It generates the center of the universal enveloping algebra of the complexified Lie algebra of SL ( 2, "'R "').
22.
This also indicates exactly when a universal enveloping algebra can be constructed for some object : if it has a bilinear operator, then the construction can go through.
23.
The universal enveloping algebra of a free Lie algebra on a set " X " is the free associative algebra generated by " X ".
24.
Universal enveloping algebras play a relatively minor role in the representation theory of Lie groups; the greatest utility is perhaps to give a precise definition for the Casimir operators.
25.
Over a field of characteristic zero, it is isomorphic to the enveloping algebra of the Lie algebra of " G " and thus gives no new information.
26.
The best-known example is, perhaps the Albert algebra, an exceptional Jordan algebra that is not enveloped by the canonical construction of the enveloping algebra for Jordan algebras.
27.
A representation of a group induces a representation of a corresponding group ring or group algebra, while representations of a Lie algebra correspond bijectively to representations of its universal enveloping algebra.
28.
*Lie Superalgebras and Enveloping Algebras Ian M . Musson, Graduate Studies in Mathematics 2012; 488 pp; hardcover Volume : 131 ISBN 978-0-8218-6867-6
29.
The " associative enveloping algebra " or " multiplication algebra " of " A " is the associative algebra generated by the left and right linear maps.
30.
The infinitesimal character is the linear form on the center " Z " of the universal enveloping algebra of the Lie algebra of " G " that the representation induces.
