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The universal gl-weight system and the chromatic polynomial
Journal of Geometry and Physics. 2026. Vol. 225. Article 105841.
Weight systems associated to the Lie algebras 𝔤𝔩(N) for N = 1,2,... can be unified into auniversal one. The construction is based on an extension of the 𝔤𝔩(N) weight systems to permutations. This universal weight system takes values in the algebra of polynomials C[N;C1,C2,...] in infinitely many variables. We show that under the substitution Cm = xNm−1, m = 1,2,..., the leading term in N of the value of the universal 𝔤𝔩 weight system becomes the chromatic polynomial of the intersection graph of the chord diagram. Moreover, we show that under the substitution Cm = pmNm−1, m = 1,2,..., the leading term in N of the value of the universal 𝔤𝔩-weight system determines a filtered Hopf algebra homomorphism from the rotational Hopf algebra of permutations, which we construct in the present paper, to the Hopf algebra of polynomials C[p1,p2,...].
Keywords: Hopf algebraweight systemchord diagramLie algebrachromatic polynomialFinite type knot invariants
Publication based on the results of:
Springer, 2026.
This book presents established and new research on the close connections between graph games and systems of logic, particularly existing and newly designed modal logics. The volume utilizes two graph games – the sabotage game and the hide-and-seek game – to demonstrate the natural interplay between designing new graph games and exploring new kinds of ...
Added: June 30, 2026
Pochinka O., Barinova M., Journal of Geometry and Physics 2026 Vol. 228 P. 1–8
In the present paper we consider an Ω-stable 3-diffeomorphism with a solid or thickened surfaced non-trivial basic set. Such basic sets include, for instance, all one-dimensional expanding attractors and those two-dimensional basic sets that are not expanding. We prove that the chain recurrent set of every such a diffeomorphism necessarily contains at least two non-trivial ...
Added: June 30, 2026
German O., Illarionov A., Известия РАН. Серия математическая 2026 Т. 90 № 3 С. 3–18
Пусть симплекс с целочисленными вершинами - содержащий ровно одну целочисленную точку, отличную от своих вершин. В работе доказывается, что если точка находится во внутренности симплекса или в относительной внутренности некоторой гиперграни симплекса, то объем симплекса ограничен величиной, зависящей только от размерности, в противном случае объем симплекса может быть сколь угодно большим. Этот результат применяется для вывода асимптотической формулы для среднего числа вершин полиэдров ...
Added: June 29, 2026
Ivchenko A., Nigmatullin R. R., Dorokhin S. V., Mathematics 2021 Vol. 9 No. 4 Article 381
n this paper, we focus on the generalization of the Hurst empirical law and suggest a set of reduced parameters for quantitative description of long-time series. These series are usually considered as a specific response of a complex system (economic, geophysical, electromagnetic and other systems), where successive fixations of external factors become impossible. We consider ...
Added: June 27, 2026
Ivchenko A., Shestoperov A. I., Fomina E. V., Microgravity Science and Technology 2025 Vol. 37 No. 19 P. 1–19
The paper is dedicated to the analysis of medico-biological data obtained during locomotor testing of astronauts. Accurate data interpretation plays a crucial role in locomotion system monitoring, prophylaxis of long-duration spaceflight negative effects and thus in the development of an autonomous medical support system for deep space expeditions. During the locomotor testing the astronaut changes ...
Added: June 26, 2026
Gadzhimirzaev S., Хельвас А. В., 2023 3rd International Conference on Innovative Research in Applied Science, Engineering and Technology (IRASET) Mohammedia, Morocco 2023 P. 1–6
The article proposes the architecture for eventdriven Emergency Operation Center with Machine Vision Component. Sources of information are analyzed and approaches to machine vision events for tactical situations detection and estimation are discussed. Messages from Machine Vision Components are converted to Common Alerting Protocol and processed by Operation Center environment for tactical situations recognition. ...
Added: June 26, 2026
Gadzhimirzaev S., Хельвас А. В., Лукьянченко П. П., Computer Research and Modeling 2023 Vol. 15 No. 1 P. 129–140
In this article we propose a new approach to the analysis of econometric industry parameters for the industry consolidation level. The research is based on the simple industry automatic control model. The state of the industry is measured by quarterly obtained econometric parameters from each industry’s company provided by the tax control regulator. An approach ...
Added: June 26, 2026
Gadzhimirzaev S., Хельвас А. В., International Frequency Sensor Association (IFSA) Publishing, 19-21 February 2025 Granada, Spain 2025 P. 172–176
The paper presents models for an innovative fully robotic warehouse for storing boxed goods. A discrete multiagent simulation of the movement of shuttles in a warehouse for a given sequence of pallet shipments has been implemented. Different strategies for placement of boxes in various areas of a warehouse are evaluated, as well as optimal routing ...
Added: June 26, 2026
Fedorov Timofey, Moscow Mathematical Journal 2026 Vol. 26 No. 1 P. 73–85
We obtain a complete list of smooth projective threefolds over C for which the dimension of the space of vanishing cycles (in H2(Y,Q) of the smooth hyperplane section Y) equals 2. We also obtain a complete list of rank 2 very ample vector bundles E on smooth projective surfaces with c2(E)=3. ...
Added: June 25, 2026
Воронеж: Издательский дом ВГУ, 2026.
В сборнике представлены материалы докладов и лекций, включенных в программу весенней математической школы. ...
Added: June 25, 2026
Kazaryan M., Krasilnikov E., Lando S. et al., Russian Mathematical Surveys 2025 Vol. 80 No. 6(486) P. 73–136
The paper is devoted to a description of the recent progress in understanding the extension of Lie algebra weight systems to permutations. Lie algebra weight systems are functions on chord diagrams arising naturally in Vassiliev's theory of finite-type knot invariants. These functions satisfy certain linear restrictions known as Vassiliev's 4-term relations.
