On the Representations of the Semisimple Lie Groups. I. The Explicit Construction of Invariants for the Unimodular Unitary Group in N Dimensions

We give a review of some group-theoretical results related to nonrelativistic holography. Our main playgrounds are the Schrödinger equation and the Schrödinger algebra. We first recall the interpretation of nonrelativistic holography as equivalence between representations of the Schrödinger algebra describing bulk fields and boundary fields. One important result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory, and that these operators and the bulk-to-boundary operators are intertwining operators. Further, we recall the fact that there is a hierarchy of equations on the boundary, invariant with respect to Schrödinger algebra. We also review the explicit construction of an analogous hierarchy of invariant equations in the bulk, and that the two hierarchies are equivalent via the bulk-to-boundary intertwining operators. The derivation of these hierarchies uses a mechanism introduced first for semisimple Lie groups and adapted to the nonsemisimple Schrödinger algebra. These require development of the representation theory of the Schrödinger algebra which is reviewed in some detail. We also recall the q-deformation of the Schrödinger algebra. Finally, the realization of the Schrödinger algebra via difference operators is reviewed.

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Representations of complex semisimple Lie groups and Lie algebras

Bulletin of the American Mathematical Society ◽

10.1090/s0002-9904-1966-11528-8 ◽

1966 ◽

Vol 72 (3) ◽

pp. 522-526 ◽

Cited By ~ 3

Author(s):

K. R. Parthasarathy ◽

R. Ranga Rao ◽

V. S. Varadarajan

Keyword(s):

Lie Groups ◽

Lie Algebras ◽

Semisimple Lie Groups ◽

Complex Semisimple

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Twisted K-theory constructions in the case of a decomposable Dixmier-Douady class

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is014007001jkt274 ◽

2014 ◽

Vol 14 (2) ◽

pp. 247-272 ◽

Cited By ~ 4

Author(s):

Antti J. Harju ◽

Jouko Mickelsson

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Lie Groups ◽

Explicit Construction ◽

Theory Model ◽

Quantum Field ◽

Compact Lie Groups ◽

Field Theory Model ◽

Cup Product ◽

K Theory

AbstractTwisted K-theory on a manifold X, with twisting in the 3rd integral cohomology, is discussed in the case when X is a product of a circle and a manifold M. The twist is assumed to be decomposable as a cup product of the basic integral one form on and an integral class in H2(M,ℤ). This case was studied some time ago by V. Mathai, R. Melrose, and I.M. Singer. Our aim is to give an explicit construction for the twisted K-theory classes using a quantum field theory model, in the same spirit as the supersymmetric Wess-Zumino-Witten model is used for constructing (equivariant) twisted K-theory classes on compact Lie groups.

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