Capelli elements for the algebra g _2

摘要

Itoh and Umeda (see Itoh, M., and T. Umeda, On Central El- ements in the Universal Enveloping Algebras of the Orthogonal Lie Algebras, Compositio Mathematica 127 (2001), 333-359) constructed central elements in the universal enveloping algebra U(oN), named Capelli elements, as sums of squares of noncommutative Pfaffians of some matrices, whose entries belong to oN. However for exceptional algebras there are no construction of this type. In the present paper we construct central elements in U(g _2) as sums of squares of Pfaffians of some matrices, whose elements belong to g _2. For U(g _2), as in the case U(oN), we give characterization of these central elements in terms of their vanishing properties. Also for U(g _2) an explicit relations between con- structed central elements and higher Casimir elements defined in Zhelobenko, D. P., Compact Lie groups and their representations," Amer. Math. Soc., Provi- dence, R.I, 1973, are obtained.