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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 $$\mathfrak D_{2,0}(L)$$.
Show Scanned Page ##### MSC:
 17B35 Universal enveloping (super)algebras 17B20 Simple, semisimple, reductive (super)algebras 17B30 Solvable, nilpotent (super)algebras
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##### References:
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