Realization of Lie algebras and superalgebras in terms of creation and annihilation operators: I
Teoretičeskaâ i matematičeskaâ fizika, Tome 124 (2000) no. 2, pp. 227-238
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For every finite-dimensional nilpotent complex Lie algebra or superalgebra $\mathfrak n$, we offer three algorithms for realizing it in terms of creation and annihilation operators. We use these algorithms to realize Lie algebras with a maximal subalgebra of finite codimension. For a simple finite-dimensional $\mathfrak g$ whose maximal nilpotent subalgebra is $\mathfrak n$, this gives its realization in terms of first-order differential operators on the big open cell of the flag manifold corresponding to the negative roots of $\mathfrak g$. For several examples, we executed the algorithms using the MATHEMATICA-based package SUPERLie. These realizations form a preparatory step in an explicit construction and description of an interesting new class of simple Lie (super)algebras of polynomial growth, generalizations of the Lie algebra of matrices of complex size.
@article{TMF_2000_124_2_a2,
author = {\v{C}. Burd{\'\i}k and P. Ya. Grozman and D. A. Leites and A. N. Sergeev},
title = {Realization of {Lie} algebras and superalgebras in terms of creation and annihilation operators: {I}},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {227--238},
publisher = {mathdoc},
volume = {124},
number = {2},
year = {2000},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2000_124_2_a2/}
}
TY - JOUR AU - Č. Burdík AU - P. Ya. Grozman AU - D. A. Leites AU - A. N. Sergeev TI - Realization of Lie algebras and superalgebras in terms of creation and annihilation operators: I JO - Teoretičeskaâ i matematičeskaâ fizika PY - 2000 SP - 227 EP - 238 VL - 124 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_2000_124_2_a2/ LA - ru ID - TMF_2000_124_2_a2 ER -
%0 Journal Article %A Č. Burdík %A P. Ya. Grozman %A D. A. Leites %A A. N. Sergeev %T Realization of Lie algebras and superalgebras in terms of creation and annihilation operators: I %J Teoretičeskaâ i matematičeskaâ fizika %D 2000 %P 227-238 %V 124 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/item/TMF_2000_124_2_a2/ %G ru %F TMF_2000_124_2_a2
Č. Burdík; P. Ya. Grozman; D. A. Leites; A. N. Sergeev. Realization of Lie algebras and superalgebras in terms of creation and annihilation operators: I. Teoretičeskaâ i matematičeskaâ fizika, Tome 124 (2000) no. 2, pp. 227-238. http://geodesic.mathdoc.fr/item/TMF_2000_124_2_a2/