The contractions of Lie groups and Lie algebras and their representations are studied geometrically. We prove they can be defined by deformations in Poisson algebras of symplectic manifolds on which the groups act. These deformations are given by Dirac constraints which induce on C∞ functions on the deformed manifold an associative twisted product, characterizing the contracted group or its representations. We treat the contractions of SO(n) to E(n) and apply this theory to thermodynamical limits in spin systems.
REFERENCES
1.
F.
Bayen
, M.
Flato
, C.
Fronsdal
, A.
Lichnerowicz
, and D.
Sternheimer
, Lett. Math. Phys.
1
, 521
–30
(1977
).2.
F.
Bayen
, M.
Flato
, C.
Fronsdal
, A.
Lichnerowicz
, and D.
Sternheimer
, Ann. Phys. (N.Y.)
111
, 61
–110
111
–51
, (1978
).3.
4.
5.
N. Bourbaki, Eléments de Mathématiques, Fasc. XXVI (Hermann, Paris, 1971).
6.
M.
Flato
, A.
Lichnerowicz
, and D.
Sternheimer
, J. Math. Phys.
17
, 1754
–62
(1976
).7.
D. A. Dubin, Solvable Models in Algebraic Statistical Mechanics (Clarendon, Oxford, 1974).
This content is only available via PDF.
© 1979 American Institute of Physics.
1979
American Institute of Physics
You do not currently have access to this content.
