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.
Similar content being viewed by others
References
V.I. Arnold, Singularities of Caustics and Wave Fronts, Kluwer Academic Publishers Group, Dordrecht, 1990.
V.M. Buchstaber, “Semigroups of maps into groups, operator doubles and complex cobordisms,” Amer. Math. Soc. Transl. (2), Vol. 170, 1995, pp. 9–35.
V.M. Buchstaber, “Groups of polynomial transformations of a line, informal symplectic manifolds, and the Landweber–Novikov algebra,” Usp. Mat. Nauk, 54, No.4, 161–162 (1999).
V. M. Buchstaber and D. V. Leykin, “Lie algebras associated with σ-functions and versal deformations,” Usp. Mat. Nauk, 57, No.3, 145–146, 2002.
V. M. Buchstaber and D. V. Leykin, “Graded Lie algebras that define hyperelliptic sigma functions,” Dokl. RAS, 385, No.5, 2002.
V. M. Buchstaber, V. Z. Enolskii, and D. V. Leykin, “Hyperelliptic Kleinian functions and applications,” In: Solitons, Geometry, and Topology: on the Crossroad, Amer. Math. Soc. Transl., Ser. 2, Vol. 179, Amer. Math. Soc., Providence, 1997, pp. 1–34.
V. M. Buchstaber, V. Z. Enolskii, and D. V. Leykin, “Kleinian functions, hyperelliptic Jacobians and applications,” Rev. Math. Math. Phys., 10, No.2, 3–120 (1997).
B. Dubrovin, Geometry of 2D topological field theories, In: Integrable systems and quantum groups (Montecatini Terme, 1993), Lect. Notes in Math., Vol. 1620, Springer-Verlag, Berlin, 1996, pp. 120–348.
B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry. Methods and applications, Part I, Graduated Text in Math., Vol. 93, Springer-Verlag, New York Berlin, 1984.
B. A. Dubrovin and S. P. Novikov, Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory,” Usp. Mat. Nauk, 44, No.6, 29–98 (1989).
B. Dubrovin and Y. Zang, “Frobenius manifold and Virasoro constraints,” Selecta Math. (N.S.), 5, No.4, 423–466 (1999).
A. B. Givental, “Convolution of invariants of groups generated b reflections and associated with simple singularities of functions,” Funkts. Anal. Prilozhen., 14, No.2, 4–14 (1980).
A. B. Givental, “Gromov-Witten invariants and quantization of quadratic Hamiltonians,” MMJ, 1, No.4, 551–568 (2001).
M. V. Karasev and V. P. Maslov, Nonlinear Poisson Brackets. Geometry and Quantization, Amer. Math. Soc., Providence, RI, 1993.
S. P. Novikov, “Various doublings of Hopf algebras. Algebras of operators on quantum groups, complex cobordisms,” Usp. Mat. Nauk, 47, No.5, 189–190 (1992).
K. Saito, “Theory of logarithmic differential forms and logarithmic vector filds,” J. Fac. Sci. Univ. Tokyo Sect. Math., 27, 263–291 (1980).
M. A. Semenov-Tian-Shansky, “Poisson Lie groups, quantum dualit principle and the quantum double,” Contemp. Math., 175, 219–248 (1994).
K. Weierstraß, Zur Theorie der elliptischen Funktionen,Mathematische Werke, Vol. 2, Teubner, Berlin, 1894, pp. 245–255.
V. M. Zakalyukin, “Reconstructions of wave fronts depending on one parameter,” Funkts. Anal. Prilozhen., 10, No.2, 69–70 (1976).
New Developments in Singularit Theory (D. Siersma, C.T.C. Wall, V. Zakalyukin, eds.), NATO Science Series II,Vol. 21., 2001.
Rights and permissions
About this article
Cite this article
Buchstaber, V.M., Leykin, D.V. Polynomial Lie Algebras. Functional Analysis and Its Applications 36, 267–280 (2002). https://doi.org/10.1023/A:1021757609372
Issue Date:
DOI: https://doi.org/10.1023/A:1021757609372