Ideals of the Enveloping AlgebraU(sl2)

AbstractIn this paper we study the two-sided ideals of the enveloping algebraU=U(sl2(K)) over an arbitrary fieldKof characteristic zero. Starting with two basic ideas, that an irreducible Lie module is generated by its highest weight vector and that the Lie module structure ofUcomes from its ring multiplication, we have found a “good” subset ofUconsisting of highest weight vectors for irreducibleU-submodules ofUso that each two-sided ideal ofUis uniquely generated by at most two elements of that set. Actually, each ideal is generated as a two-sided ideal by just one element. By uniqueness, all the information about the ideal is encoded in the formula for its generator(s). For example, we can list and classify all the prime ideals by height, determine the intersection of an ideal with the center, find the radical ideals and the radical of an ideal, and determine when two ideals are included one in another. An interesting property is that each ideal ofUcan be uniquely written as a product of primes. We also obtain the “least common multiple” and the “greatest common divisor” formulas for the prime ideal factorizations of the intersection and the sum of two ideals. This paper contains many other results of this nature