Chord diagrams can be interpreted as fixed-point-free involutions in ...
Added: December 4, 2025
Vassiliev V., Pacific Journal of Mathematics 2023 Vol. 326 No. 1 P. 135–160
The topology of the space of subalgebras of the function space specified by collections of equality conditins f(x)=f(y) is studied and applied to the problems of interpolation theory and knot theory ...
Added: October 31, 2025
D. A. Matveev, Siberian Mathematical Journal 2025 Vol. 66 No. 3 P. 715–727
We consider an affine algebraic variety with a torus action of complexity one. It is known
that in this case homogeneous locally nilpotent derivations on the algebra of functions of this variety
are defined in terms of a polyhedral divisor. In the present paper, a formula is obtained for multiple
commutators of two homogeneous locally nilpotent derivations with ...
Added: July 21, 2025
Arzhantsev I., Russian Mathematical Surveys 2001 Vol. 56 No. 3 P. 569–571
Added: June 13, 2025
Arzhantsev I., Gayfullin S., Lopatkin V., Izvestiya. Mathematics 2025 Vol. 89 No. 3 P. 425–441
For a finite set of homogeneous locally nilpotent derivations of the algebra of polynomials in several variables, a finite dimensionality criterion for the Lie algebra generated by these derivations is known. The structure of the corresponding finite-dimensional Lie algebras was also described in previous works. In this paper, we obtain a finite dimensionality criterion for ...
Added: June 12, 2025
Arzhantsev I., Gayfullin S., Lopatkin V., Известия РАН. Серия математическая 2025 Т. 89 № 3 С. 5–22
For a finite set of homogeneous locally nilpotent derivations of the algebra of polynomials
in several variables, a finite dimensionality criterion for the Lie algebra generated by these
derivations is known. Also the structure of the corresponding finite-dimensional Lie algebras is
described in previous works. In this paper, we obtain a finite dimensionality criterion for a Lie
algebra generated ...
Added: May 31, 2025
Lando S., Yang Z., Annales de l'Institut Henri Poincare (D) Combinatorics, Physics and their Interactions 2025
n a recent paper by M. Kazarian and the second author, a recurrence for the Lie algebras so(N) weight systems has been suggested; the recurrence allows one to construct the universal so weight system. The construction is based on an extension of the so weight systems to permutations. Another recent paper, by M. Kazarian, N. Kodaneva, and the first author, shows that under the ...
Added: May 15, 2025
Zinova P., Kazaryan M., Математический сборник 2023 Т. 214 № 6 С. 87–109
В теории Васильева инварианты узлов конечного порядка описываются в терминах весовых систем – функций на хордовых диаграммах, удовлетворяющих четырехчленным соотношениям. В частности, крашеному многочлену Джонса соответствует весовая система, описываемая в терминах алгебры Ли sl2sl2. Согласно теореме Чмутова–Ландо значение этой весовой системы зависит лишь от графа пересечений хордовой диаграммы, что позволяет говорить о ее значениях на графах пересечений.
В настоящей статье мы ...
Added: February 4, 2025
Kodaneva N., Lando S., Journal of Geometry and Physics 2025 Vol. 210 Article 105421
Weight systems are functions on chord diagrams satisfying so-called Vassiliev’s 4-term relations. They are closely related to finite type knot invariants, see [31 Certain weight systems can be derived from graph invariants, see a recent account in [19]. Another main source of weight systems are Lie algebras, the construction due to D. Bar-Natan [3] and ...
Added: January 23, 2025
Stasenko R., Математический сборник 2023 Т. 214 № 4 С. 132–180
n Vinberg's works certain non-Abelian gradings of simple Lie algebras were introduced and investigated, namely, short SO3- and SL3-structures. We investigate a different kind of these, short SL2-structures. The main results refer to the one-to-one correspondence between such structures and certain special Jordan algebras.
Bibliography: 8 titles. ...
Added: March 18, 2024
Yang Z., Journal of Geometry and Physics 2023 Vol. 187 Article 104808
To a finite type knot invariant, a weight system can be associated, which is a function on chord diagrams satisfying so-called 4-term relations. In the opposite direction, each weight system determines a finite type knot invariant. In particular, a weight system can be associated to any metrized Lie algebra, and any metrized Lie superalgebra. However, computation ...
Added: March 24, 2023
Akbarov S. S., De Gruyter, 2022.
The term "stereotype space" was introduced in 1995 and is used for a category of locally convex spaces with surprisingly elegant properties. Its study gives an unexpected point of view on functional analysis that brings this field closer to other main branches of mathematics, namely, to algebra and geometry.
This volume contains:
the foundations of the theory ...
Added: June 1, 2022
Glutsyuk A., Israel Journal of Mathematics 2023 Vol. 258 No. 1 P. 137–184
Reflections from hypersurfaces act by symplectomorphisms on the space of oriented lines with respect to the canonical symplectic form. We consider an arbitrary infinitely-smooth hypersurface in Euclidean space that is either a global strictly convex closed hypersurface, or a germ of hypersurface. We deal with the pseudogroup generated by compositional ratios of reflections from it ...
Added: November 4, 2021
Krasilnikov E., Arnold Mathematical Journal 2021 Vol. 7 No. 4 P. 609–618
Chord diagrams and 4-term relations were introduced by Vassiliev in the late 1980. Various constructions of weight systems are known, and each of such constructions gives rise to a knot invariant. In particular, weight systems may be constructed from Lie algebras as well as from the so-called 4-invariants of graphs. A Chmutov–Lando theorem states that ...
Added: September 29, 2021