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Funktsional. Anal. i Prilozhen., 2002, Volume 36, Issue 4, Pages 18–34 (Mi faa216)  

This article is cited in 28 scientific papers (total in 29 papers)

Polynomial Lie Algebras

V. M. Buchstabera, D. V. Leikinb

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

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 $k[x_1,…,x_n]/(f_1,…,f_n)$ 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


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English version:
Functional Analysis and Its Applications, 2002, 36:4, 267–280

Bibliographic databases:

UDC: 512.554.32+517
Received: 05.05.2002

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
\by V.~M.~Buchstaber, D.~V.~Leikin
\paper Polynomial Lie Algebras
\jour Funktsional. Anal. i Prilozhen.
\yr 2002
\vol 36
\issue 4
\pages 18--34
\jour Funct. Anal. Appl.
\yr 2002
\vol 36
\issue 4
\pages 267--280

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    This publication is cited in the following articles:
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    2. V. M. Buchstaber, D. V. Leikin, “Heat Equations in a Nonholonomic Frame”, Funct. Anal. Appl., 38:2 (2004), 88–101  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. V. M. Buchstaber, D. V. Leikin, “Addition Laws on Jacobian Varieties of Plane Algebraic Curves”, Proc. Steklov Inst. Math., 251 (2005), 49–120  mathnet  mathscinet  zmath
    4. E. D. Belokolos, V. Z. Ènol'skii, M. Salerno, “Wannier Functions for Quasiperiodic Finite-Gap Potentials”, Theoret. and Math. Phys., 144:2 (2005), 1081–1099  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. E. Previato, V. Z. Ènol'skii, “Ultra-elliptic solitons”, Russian Math. Surveys, 62:4 (2007), 796–798  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. V. M. Buchstaber, D. V. Leikin, “Differentiation of Abelian functions with respect to parameters”, Russian Math. Surveys, 62:4 (2007), 787–789  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    7. Eilbeck, JC, “Abelian Functions for Trigonal Curves of Genus Three”, International Mathematics Research Notices, 2007, rnm140  crossref  isi  elib  scopus
    8. V. M. Buchstaber, D. V. Leikin, “Solution of the Problem of Differentiation of Abelian Functions over Parameters for Families of $(n,s)$-Curves”, Funct. Anal. Appl., 42:4 (2008), 268–278  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    9. Baldwin, S, “Abelian functions for cyclic trigonal curves of genus 4”, Journal of Geometry and Physics, 58:4 (2008), 450  crossref  mathscinet  zmath  adsnasa  isi  scopus
    10. E. Yu. Bunkova, V. M. Buchstaber, “Heat Equations and Families of Two-Dimensional Sigma Functions”, Proc. Steklov Inst. Math., 266 (2009), 1–28  mathnet  crossref  mathscinet  zmath  isi  elib
    11. Buchstaber V.M., “Heat Equations and Sigma Functions”, Geometric Methods in Physics, AIP Conference Proceedings, 1191, 2009, 46–58  crossref  adsnasa  isi  scopus
    12. Petravchuk A.P., “On pairs of commuting derivations of the polynomial ring in one or two variables”, Linear Algebra Appl, 433:3 (2010), 574–579  crossref  mathscinet  zmath  isi  scopus
    13. E. Yu. Bun'kova, “The differential-geometric structure of the universal bundle of elliptic curves”, Russian Math. Surveys, 66:4 (2011), 816–818  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    14. Arzhantsev I.V., Makedonskii E.A., Petravchuk A.P., “Finite-Dimensional Subalgebras in Polynomial Lie Algebras of Rank One”, Ukrainian Math J, 63:5 (2011), 827–832  crossref  mathscinet  zmath  isi  elib  scopus
    15. E. Yu. Netay, “Geometric differential equations on bundles of Jacobians of curves of genus 1 and 2”, Trans. Moscow Math. Soc., 74 (2013), 281–292  mathnet  crossref  mathscinet  zmath  elib
    16. Makedonskyi I., “on Noncommutative Bases of Free Modules of Derivations Over Polynomial Rings”, Commun. Algebr., 44:1 (2016), 11–25  crossref  mathscinet  zmath  isi  elib  scopus
    17. V. M. Buchstaber, “Polynomial dynamical systems and the Korteweg–de Vries equation”, Proc. Steklov Inst. Math., 294 (2016), 176–200  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    18. Berkeley G., Mikhailov A.V., Xenitidis P., “Darboux transformations with tetrahedral reduction group and related integrable systems”, J. Math. Phys., 57:9 (2016), 092701  crossref  mathscinet  zmath  isi  elib  scopus
    19. 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  mathnet  crossref  crossref  mathscinet  isi  elib
    20. V. M. Buchstaber, “Polynomial Lie algebras and the Zelmanov–Shalev theorem”, Russian Math. Surveys, 72:6 (2017), 1168–1170  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    21. Bunkova E.Yu., “Differentiation of Genus 3 Hyperelliptic Functions”, Eur. J. Math., 4:1, 1, SI (2018), 93–112  crossref  mathscinet  zmath  isi  scopus
    22. D. V. Millionshchikov, “Polynomial Lie algebras and growth of their finitely generated Lie subalgebras”, Proc. Steklov Inst. Math., 302 (2018), 298–314  mathnet  crossref  crossref  mathscinet  isi  elib
    23. Bernatska J. Leykin D., “On Degenerate SIGMA-Functions in Genus 2”, Glasg. Math. J., 61:1 (2019), 169–193  crossref  mathscinet  zmath  isi  scopus
    24. Bunkova E.Yu., “On the Problem of Differentiation of Hyperelliptic Functions”, Eur. J. Math., 5:3, SI (2019), 712–719  crossref  mathscinet  isi
    25. V. V. Gorbatsevich, “Polinomialnye realizatsii konechnomernykh algebr Li”, Funkts. analiz i ego pril., 54:2 (2020), 25–34  mathnet  crossref  mathscinet
    26. V. M. Buchstaber, E. Yu. Bunkova, “Lie Algebras of Heat Operators in a Nonholonomic Frame”, Math. Notes, 108:1 (2020), 15–28  mathnet  crossref  crossref  mathscinet  isi  elib
    27. V. M. Buchstaber, E. Yu. Bunkova, “Sigma Functions and Lie Algebras of Schrödinger Operators”, Funct. Anal. Appl., 54:4 (2020), 229–240  mathnet  crossref  crossref  mathscinet  isi  elib
    28. V. M. Bukhshtaber, E. Yu. Bunkova, “Giperellipticheskie sigma-funktsii i polinomy Adlera–Mozera”, Funkts. analiz i ego pril., 55:3 (2021), 3–25  mathnet  crossref
    29. D. V. Millionshchikov, S. V. Smirnov, “Characteristic algebras and integrable exponential systems”, Ufa Math. J., 13:2 (2021), 41–69  mathnet  crossref  isi
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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