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

The main objectives of the COIMBRA project were to: i) study basic open questions on infinite dimensional associative noncommutative algebras and ii) tackle problems from a number of other areas of mathematics using noncommutative ring theory. A significant portion of the project was devoted to studying the structural impact of nilpotency and interactions between

the notions of the nil, the nilpotent and the Jacobson radical. Recall that the Jacobson radical is important for structure theorems in noncommutative ring theory,

where one divides a given ring by its Jacobson radical before applying the main structure theorems. It is worth noticing that Jacobson radical rings have found applications in solutions for the quantum Yang-Baxter equation as a part of brace theory (introduced by Rump in 2005).

The problems that we investigated involved various other topics in noncommutative ring theory, such as free algebras, differential polynomial rings, growth of algebras and division rings.

With respect to the first objective, we solved several basic long standing open questions on infinite dimensional associative noncommutative algebras. These questions had been left unanswered for about 20 years and were as follows:

(1) Kontsevich conjecture on noncommutative birational transformations (PDRA Iyudu and S. Shkarin)

(2) Amberg-Sysak question on the adjoint group of a nil ring which was not an Engel group (the PI)

(3) question of existence of analogon of Amitsur's theorem on polynomial rings for differential polynomial rings (the PI)

(4) Ufnarovski's question on automaton algebras (PDRA Iyudu and S. Shkarin) (5) Polishchuk and Positselski's questions from their book on 'Quadratic algebras' (PDRA Iyudu and Skharin)

(6) Puczylowski's question on when tensor product of algebras is nil and not nilpotent (the PI),

(7) An analogon of Berman's result on the Jacobson radical of graded rings for nil ideals (the PI) and (8) Latyshev's problem on nilpotency

(PDRA Malev and I. Ivanov-Pogodayev).

We also obtained several results on multiplicative commutators in division rings, and showed how they influence the structure of division rings (PDRA Aghabali and his several collaborators). With our collaborators we solved several open questions on interactions between the notions of the nil, the nilpotent and the Jacobson radical, for example Zelmanov's question on growth of nil algebras, Shestakov's question on the Jacobson radical of differential polynomial rings and Nielsen-Ziembowski question on polynomial rings over Baer radical rings. We also partially solved Anick's questions on growth of algebras and nilpotent algebras (asked in 1980's) and several other open questions.

Regarding the second objective, we used algebraic methods to investigate topics from several other research areas.

In particular we investigated noncommutative rings and other structures originating in mathematical physics, Hopf-Galois extensions and geometry.

We developed new conceptual tools for studying such structures

as follows:

(1) The PI solved an open problem of W. Donovan and M. Wemyss related to geometric argument and Toda by showing that the degree of a potential algebra and of an Acon is always at least 8.

(2) The PDRA Iyudu, jointly with S. Shkarin, showed that potential algebras are infinitely dimensional provided that the potential has only terms of degree 4 or higher.

(3) We obtained several results related to braces and skew braces, Hopf-Galois extensions and set-theoretic solutions of the Yang-Baxter equation which are too numerous to list here (the PI, PDRA Vendramin, PDRA Zenouz and their collaborators).

The above results comprise 26 peer reviewed and published papers and a further 3 accepted for publication obtained during the COIMBRA project.

the notions of the nil, the nilpotent and the Jacobson radical. Recall that the Jacobson radical is important for structure theorems in noncommutative ring theory,

where one divides a given ring by its Jacobson radical before applying the main structure theorems. It is worth noticing that Jacobson radical rings have found applications in solutions for the quantum Yang-Baxter equation as a part of brace theory (introduced by Rump in 2005).

The problems that we investigated involved various other topics in noncommutative ring theory, such as free algebras, differential polynomial rings, growth of algebras and division rings.

With respect to the first objective, we solved several basic long standing open questions on infinite dimensional associative noncommutative algebras. These questions had been left unanswered for about 20 years and were as follows:

(1) Kontsevich conjecture on noncommutative birational transformations (PDRA Iyudu and S. Shkarin)

(2) Amberg-Sysak question on the adjoint group of a nil ring which was not an Engel group (the PI)

(3) question of existence of analogon of Amitsur's theorem on polynomial rings for differential polynomial rings (the PI)

(4) Ufnarovski's question on automaton algebras (PDRA Iyudu and S. Shkarin) (5) Polishchuk and Positselski's questions from their book on 'Quadratic algebras' (PDRA Iyudu and Skharin)

(6) Puczylowski's question on when tensor product of algebras is nil and not nilpotent (the PI),

(7) An analogon of Berman's result on the Jacobson radical of graded rings for nil ideals (the PI) and (8) Latyshev's problem on nilpotency

(PDRA Malev and I. Ivanov-Pogodayev).

We also obtained several results on multiplicative commutators in division rings, and showed how they influence the structure of division rings (PDRA Aghabali and his several collaborators). With our collaborators we solved several open questions on interactions between the notions of the nil, the nilpotent and the Jacobson radical, for example Zelmanov's question on growth of nil algebras, Shestakov's question on the Jacobson radical of differential polynomial rings and Nielsen-Ziembowski question on polynomial rings over Baer radical rings. We also partially solved Anick's questions on growth of algebras and nilpotent algebras (asked in 1980's) and several other open questions.

Regarding the second objective, we used algebraic methods to investigate topics from several other research areas.

In particular we investigated noncommutative rings and other structures originating in mathematical physics, Hopf-Galois extensions and geometry.

We developed new conceptual tools for studying such structures

as follows:

(1) The PI solved an open problem of W. Donovan and M. Wemyss related to geometric argument and Toda by showing that the degree of a potential algebra and of an Acon is always at least 8.

(2) The PDRA Iyudu, jointly with S. Shkarin, showed that potential algebras are infinitely dimensional provided that the potential has only terms of degree 4 or higher.

(3) We obtained several results related to braces and skew braces, Hopf-Galois extensions and set-theoretic solutions of the Yang-Baxter equation which are too numerous to list here (the PI, PDRA Vendramin, PDRA Zenouz and their collaborators).

The above results comprise 26 peer reviewed and published papers and a further 3 accepted for publication obtained during the COIMBRA project.