Geometric Description of an Induced Representation of a Lie Algebra
by
Alexander Levichev
Boston University and Sobolev Institute of Mathematics

Induced (in the sense of G. Mackey) representation U of a Lie group G proved to be a major tool in description of a quantum mechanical particle. In my survey article [Le95] (see its URL address below) U has been described using modern terminology. The adequate Lie-algebraic version seemed to be missing.

Theorem. Let g, h, k stand for Lie algebras of G, H, K, respectively. Let r be the inducing representation of h (be aware that g is a (vector space) direct sum of a (vertical) h and a (horizontal) k subalgebras). (Up to an equivalence) the tangent representation dU acts as follows: the image of a vector v from g under dU is the sum of a respective vector field v on K and of an r(l) where l is a vertical component of A_xv, A_x is the respective operator of the adjoint action of G on g, x being a point of (the base) K.

Technically, the second term is a (point-dependent) matrix which (together with the first term) acts on (K-parallelized) sections of the induced vector bundle over K. The above result facilitates a simpler treatment of properties of induced representations. Of fundamental importance in theoretical physics are cases: 1) G is the Poincare group, H is the Lorentz group, K consists of space-time translations; 2) G is the conformal group, N is its maximal (essentially) compact subgroup, H is the (scale-extended) Poincare group.

http://math.bu.edu/people/levit/chronometry.ps

Date received: February 20, 2000