using these methods it was shown that the adjoint group of a nil algebra need not be an Engel group. This answered an open question of Amberg and Sysak from 1998, and in the case of an uncountable field a recent question by Zelmanov.

4. For constructing algebras with a given growth function satisfying prescribed relations, provided that the number of prescribed relations of a given degree are small enough; by applying these methods, a first example of finitely generated algebraic algebras with growth strictly between polynomial and exponential was constructed, thus answering a recent question by Zelmanov.We described the growth and the Gelfand-Kirillov dimension of homomorphic images of the Golod Shafarevich algebras.

5. New and unconventional ways to use the Diamond lemma which allowed us to reprove in a constructive, combinatorial way some old results on Sklyanin algebras. This disproved the commonly accepted belief that it is impossible to achieve such objectives by purely algebraic and combinatorial means such as the Groebner basis technique.

We have also obtained the following results and observations.

6. That there exists a finitely generated nil algebra A such that the tensor product of this algebra with itself is not nil nil; hence we have obtained a `tensor product' analogon of the Golod construction of nil not nilpotent algebra. This answers a question by Puczylowski from 1993. We have also solved another old open problem from noncommutative ring theory by showing that the nil radical in Z-graded rings is homogeneous. This is an analogon of Bergman`s result that the Jacobson radical of a Z-graded ring is homogeneous.

7. We studied questions related to Anick's conjecture on quadratic algebras. Our main result deals with the asymptotic version of Anick's conjecture, and observation that for step nilpotent quadratic algebras this asymptotic result become precise when n is from the Fibonacci sequence.

8. New algebraic ideas were developed to answer a question of Shestakov on differential polynomial rings, by showing that differential polynomial rings over locally nilpotent rings need not be Jacobson radical.

9. That graded domains with cubic growth have a classical Krull dimension.

10. We have investigated some topics in numerical analysis (with collaborators). It was shown that some commonly used algorithms are mixed forward-backward stable.

11. We express the K-theory of graph algebras via combinatorial characteristics of underlying graphs.

12. We have investigated the structure of braces. Braces are a generalisation of Jacobson radical rings and have been introduced by Rump in 2005 to investigate set-theoretical, involutive, non-degenerate solutions of the Yang-Baxter equation. We showed that the adjoint group of a finite left brace is nilpotent if and only if this brace is nilpotent. We also showed that the structure group of a finite brace cannot be an Engel group. It was also shown that every finite solution of the Yang-Baxter equation can be in a convenient way embedded into a finite brace. And finally that every non-degenerate involutive set theoretic solution of the Yang-Baxter equation whose structure group has the cardinality which is a cube free number is a multipermutation solution and hence could be well understood.

Study basic open questions on infinite dimensional associative noncommutative algebras; tackle problems from a number of related areas of mathematics using noncommutative ring theory; develop new ways to approach existing problems that could benefit future researchers. In particular, the project members aim to create conceptual tools to help with the study of a variety of fundamental algebraic structures, for example tools related to the Diamond Lemma and the Golod-Shafarevich theorem.

Project Achievements: We have developed several new conceptual tools for the study of a variety of fundamental algebraic structures, some of these directed toward the Diamond Lemma and the Golod-Shafarevich theorem. We have then used these methods to solve several longstanding open questions in noncommutative ring theory. We have also used algebraic methods to investigate topics from other research areas; for example mathematical physics, group theory and numerical analysis.

A brief summary of new techniques which we have developed to-date now follows, along with results obtained by us in consequence.

1. For working with noncommutative rational identities; this facilitated proof of a longstanding Kontsevich conjecture on noncommutative birational transformations. Our result shows that matrix rings over quotients of free algebras satisfy some quite surprising and strong properties.

2. For working with differential polynomial rings; by applying these methods we have shown that Amitsur's theorem on polynomial rings cannot be generalised to differential polynomial rings. This answers an open question which goes back to the 1980's.

3. For the study of adjoint groups of Jacobson radical rings, and adjoint groups of nil rings and nilpotent rings; using these methods it was shown that the adjoint group of a nil algebra need not be an Engel group. This answered an open question of Amberg and Sysak from 1998, and in the case of an uncountable field a recent question by Zelmanov.

4. For constructing algebras with a given growth function satisfying prescribed relations, provided that the number of prescribed relations of a given degree are small enough; by applying these methods, a first example of finitely generated algebraic algebras with growth strictly between polynomial and exponential was constructed, thus answering a recent question by Zelmanov.We described the growth and the Gelfand-Kirillov dimension of homomorphic images of the Golod Shafarevich algebras.

5. New and unconventional ways to use the Diamond lemma which allowed us to reprove in a constructive, combinatorial way some old results on Sklyanin algebras. This disproved the commonly accepted belief that it is impossible to achieve such objectives by purely algebraic and combinatorial means such as the Groebner basis technique.

We have also obtained the following results and observations.

6. That there exists a finitely generated nil algebra A such that the tensor product of this algebra with itself is not nil nil; hence we have obtained a `tensor product' analogon of the Golod construction of nil not nilpotent algebra. This answers a question by Puczylowski from 1993. We have also solved another old open problem from noncommutative ring theory by showing that the nil radical in Z-graded rings is homogeneous. This is an analogon of Bergman`s result that the Jacobson radical of a Z-graded ring is homogeneous.

7. We studied questions related to Anick's conjecture on quadratic algebras. Our main result deals with the asymptotic version of Anick's conjecture, and observation that for step nilpotent quadratic algebras this asymptotic result become precise when n is from the Fibonacci sequence.

8. New algebraic ideas were developed to answer a question of Shestakov on differential polynomial rings, by showing that differential polynomial rings over locally nilpotent rings need not be Jacobson radical.

9. That graded domains with cubic growth have a classical Krull dimension.

10. We have investigated some topics in numerical analysis (with collaborators). It was shown that some commonly used algorithms are mixed forward-backward stable.

11. We express the K-theory of graph algebras via combinatorial characteristics of underlying graphs.

12. We have investigated the structure of braces. Braces are a generalisation of Jacobson radical rings and have been introduced by Rump in 2005 to investigate set-theoretical, involutive, non-degenerate solutions of the Yang-Baxter equation. We showed that the adjoint group of a finite left brace is nilpotent if and only if this brace is nilpotent. We also showed that the structure group of a finite brace cannot be an Engel group. It was also shown that every finite solution of the Yang-Baxter equation can be in a convenient way embedded into a finite brace. And finally that every non-degenerate involutive set theoretic solution of the Yang-Baxter equation whose structure group has the cardinality which is a cube free number is a multipermutation solution and hence could be well understood.