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Algebra  2013

# Wronskian Envelope of a Lie Algebra

Abstract:

The famous Poincaré-Birkhoff-Witt theorem states that a Lie algebra, free as a module, embeds into its associative envelope—its universal enveloping algebra—as a sub-Lie algebra for the usual commutator Lie bracket. However, there is another functorial way—less known—to associate a Lie algebra to an associative algebra and inversely. Any commutative algebra equipped with a derivation , that is, a commutative differential algebra, admits a Wronskian bracket under which it becomes a Lie algebra. Conversely, to any Lie algebra a commutative differential algebra is universally associated, its Wronskian envelope, in a way similar to the associative envelope. This contribution is the beginning of an investigation of these relations between Lie algebras and differential algebras which is parallel to the classical theory. In particular, we give a sufficient condition under which a Lie algebra may be embedded into its Wronskian envelope, and we present the construction of the free Lie algebra with this property. 1. Introduction Any (associative) algebra (and more generally a Lie-admissible algebra), say , admits a derived structure of Lie algebra under the commutator bracket , . This actually describes a forgetful functor, more precisely an algebraic functor, from associative to Lie algebras. This functor admits a left adjoint that enables to associate to any Lie algebra its universal associative envelope. In this way the theory of Lie algebras may be explored through (but not reduced to) that of associative algebras. A Lie algebra which embeds into its universal enveloping algebra is referred to as special. The famous Poincaré-Birkhoff-Witt theorem states that any Lie algebra which is free as a module (and therefore, any Lie algebra over a field) is special. When the Lie algebra is Abelian, then its universal enveloping algebra reduces to the symmetric algebra of its underlying module structure, and thus any commutative Lie algebra is trivially special. However, there is another way to associate an associative algebra to a Lie algebra, and reciprocally, in a functorial way. The idea does not consist anymore to consider noncommutative algebras under commutators but differential commutative algebras together with the so-called Wronskian determinant. A derivation of an algebra is a linear map that satisfies the usual Leibniz rule. An algebra with a distinguished derivation is said to be a differential algebra. For any such pair may be defined a bilinear map by . When the algebra is commutative, then is alternating and satisfies the Jacobi identity so that it

References

  I. Kaplansky, An Introduction to Differential Algebra, Publications de l’Institut Mathématiques de l’Université de Nancago, Hermann, 1957.  S. MacLane, Categories for the Working Mathematician, vol. 5 of Graduate Texts in Mathematics, Springer, 1971.  N. Bourbaki, Elements of Mathematics, Algebra, chapter 1–3, Springer, 1998.  M. van der Put and M. F. Singer, Galois Theory of Linear Differential Equations, vol. 328 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, 2003.  J. F. Ritt, Differential Equations from the Algebraic Standpoint, vol. 14 of American Mathematical Society, Colloquium Publications, 1932.  Y. Chen, Y. Chen, and Y. Li, “Composition-diamond lemma for differential algebras,” The Arabian Journal for Science and Engineering A, vol. 34, no. 2, pp. 135–145, 2009.  P. M. Cohn, Universal Algebra, vol. 6 of Mathematics and its Applications, Kluwer, 1981.  N. Bourbaki, Elements of Mathematics, Lie Groups and Lie Algebras, chapter 1, Springer, 1998.  A. S. Dzhumadil'cprimedaev, “ -Lie structures generated by Wronskians,” Sibirski？ Matematicheski？ Zhurnal, vol. 46, no. 4, pp. 601–612, 2005.  E. G. Manes, Algebraic Theories, vol. 26 of Graduate Texts in Mathematics,, Springer, 1976.  Yu. Bahturin, Identical Relations in Lie Algebras, VNU Science Press, 1987.  Yu. P. Razmyslov, “Simple Lie algebras satisfying the standard Lie identity of degree 5,” Mathematics of the USSR-Izvestiya, vol. 26, no. 1, 1986.  A. A. Kirillov, V. Yu. Ovsienko, and O. D. Udalova, “Identities in the Lie algebra of vector fields on the real line,” Selecta Mathematica Formerly Sovietica, vol. 10, no. 1, pp. 7–17, 1991.  G. M. Bergman, The Lie Algebra of Vector Fields in ？n Satisfies Polynomial Identities, University of California, Berkeley, Calif, USA, 1979.  F. W. Lawvere, Functorial semantics of algebraic theories [Ph.D. thesis], Columbia University, 1963.  Yu. P. Razmyslov, Identities of Algebras and Their Representations, vol. 138 of Translations of Mathematical Monographs, American Mathematical Society, 1994.  P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, Springer, 1993.

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