Sur les corps liés aux algèbres enveloppantes des algèbres de Lie
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 \(\mathfrak D_{2,0}(L)\).
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