Geodesic
Capelli Elements for the Algebra g2
Journal of Lie theory, Tome 23 (2013) no. 2, pp. 589-606 Cet article a éte moissonné depuis la source Heldermann Verlag

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\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.
Classification : 17B25, 16S30
Mots-clés : Central elements, universal enveloping algebra, pfaffian 
@article{JLT_2013_23_2_JLT_2013_23_2_a11,
     author = {D. V. Artamonov and V. 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/JLT_2013_23_2_JLT_2013_23_2_a11/}
}
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D. V. Artamonov; V. 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/JLT_2013_23_2_JLT_2013_23_2_a11/