A unitary representation of a, possibly infinite dimensional, Lie group G is called semibounded if the corresponding operators idπ(x) from the derived representations are uniformly bounded from above on some non-empty open subset of the Lie algebra 𝔤. In the first part of the present paper we explain how this concept leads to a fruitful interaction between the areas of infinite dimensional convexity, Lie theory, symplectic geometry (momentum maps) and complex analysis. Here open invariant cones in Lie algebras play a central role and semibounded representations have interesting connections to C*-algebras which are quite different from the classical use of the group C*-algebra of a finite dimensional Lie group. The second half is devoted to a detailed discussion of semibounded representations of the diffeomorphism group of the circle, the Virasoro group, the metaplectic representation on the bosonic Fock space and the spin representation on fermionic Fock space.
@article{CML_2010__2_1_37_0,
author = {Karl-Hermann Neeb},
title = {Semibounded representations and invariant cones in infinite dimensional lie algebras},
journal = {Confluentes Mathematici},
pages = {37--134},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {2},
number = {1},
year = {2010},
doi = {10.1142/S1793744210000132},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744210000132/}
}
TY - JOUR AU - Karl-Hermann Neeb TI - Semibounded representations and invariant cones in infinite dimensional lie algebras JO - Confluentes Mathematici PY - 2010 SP - 37 EP - 134 VL - 2 IS - 1 PB - World Scientific Publishing Co Pte Ltd UR - https://cml.centre-mersenne.org/articles/10.1142/S1793744210000132/ DO - 10.1142/S1793744210000132 LA - en ID - CML_2010__2_1_37_0 ER -
%0 Journal Article %A Karl-Hermann Neeb %T Semibounded representations and invariant cones in infinite dimensional lie algebras %J Confluentes Mathematici %D 2010 %P 37-134 %V 2 %N 1 %I World Scientific Publishing Co Pte Ltd %U https://cml.centre-mersenne.org/articles/10.1142/S1793744210000132/ %R 10.1142/S1793744210000132 %G en %F CML_2010__2_1_37_0
Karl-Hermann Neeb. Semibounded representations and invariant cones in infinite dimensional lie algebras. Confluentes Mathematici, Tome 2 (2010) no. 1, pp. 37-134. doi : 10.1142/S1793744210000132. https://cml.centre-mersenne.org/articles/10.1142/S1793744210000132/
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