\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.
1
Moscow State University, Leninskie gory 1, 119421 Moscow GSP-1, Russia
2
Moscow Aviation Institute, Volokolamskoe Shosse 4, 125993 Moscow GSP-3 A-80, Russia
Dmitry V. Artamonov; Valentina A. Golubeva. Capelli Elements for the Algebra g2. Journal of Lie Theory, Tome 23 (2013) no. 2, pp. 589-606. http://geodesic.mathdoc.fr/item/JOLT_2013_23_2_a11/
@article{JOLT_2013_23_2_a11,
author = {Dmitry V. Artamonov and Valentina A. Golubeva},
title = {Capelli {Elements} for the {Algebra} g\protect\textsubscript{2}},
journal = {Journal of Lie Theory},
pages = {589--606},
year = {2013},
volume = {23},
number = {2},
url = {http://geodesic.mathdoc.fr/item/JOLT_2013_23_2_a11/}
}
TY - JOUR
AU - Dmitry V. Artamonov
AU - Valentina A. Golubeva
TI - Capelli Elements for the Algebra g2
JO - Journal of Lie Theory
PY - 2013
SP - 589
EP - 606
VL - 23
IS - 2
UR - http://geodesic.mathdoc.fr/item/JOLT_2013_23_2_a11/
ID - JOLT_2013_23_2_a11
ER -
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%A Valentina A. Golubeva
%T Capelli Elements for the Algebra g2
%J Journal of Lie Theory
%D 2013
%P 589-606
%V 23
%N 2
%U http://geodesic.mathdoc.fr/item/JOLT_2013_23_2_a11/
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