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Journal of Lie Theory 23 (2013), No. 2, 589--606
Copyright Heldermann Verlag 2013

Capelli Elements for the Algebra g2

Dmitry V. Artamonov
Moscow State University, Leninskie gory 1, 119421 Moscow GSP-1, Russia

Valentina A. Golubeva
Moscow Aviation Institute, Volokolamskoe Shosse 4, 125993 Moscow GSP-3 A-80, Russia


\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) 333--359] 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

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