V. M. Buchstaber, D. V. Leikin, “Polynomial Lie Algebras”, Funktsional. Anal. i Prilozhen., 36:4 (2002), 18–34; Funct. Anal. Appl., 36:4 (2002), 267

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Funktsional'nyi Analiz i ego Prilozheniya, 2002, Volume 36, Issue 4, Pages 18–34
DOI: https://doi.org/10.4213/faa216 (Mi faa216)

This article is cited in 35 scientific papers (total in 36 papers)

Polynomial Lie Algebras

V. M. Buchstaber^a, D. V. Leikin^b

^a Steklov Mathematical Institute, Russian Academy of Sciences
^b Institute of Magnetism, National Academy of Sciences of Ukraine

Full-text PDF (240 kB) Citations (36)

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DOI: https://doi.org/10.4213/faa216

Abstract: We introduce and study a special class of infinite-dimensional Lie algebras that are finite-dimensional modules over a ring of polynomials. The Lie algebras of this class are said to be polynomial. Some classification results are obtained. An associative co-algebra structure on the rings

$k[x_1,\dots,x_n]/(f_1,\dots,f_n)$ is introduced and, on its basis, an explicit expression for convolution matrices of invariants for isolated singularities of functions is found. The structure polynomials of moving frames defined by convolution matrices are constructed for simple singularities of the types

$A$ ,

$B$ ,

$C$ ,

$D$ , and

$E_6$ .

Keywords: Lie algebra, moving frame, convolution of invariants, co-algebra.

Received: 05.05.2002

English version:
Functional Analysis and Its Applications, 2002, Volume 36, Issue 4, Pages 267–280
DOI: https://doi.org/10.1023/A:1021757609372

Bibliographic databases:

Document Type: Article

UDC: 512.554.32+517

Language: Russian

Citation: V. M. Buchstaber, D. V. Leikin, “Polynomial Lie Algebras”, Funktsional. Anal. i Prilozhen., 36:4 (2002), 18–34; Funct. Anal. Appl., 36:4 (2002), 267–280

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/faa216

https://doi.org/10.4213/faa216

https://www.mathnet.ru/eng/faa/v36/i4/p18

This publication is cited in the following 36 articles:

Julia Bernatska, “Abelian Function Fields on Jacobian Varieties”, Axioms, 14:2 (2025), 90
J. Chris Eilbeck, John Gibbons, Yoshihiro Ônishi, Seidai Yasuda, “Theory of heat equations for sigma functions”, Glasgow Math. J., 2025, 1
V. M. Buchstaber, E. Yu. Bunkova, “Formulas for Differentiating Hyperelliptic Functions with Respect to Parameters and Periods”, Proc. Steklov Inst. Math., 325 (2024), 60–73
E. Yu. Bunkova, V. M. Buchstaber, “Explicit Formulas for Differentiation of Hyperelliptic Functions”, Math. Notes, 114:6 (2023), 1151–1162
Julia Bernatska, Dmitry Leykin, “Solution of the Jacobi inversion problem on non-hyperelliptic curves”, Lett Math Phys, 113:5 (2023)
V. M. Buchstaber, E. Yu. Bunkova, “Hyperelliptic Sigma Functions and Adler–Moser Polynomials”, Funct. Anal. Appl., 55:3 (2021), 179–197
D. V. Millionshchikov, S. V. Smirnov, “Characteristic algebras and integrable exponential systems”, Ufa Math. J., 13:2 (2021), 41–69
V. V. Gorbatsevich, “Polinomialnye realizatsii konechnomernykh algebr Li”, Funkts. analiz i ego pril., 54:2 (2020), 25–34
V. M. Buchstaber, E. Yu. Bunkova, “Lie Algebras of Heat Operators in a Nonholonomic Frame”, Math. Notes, 108:1 (2020), 15–28
V. M. Buchstaber, E. Yu. Bunkova, “Sigma Functions and Lie Algebras of Schrödinger Operators”, Funct. Anal. Appl., 54:4 (2020), 229–240
Buchstaber V.M. Enolski V.Z. Leykin D.V., “SIGMA-Functions: Old and New Results”, Integrable Systems and Algebraic Geometry: a Celebration of Emma Previato'S 65Th Birthday, Vol 2, London Mathematical Society Lecture Note Series, 459, ed. Donagi R. Shaska T., Cambridge Univ Press, 2020, 175–214
V. V. Gorbatsevich, “Polynomial Realizations of Finite-Dimensional Lie Algebras”, Funct Anal Its Appl, 54:2 (2020), 93
Bernatska J. Leykin D., “On Degenerate SIGMA-Functions in Genus 2”, Glasg. Math. J., 61:1 (2019), 169–193
Bunkova E.Yu., “On the Problem of Differentiation of Hyperelliptic Functions”, Eur. J. Math., 5:3, SI (2019), 712–719
Bunkova E.Yu., “Differentiation of Genus 3 Hyperelliptic Functions”, Eur. J. Math., 4:1, 1, SI (2018), 93–112
D. V. Millionshchikov, “Polynomial Lie algebras and growth of their finitely generated Lie subalgebras”, Proc. Steklov Inst. Math., 302 (2018), 298–314
V. M. Buchstaber, A. V. Mikhailov, “Infinite-Dimensional Lie Algebras Determined by the Space of Symmetric Squares of Hyperelliptic Curves”, Funct. Anal. Appl., 51:1 (2017), 2–21
V. M. Buchstaber, “Polynomial Lie algebras and the Zelmanov–Shalev theorem”, Russian Math. Surveys, 72:6 (2017), 1168–1170
Makedonskyi I., “on Noncommutative Bases of Free Modules of Derivations Over Polynomial Rings”, Commun. Algebr., 44:1 (2016), 11–25
V. M. Buchstaber, “Polynomial dynamical systems and the Korteweg–de Vries equation”, Proc. Steklov Inst. Math., 294 (2016), 176–200

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Functional Analysis and Its Applications

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