A5012001341- Advanced Topics in Algebra
- Algebraic structures and combinatorial models
- Geometric and Algebraic Topology
- Homotopy and Cohomology in Algebraic Topology
- Finite Group Theory Research
- Nonlinear Waves and Solitons
- Rings, Modules, and Algebras
- semigroups and automata theory
- Advanced Algebra and Geometry
- Polynomial and algebraic computation
- Advanced Algebra and Logic
- Algebraic Geometry and Number Theory
- Coding theory and cryptography
- Mathematics and Applications
- Advanced Operator Algebra Research
- Advanced Theoretical and Applied Studies in Material Sciences and Geometry
- graph theory and CDMA systems
- Computability, Logic, AI Algorithms
- Advanced Combinatorial Mathematics
- Mathematical Dynamics and Fractals
- Matrix Theory and Algorithms
- Natural Language Processing Techniques
- Fusion and Plasma Physics Studies
- advanced mathematical theories
- Logic, programming, and type systems
Novosibirsk State University
2008-2024
Sobolev Institute of Mathematics
2013-2024
Russian Academy of Sciences
1991-2020
South China Normal University
2010-2020
Siberian Branch of the Russian Academy of Sciences
1996-2020
University of Hong Kong
2008
Institute of Mathematics and Informatics
2003
Czech Academy of Sciences, Institute of Mathematics
2003
Chang Jung Christian University
2003
Korea Institute for Advanced Study
1999
In this survey, we formulate the Gröbner-Shirshov bases theory for associative algebras and Lie algebras. Some new Composition-Diamond lemmas applications are mentioned.
In this paper, we define the Gröbner–Shirshov basis for a dialgebra. The Composition–Diamond lemma dialgebras is given then. As result, give bases universal enveloping algebra of Leibniz algebra, bar extension dialgebra, free product two dialgebras, and Clifford We obtain some normal forms algebras mentioned above.
Abstract *Supported in part by the Russia's Fund of Fundamental Research. Acknowledgments Notes
A Gröbner-Shirshov basis for the Lie algebra n , abstractly defined by generators h i x y i=1,..., and Serre relations Cartan matrix over a field k of characteristic ≠2 is constructed. It consists together with following relations: [Formula: see text] j≥1, i≥2, i+j≤n same 1 ,…, where [z z 2 …z m ] we mean … ]]. As an application get direct proof that as defined, isomorphic to sℓ +1 (k).
In this paper, we review Shirshov's method for free Lie algebras invented by him in 1962 which is now called the Groebner-Shirshov bases theory.
In this paper we prove the existence of a finitely presented Lie algebra over an arbitrary field in which word problem is unsolvable.
In this paper, by using Gröbner–Shirshov bases, we show that in the following classes, each (respectively, countably generated) algebra can be embedded into a simple two-generated) algebra: associative differential algebras, Ω-algebras, λ-differential algebras. We generated over countable field k two-generated semigroups, Lie give another proofs of well known theorems: group algebra, semigroup, algebra) algebra).
In this paper, we firstly establish Composition-Diamond lemma for $\Omega$-algebras. We give a Gr\"{o}bner-Shirshov basis of the free $L$-algebra as quotient algebra $\Omega$-algebra, and then normal form is obtained. secondly $L$-algebras. As applications, bases dialgebra product two $L$-algebras, show four embedding theorems $L$-algebras: 1) Every countably generated can be embedded into two-generated $L$-algebra. 2) simple 3) over countable field 4) Three arbitrary $L$-algebras $A$, $B$,...
We establish Gr\"{o}bner-Shirshov bases theory for Gelfand-Dorfman-Novikov algebras over a field of characteristic $0$. As applications, PBW type theorem in Shirshov form is given and we provide an algorithm solving the word problem with finite homogeneous relations. also construct subalgebra one generated free algebra which not free.