The history of the laboratory of  “algebra and its applications” goes back to the department of “mathematical analysis and mechanics”, which began its work when the Institute of Mathematics was founded. The department of “mathematical analysis and mechanics” was founded on January 16, 1944. From 1944 to 1960 the department was staffed by such scientists as N.N. Nazarov, M.T.Urazbaev, D.K. Karimov, I.S. Arjanikh, M.S. Areshev, M.F. Shulgin.

In the early years, the department of “mathematical analysis and mechanics” was headed by M.S.Areshev, but in 1956 the department was divided into “mathematical analysis” and “theoretical mechanics”. The department of “mathematical analysis” was headed by S.Kh.Sirojiddinov, and the department of “theoretical mechanics” was headed by I.S. Arjanikh. In 1956-1958 S.Kh.Sirojiddinov, in 1958-1960 N.P. Romanov, in 1960-1980 I.S. Arjanikh, and in 1980-1985 G.P. Matviyevskaya worked in the department of "Mathematical analysis" as its director.

In 1979, the department of "functional analysis" was established at the Institute of Mathematics, headed by T.A. Sarimsakov from 1979 to 1986. In 1986, “mathematical analysis” and “functional analysis” were merged to form “algebra and analysis”. In 1986-1988 T.A. Sarimsakov was the head of the department "algebra and analysis", in 1988-2018 Sh.A. Ayupov was the head of the department. From 2018 to 2019, U.A. Rozikov headed the department.

On July 9, 2019, the President of the Republic of Uzbekistan issued a decree “On state support for the further development of mathematics education and science, as well as measures to radically improve the activities of V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences.” According to the PP-4387, the department “algebra and analysis” was renamed to the laboratory “algebra and its applications”.

At present, in the laboratory “algebra and its applications” acad. Sh.A.Ayupov, acad. Dj.Khadjiev, prof. U.A.Rozikov, prof. B.A.Omirov, DSc. I.S.Rakhimov, DSc. U.U.Jamilov, DSc. F.Kh.Eshmatov, DSc. A.Kh.Khudoyberdiyev, PhD. R.A.Dadaxodjaev, PhD. J.Q.Adashev, PhD. O.N.Khakimov, PhD. R.M.Turdibaev, PhD. K.K. Abdurasulov carry out scientific activities.

Researchers in the Laboratory of “Algebra and its applications” are currently working in the following areas:

operator algebras, Jordan algebras of joint operators and Jordan Banach algebras, derivations in operator algebras, derivations and automorphisms in unlimited operator algebras in Gilbert spaces and their applications in quantum dynamics, non-associative algebras, Lie and Jordan algebra algebras and superalgebras, Zinbiel algebras, deformations and cohomologies of algebras, p-adic analysis, evolution algebras and their applications, algebraic geometry, group and graph theory, Gibbs measures, random variables in disordered fields, p-adic pop static physics, p-adic statistical physics.

Operator algebras:

·        The derivations of the algebra of all local dimensional operators, the algebra of all dimensional operators, Arens algebras associated with von Neuemann algebras and the exact normal trace are classified.

·        In the algebra of T-compact operators connected to a half-finite infinite specific background Neumann algebra, the operation of derivation is proved to be automatically continuous.

·        It is shown that derivation is formed by the elements of the algebra of all T-dimensional operators connected to von-Neumann algebras.

·        Description of local derivations of regular commutative algebras and algebras of dimensional operators connected to type I von-Neumann algebras are given.

·        Conditions for the existence of non-derivation local derivation in commutative unit regular algebras are found.

·        Local derivations of algebras of dimensional operators with respect to type I von-Neumann algebras without Abel component is described.

Dynamic systems:

·        All extreme Volterra operators identified in three- and four-dimensional simplexes are classified as ergodic or non-ergodic. It is shown that quadratic stochastic operators corresponding to graphs have a single fixed point, and that the arbitrary trajectory of such operators tends to a fixed point faster than any geometric progression.

·        The arbitrary trajectory of quadratic stochastic operators corresponding to any finite Abel group or the approximation of a periodic or fixed point is proved. It is shown that an arbitrary fixed no Volterra operator in a two-dimensional simplex has a single fixed point, and that such operators have two- and three-period trajectories.

·        The quadratic stochastic operator Volterra with a two-period trajectory of a bisexual population is constructed. A sufficient condition for the inertia of quadratic stochastic operators has been found for such a population, and this condition, found in a small-sized simplex, is also shown to be a necessary condition for inertia.

·        The infinite-dimensional simplex includes the quadratic stochastic operators Volterra and the block Volterra, for which a set of limit points on the strong and weak approximations of the trajectory is defined. The existence of nonergodic Volterra operators has been proven using the classification of these two types of limit points.

