4. Intertwining operators: X-space realization

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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State-space realization and inversion of 2-D systems

IEEE Transactions on Circuits and Systems ◽

10.1109/tcs.1980.1084865 ◽

1980 ◽

Vol 27 (7) ◽

pp. 612-619 ◽

Cited By ~ 47

Author(s):

R. Eising

Keyword(s):

State Space ◽

State Space Realization ◽

Space Realization ◽

And Inversion

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Two-dimensional B-CFE: Circuit and state space realization

International Journal of Systems Science ◽

10.1080/002077299292579 ◽

1999 ◽

Vol 30 (2) ◽

pp. 219-222 ◽

Cited By ~ 2

Author(s):

George E. Antoniou

Keyword(s):

State Space ◽

Two Dimensional ◽

State Space Realization ◽

Space Realization

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On a construction of unitary cocycles and the representation theory of amenable groups

Ergodic Theory and Dynamical Systems ◽

10.1017/s014338570000184x ◽

1983 ◽

Vol 3 (1) ◽

pp. 129-133 ◽

Cited By ~ 1

Author(s):

Colin E. Sutherland

Keyword(s):

Representation Theory ◽

Probability Space ◽

Amenable Group ◽

Unitary Representations ◽

Intertwining Operators ◽

Type I ◽

Amenable Groups

AbstractIf K is a countable amenable group acting freely and ergodically on a probability space (Γ, μ), and G is an arbitrary countable amenable group, we construct an injection of the space of unitary representations of G into the space of unitary 1-cocyles for K on (Γ, μ); this injection preserves intertwining operators. We apply this to show that for many of the standard non-type-I amenable groups H, the representation theory of H contains that of every countable amenable group.

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Combinatorial bases of modules for affine Lie algebra B 2(1)

Open Mathematics ◽

10.2478/s11533-012-0111-x ◽

2013 ◽

Vol 11 (2) ◽

Cited By ~ 2

Author(s):

Mirko Primc

Keyword(s):

Lie Algebra ◽

Vertex Operator ◽

Vertex Operator Algebra ◽

Intertwining Operators ◽

Type B ◽

Highest Weight Module ◽

Highest Weight ◽

Affine Lie Algebra ◽

Algebra Theory ◽

Simple Currents

AbstractWe construct bases of standard (i.e. integrable highest weight) modules L(Λ) for affine Lie algebra of type B 2(1) consisting of semi-infinite monomials. The main technical ingredient is a construction of monomial bases for Feigin-Stoyanovsky type subspaces W(Λ) of L(Λ) by using simple currents and intertwining operators in vertex operator algebra theory. By coincidence W(kΛ0) for B 2(1) and the integrable highest weight module L(kΛ0) for A 1(1) have the same parametrization of combinatorial bases and the same presentation P/I.

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Free Space Realization of the Symmetrical Tunable Auto‐Focusing Lommel Gaussian Vortex Beam

Annalen der Physik ◽

10.1002/andp.202100419 ◽

2021 ◽

pp. 2100419

Author(s):

Jialong Tu ◽

Xinyue Wang ◽

Xing Yu ◽

Haonan Wang ◽

Dongmei Deng

Keyword(s):

Free Space ◽

Vortex Beam ◽

Space Realization ◽

Auto Focusing

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Casselman’s Basis of Iwahori Vectors and the Bruhat Order

Canadian Journal of Mathematics ◽

10.4153/cjm-2011-042-3 ◽

2011 ◽

Vol 63 (6) ◽

pp. 1238-1253 ◽

Cited By ~ 3

Author(s):

Daniel Bump ◽

Maki Nakasuji

Keyword(s):

Weyl Group ◽

Bruhat Order ◽

Group Element ◽

The Other ◽

Intertwining Operators ◽

Transition Matrices ◽

Natural Basis ◽

Spherical Representation ◽

Langlands Parameters ◽

Combinatorial Condition

AbstractW. Casselman defined a basis fu of Iwahori fixed vectors of a spherical representation of a split semisimple p-adic group G over a nonarchimedean local field F by the condition that it be dual to the intertwining operators, indexed by elements u of the Weyl group W. On the other hand, there is a natural basis , and one seeks to find the transition matrices between the two bases. Thus, let and . Using the Iwahori–Hecke algebra we prove that if a combinatorial condition is satisfied, then , where z are the Langlands parameters for the representation and α runs through the set S(u, v) of positive coroots (the dual root systemof G) such that with rα the reflection corresponding to α. The condition is conjecturally always satisfied if G is simply-laced and the Kazhdan–Lusztig polynomial Pw0v,w0u = 1 with w0 the long Weyl group element. There is a similar formula for conjecturally satisfied if Pu,v = 1. This leads to various combinatorial conjectures.

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