Trace formulas for the Casimir operators of the unextended Schrödinger algebra S(N)

R. Campoamor-Stursberg

doi:10.1063/1.5141091

Trace formulas for the Casimir operators of the unextended Schrödinger algebra S(N)

Journal of Mathematical Physics ◽

10.1063/1.5141091 ◽

2020 ◽

Vol 61 (4) ◽

pp. 043508

Author(s):

R. Campoamor-Stursberg

Keyword(s):

Trace Formulas ◽

Casimir Operators ◽

Schrödinger Algebra

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NONRELATIVISTIC HOLOGRAPHY — A GROUP-THEORETICAL PERSPECTIVE

International Journal of Modern Physics A ◽

10.1142/s0217751x14300014 ◽

2014 ◽

Vol 29 (03n04) ◽

pp. 1430001 ◽

Cited By ~ 10

Author(s):

V. K. DOBREV

Keyword(s):

Representation Theory ◽

Theoretical Perspective ◽

Explicit Construction ◽

Intertwining Operators ◽

Difference Operators ◽

Semisimple Lie Groups ◽

Boundary Operators ◽

Boundary Fields ◽

Theoretical Results ◽

Schrödinger Algebra

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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