zbMATH — the first resource for mathematics

Sur les corps liés aux algèbres enveloppantes des algèbres de Lie. (French) Zbl 0144.02104
Let \(G\) be a Lie algebra over a field \(L\) of characteristic 0. \(\mathfrak U(G)\) be the enveloping algebra of \(G\). \(\mathfrak U(G)\) is, in general, non-commutative and is an Ore ring without zero divisors, i.e., for any two non-zero elements, there exist common left and right multiples. Therefore, \(\mathfrak U(G)\) has a quotient field \(\mathfrak D(G)\) which is called Lie field of \(G\). In this note, the authors construct for any pair of non-negative integers \((n, k)\), a field \(\mathfrak D_{n,k}(L)\) containing \(L\) such that the dimension of it over \(L\) is \(2n+k\) and the dimension of its center over \(L\) is \(k\). Then they state a conjecture that if \(G\) is an algebraic Lie algebra over \(L\), then \(\mathfrak D(G)\) is isomorphic to \(\mathfrak D_{n,k}(L)\) for some pair \((n, k)\). Here, the conjecture is solved affirmatively for the Lie algebras \(\mathfrak{gl}(n, L)\), \(\mathfrak{sl}(n, L)\) and the nilpotent Lie algebras. Further, they give examples of two Lie algebras over the field \(R\) of real numbers which are both not algebraic such that the Lie field of the one is not isomorphic to any \(\mathfrak D_{n,k}(R)\) and that of the other is isomorphic to \(\mathfrak D_{2,0}(L)\).

17B35 Universal enveloping (super)algebras
17B20 Simple, semisimple, reductive (super)algebras
17B30 Solvable, nilpotent (super)algebras
Full Text: DOI Numdam EuDML
[1] Séminaire Sophus Lie, Paris, 1955.
[2] I. M. Gelfand, Центр инфинитезимального группового кольца,Mat. Sbornik, 1950, t. 26, 103–112.
[3] J. Dixmier, Sur les représentations unitaires des groupes de Lie nilpotents, II,Bull. Soc. Math. France, 85 (1957), 325–388. · Zbl 0085.10303
[4] J. Dixmier, Sur l’algèbre enveloppante d’une algèbre de Lie nilpotente,Arch. Math., 1959, vol. 10, 321–326. · Zbl 0146.26102
[5] A. A. Kirillov, Унитарные представления нильпотентных групп Ли,Uspekhi Matem. Nauk, 1962, t. 17, 57–101.
[6] P. Bernat, Sur le corps enveloppant d’une algèbre de Lie résoluble,C. r. Acad. Sci., t. 258, 2713–2715. · Zbl 0131.27202
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.