
Journal of Lie Theory 23 (2013), No. 2, 589606 Copyright Heldermann Verlag 2013 Capelli Elements for the Algebra g_{2} Dmitry V. Artamonov Moscow State University, Leninskie gory 1, 119421 Moscow GSP1, Russia artamonov.dmitri@gmail.com Valentina A. Golubeva Moscow Aviation Institute, Volokolamskoe Shosse 4, 125993 Moscow GSP3 A80, Russia goloubeva@yahoo.com [Abstractpdf] \def\g{{\frak g}} \def\o{{\frak o}} M. Itoh and T. Umeda [On Central Elements in the Universal Enveloping Algebras of the Orthogonal Lie Algebras, Compositio Mathematica 127 (2001) 333359] constructed central elements in the universal enveloping algebra $U(\o_N)$, named Capelli elements, as sums of squares of noncommutative Pfaffians of some matrices, whose entries belong to $\o_N$. 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 $U(\o_N)$, we give characterization of these central elements in terms of their vanishing properties. Also for $U(\g_2)$ an explicit relations between constructed central elements and higher Casimir elements defined by D. P. Zhelobenko [Compact Lie groups and their representations, Amer. Math. Soc., Providence, R.I. (1973)] are obtained. Keywords: Central elements, universal enveloping algebra, pfaffian. MSC: 17B25, 16S30 [ Fulltextpdf (373 KB)] for subscribers only. 