Abstract
We define a new class of unitary solutions to the classical Yang--Baxter equation (CYBE). These ‘boundary solutions’ are those which lie in the closure of the space of unitary solutions of the modified classical Yang--Baxter equation (MCYBE). Using the Belavin--Drinfel'd classification of the solutions to the MCYBE, we are able to exhibit new families of solutions to the CYBE. In particular, using the Cremmer--Gervais solution to the MCYBE, we explicitly construct for all n ≥ 3 a boundary solution based on the maximal parabolic subalgebra of \({\mathfrak{s}}{\mathfrak{l}}\left( n \right)\) obtained by deleting the first negative root. We give some evidence for a generalization of this result pertaining to other maximal parabolic subalgebras whose omitted root is relatively prime to n. We also give examples of nonboundary solutions for the classical simple Lie algebras.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Belavin, A. A. and Drinfel'd, V. G.: Solutions of the classical Yang-Baxter equations for simple Lie algebras, Funct. Anal. Appl. 16 (1982), 159-180.
Chari, V. and Pressley, A.: A Guide to Quantum Groups, Cambridge University Press, New York, 1994.
Cremmer, E. and Gervais, J. L.: The quantum group structure associated with non-linearly extended Virasoro algebras, Comm. Math. Phys. 134 (1990), 619-632.
Elashvili, A. G.: Frobenius Lie algebras, Funct. Anal. Appl. 16 (1982), 94-95.
Elashvili, A. G.: Frobenius Lie algebras II, Proc.Math. Inst. Georgia Acad. Sci. (1986), 126-137 (Russian).
Fronsdal, C. and Galindo, A.: The universal T-matrix, Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups, Contemporary Mathematics 175, Amer.Math. Soc., Providence, R.I., 1994.
Fronsdal, C. and Galindo, A.: Deformations of multiparameter quantum gl(n), Lett. Math. Phys. 34 (1995), 25-36.
Fuchs, D.: Cohomology of Infinite Dimensional Lie Algebras, Consultants Bureau, New York, 1987.
Gerstenhaber, M., Giaquinto, A. and Schack, S. D.: Quantum symmetry, in: P. P. Kulish (ed.), Quantum Groups, Lecture Notes in Math. 1510, Springer, Berlin, New York, 1992, pp. 9-46.
Gerstenhaber, M., Giaquinto, A. and Schack, S. D.: Quantum groups, cohomology, and preferred deformations, in: S. Catto and A. Rocha (eds), Proc. XX Conference on Differential Geometric Techniques in Mathematical Physics, World Scientific, Singapore, 1992, pp. 529-538.
Gerstenhaber, M., Giaquinto, A. and Schack, S. D.: Construction of quantum groups from Belavin-Drinfel'd infinitesimals, in: A. Joseph and S. Shnider (eds), Quantum Deformation of Algebras and Their Representations, Israel Mathematical Conference Proc. 7, Amer. Math. Soc., Providence, 1993, pp. 45-64.
Stolin, A.: On rational solutions of Yang-Baxter equation for \({\mathfrak{s}}{\mathfrak{l}}\)(n), Math. Scand. 69 (1991), 57-80.
Stolin, A.: Constant solutions of Yang-Baxter equation for \({\mathfrak{s}}{\mathfrak{l}}\)(2) and \({\mathfrak{s}}{\mathfrak{l}}\)(3), Math. Scand. 69 (1991), 81-88.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gerstenhaber, M., Giaquinto, A. Boundary Solutions of the Classical Yang--Baxter Equation. Letters in Mathematical Physics 40, 337–353 (1997). https://doi.org/10.1023/A:1007363911649
Issue Date:
DOI: https://doi.org/10.1023/A:1007363911649