·        Criteria have been found for the relationship between the susceptibility of Markov operators and the preservation of orthogonality in infinite-dimensional simplex. It has been proven that such properties of Markov's operators do not overlap with those of finite-dimensional simplex. It is also shown that the subjectivity of Markov operators has applications such as finding positive solutions of some integral equations.

·     Quadratic stochastic operators in space  have been identified and their relationship to discrete quadratic stochastic operators in infinite-dimensional simplex has been studied. It is shown that the projective surrogate of quadratic stochastic operators is derived from the surrealism of discrete operators.

 Leibniz algebras:

·        Consequences such as Engel's theorem, Lie's theorem, Cartan's solvability criteria, and the addition of Cartan subalgebras, which are classical results in the theory of finite-dimensional nilpotent Lie algebras, are also valid for Leibniz algebras.

·        Classification of zero-filiform, naturally graded filiform, quasi-filiform and p-filiform Leibniz algebras are obtained.

·        All finite-dimensional nilpotent Leibniz superalgebras with a maximum nil-index and nilindex equal to the dimension of the superalgebra are classified.    

·        Several classes of solvable Leibniz algebras with nilradicals have been classified, and it is proved that solvable Leibniz algebras with codimension is equal to the number of generators are cohomologically rigid.

·        Lower-level algebras are studied and classification of algebras of level one and two are obtained.

·        Algebraic and geometric classifications of 4-dimensional nilpotent Novikov, right commutative and left symmetric algebras are obtained.

·        It is proved that all local and 2-local derivations of simple and semi-simple Lie algebras are derivations in the simplest sense.

·        It is shown that an arbitrary nilpotent Lie algebra has 2-local derivation, which is not a derivation.  

Gibbs measures:

·        The Gibbs measures theory was developed for the models of statistical physics given in the Cayley tree, such as Potts, SOS, HC, XY, Ising-Vannimenus. In particular, all translational-invariant Gibbs measures were classified for the Potts model.

·        Periodic, weak periodic Gibbs measures were built for HC, SOS, and Potts models. Critical temperature points are defined so that the constructed measurements (for a given model) are the endpoints of all Gibbs sets of measures.

·        Gibbs measures were constructed for models with infinite spin values ​​and phase shifts were determined depending on the temperature. The Gibbs gradient measures were constructed for an SOS model with a finite spin value. For the SOS model given in the 2nd order Kelly tree, all translation-invariant and 4 periodic gradient Gibbs set of measures were described in terms of temperature variation.

·        Introduced the concept of quantum Markov states in a tree, which is a generalization of the Cayley tree.

·        In the theory of noarhimed Gibbs measures, the concept of a generalized p-adic Gibbs measure was introduced. Generalized p-adic Gibbs measures for Ising-Vannimenus, Potts models.

·        In contrast to the actual Gibbs measures for the Potts model given in the Cayley tree, it has been shown that a set of Gibbs measures with a p-adic value can include any periodic measure. The translational-invariant Gibbs measure classification for this model also proved to differ sharply depending on the tree order in the field of real and p-adic numbers.

Algebraic geometry:

•                    For Koszul Calabi-Yau algebras, the existence of a shifted bi-simple structure meaning Cravley-Boevey-Etingof-Ginzburg is shown.

•                    Negative cyclic homology of several classes of algebra or gravity of algebraic structure by cyclic cohomology.

•                    Relationships between Calabi-Yau algebras, symmetric Frobenius algebras, Poisson algebras of the same modulus, and Frobenius Poisson algebras of the same modulus have been identified.

•                    In 2017, Sh.A.Ayupov, K.K.Kudaybyergenov, B.A.Omirov U.A.Rozikov were awarded the State Prize of the Republic of Uzbekistan of the 1st degree.

•                    In 2021, Sh.A.Ayupov was awarded the title of "Hero of Uzbekistan" and the highest award - "Gold Star" medal.

•                    U.A. Rozikov 2017 Springer Nature Top Author International Award; In 2018 he was elected an academician of the World Academy of Sciences (TWAS) and in 2020 was awarded the COMSTECH International Award "Best Scientific Article" (Rozikov UA Jour. Math. Biology, 2017. V.75, No. 6-7, p.1715—1733). ) was awarded.

The department actively cooperates with many institutes and universities, including:

•                    University of San Diego (USA)

•                    University of São Paulo (Brazil)

•                    University of Bonn (Germany)

•                    Bielefeld University (Germany)

•                    Rhein University  (Germany)

•                    University of Trieste (Italy)

•                    University of Cambridge (Great Britain)

•                    University of Leeds (Great Britain)

•                    University of Seville (Spain)

•                    University of Santiago de Compostela (Spain)

•                    University of Granada (Spain)

•                    University of Adelaide (Australia)

•                    University of Sydney (Australia)

•                    University of Strasbourg (France)

•                    Aix-Marseille University (France)

•                    University of Nantes (France)

•                    University of Paris (France)

•                    Kazan Federal University (Russia)