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We generalize the classical construction principles of infinite-dimensional real (and complex) Lie groups to the case of Lie groups over non-discrete topological fields. In particular, we discuss linear Lie groups, mapping groups, test…

Theory of representations of universal algebra is a natural development of the theory of universal algebra. In the book, I considered representation of universal algebra, diagram of representations and examples of representation. Morphism…

The goal of this paper is to consider some relations between varieties of representations of groups and varieties of associative algebras. The main emphasis is put on the varieties of representations of groups induced by the varieties of…

In this paper, first we give the notion of a representation of a relative Rota-Baxter Lie algebra and introduce the cohomologies of a relative Rota-Baxter Lie algebra with coefficients in a representation. Then we classify abelian…

In order to compute Hermitian forms on representations of real reductive groups, in the unequal rank case, it is necessary to compute twisted Kazhdan-Lusztig-Vogan polynomials. These were defined by Lusztig and Vogan (Quasisplit Hecke…

This paper introduces Lie groups in degenerate geometric (Clifford) algebras that preserve four fundamental subspaces determined by the grade involution and reversion under the adjoint and twisted adjoint representations. We prove that…

Cylindrical algebraic decompositions (CADs) are a key tool in real algebraic geometry, used primarily for eliminating quantifiers over the reals and studying semi-algebraic sets. In this paper we introduce cylindrical algebraic…

In this paper we study certain category of smooth modules for reductive $p$--adic groups analogous to the usual smooth complex representations but with the field of complex numbers replaced by a $\mathbb Q$--algebra. We prove some…

This thesis studies arithmetic of linear algebraic groups. It involves studying the properties of linear algebraic groups defined over global fields, local fields and finite fields, or more generally the study of the linear algebraic groups…

Abstract separation systems provide a simple general framework in which both tree-shape and high cohesion of many combinatorial structures can be expressed, and their duality proved. Applications range from tangle-type duality and tree…

Representation determines how we can reason about a specific problem. Sometimes one representation helps us find a proof more easily than others. Most current automated reasoning tools focus on reasoning within one representation. There is,…

We investigate the representations and the structure of Hecke algebras associated to certain finite complex reflection groups. We first describe computational methods for the construction of irreducible representations of these algebras,…

We give parameterizations of the irreducible representations of finite groups of Lie type in their defining characteristic.

While concepts and tools from Theoretical Computer Science are regularly applied to, and significantly support, software development for discrete problems, Numerical Engineering largely employs recipes and methods whose correctness and…

The scope of this teaching package is to make a brief induction to Artificial Neural Networks (ANNs) for people who have no previous knowledge of them. We first make a brief introduction to models of networks, for then describing in general…

We give a detailed description of the torsors that correspond to multiloop algebras. These algebras are twisted forms of simple Lie algebras extended over Laurent polynomial rings. They play a crucial role in the construction of Extended…

In this paper it is shown that the wave front set of a direct integral of singular, irreducible representations of a real, reductive algebraic group is contained in the singular set. Combining this result with the results of the first paper…

This paper is a sequel to arXiv:1401.6321. We define and study representation categories based on Deligne categories Rep(GL_t), Rep(O_t), Rep(Sp_2t), where t is any (non-integer) complex number. Namely, we define complex rank analogs of the…

\noindent 1. Generalities\hfil\break 2. Lie groups and Lie algebras\hfil\break 3. The unitary groups\hfil\break 4. Representations of the SU(n) groups (and of their algebras)\hfil\break 5. The tensor method for unitary groups, and\hb the…

Complexity theory offers a variety of concise computational models for computing boolean functions - branching programs, circuits, decision trees and ordered binary decision diagrams to name a few. A natural question that arises in this…