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We define root graded Lie superalgebras and study their connection with centerless cores of extended affine Lie superalgebras; our definition generalizes the known notions of root graded Lie superalgebras.

In this article, we give the classification of normal del Pezzo surfaces of rank one with at most log canonical singularities containing the affine plane defined over an algebraically non-closed field of characteristic zero. As an…

We study the combinatorics of fully commutative elements in Coxeter groups of type $H_n$ for any $n > 2$. Using the results, we construct certain canonical bases (i.e., IC bases) for non-simply-laced generalized Temperley--Lieb algebras and…

There was proposed the method of a factorization of PDE. The method is based on reduction of complicated systems to more easy ones (for example, due to dimension decrease). This concept is proposed in general case for the arbitrary PDE…

We are concerned with the arithmetic of solutions to ordinary or partial nonlinear differential equations which are algebraic in the indeterminates and their derivatives. We call these solutions D-algebraic functions, and their equations…

In this article algebraic constructions are introduced in order to study the variety defined by a radical parametrization (a tuple of functions involving complex numbers, $n$ variables, the four field operations and radical extractions). We…

We establish combinatorial formulas for the index of a class of matrix Lie algebras whose matrix forms are encoded by strict partial orderings.

To each associative unitary finite-dimensional algebra over a normal base, we associative a canonical multiplicative function called its determinant. We give various properties of this construction, as well as applications to the topology…

For any finite dimensional basic associative algebra, we study the presentation spaces and their relation with the representation spaces. We prove two propositions about a general presentation, one on its subrepresentations and the other on…

It is well-known that any solution of the Laplace equation is a real or imaginary part of a complex holomorphic function. In this paper, in some sense, we extend this property into four order hyperbolic and elliptic type PDEs. To be more…

The exactly integrable systems connected with semisimple series $A$ for arbitrary grading are presented in explicit form. Their general solutions are expressed in terms of the matrix elements of various fundamental representations of $A_n$…

We provide a clarification of the classification of two-dimensional algebras over an arbitrary base field. Using this clarification, we determine the number of non-isomorphic two-dimensional algebras over a finite field.

Using the skew-symmetry of the differential operators and multiplication operators in the canonical representations of finite-dimensional classical Lie algebras, we obtain some noncanonical polynomial representations of the classical Lie…

Let k be a commutative algebra with the field of the rational numbers included in k and let (E,p,i) be a cleft extension of A. We obtain a new mixed complex, simpler than the canonical one, giving the Hochschild and cyclic homologies of E…

We study the central extensions of Lie algebras graded by an irreducible locally finite root system.

This paper develops a general theory of canonical bases, and how they arise naturally in the context of categorification. As an application, we show that Lusztig's canonical basis in the whole quantized universal enveloping algebra is given…

We describe the affine canonical basis elements in the case when the affine quiver has arbitrary orientation. This generalizes Lusztig's description.

The enhanced group classification of a semi-linear generalization of a general bond-pricing equation is carried out by employing the underlying equivalence and additional equivalence transformations. The knowledge of the sub classes with…

Recently, a method to compute the implicit equation of a parametrized hypersurface has been developed by the authors. We address here some questions related to this method. First, we prove that the degree estimate for the stabilization of…

Using geometric methods for linearizing systems of second order cubically semi-linear ordinary differential equations and third order quintically semi-linear ordinary differential equations, we extend to the fourth order by differentiating…