Exploring Representation Theory of Unitary Groups via Linear Optical Passive Devices

2006 ◽  
Vol 13 (04) ◽  
pp. 415-426 ◽  
Author(s):  
P. Aniello ◽  
C. Lupo ◽  
M. Napolitano
Keyword(s):  
Lie Algebras ◽  
Fock Space ◽  
Fixed Number ◽  
Unitary Groups ◽  
Space Representation ◽  
Remarkable Case ◽  
Mathematical Structures ◽  
Passive Devices ◽  
Optical Modes ◽  
Linear Optical

In this paper, we investigate some mathematical structures underlying the physics of linear optical passive (LOP) devices. We show, in particular, that with the class of LOP transformations on N optical modes one can associate a unitary representation of U (N) in the N-mode Fock space, representation which can be decomposed into irreducible sub-representations living in the subspaces characterized by a fixed number of photons. These (sub-)representations can be classified using the theory of representations of semi-simple Lie algebras. The remarkable case where N = 3 is studied in detail.

scholarly journals Fock Space Representation of the Circle Quantum Group

2019 ◽  
Author(s):  
Francesco Sala ◽  
Olivier Schiffmann
Keyword(s):  
Vector Space ◽  
Quantum Group ◽  
Lie Algebras ◽  
Quantum Groups ◽  
Fock Space ◽  
Type A ◽  
Space Representation ◽  
Affine Type

Abstract In [12] we have defined quantum groups $\mathbf{U}_{\upsilon }(\mathfrak{sl}(\mathbb{R}))$ and $\mathbf{U}_{\upsilon }(\mathfrak{sl}(S^1))$, which can be interpreted as continuum generalizations of the quantum groups of the Kac–Moody Lie algebras of finite, respectively affine type $A$. In the present paper, we define the Fock space representation $\mathcal{F}_{\mathbb{R}}$ of the quantum group $\mathbf{U}_{\upsilon }(\mathfrak{sl}(\mathbb{R}))$ as the vector space generated by real pyramids (a continuum generalization of the notion of partition). In addition, by using a variant version of the “folding procedure” of Hayashi–Misra–Miwa, we define an action of $\mathbf{U}_{\upsilon }(\mathfrak{sl}(S^1))$ on $\mathcal{F}_{\mathbb{R}}$.

glq(n)-COVARIANT OSCILLATORS REALIZATION OF CERTAIN DEFORMED ALGEBRAS

2000 ◽  
Vol 15 (05) ◽  
pp. 667-677
Author(s):  
E. H. EL KINANI
Keyword(s):  
Lie Algebras ◽  
Fock Space ◽  
Space Representation ◽  
Physical Behavior ◽  
Infinite Dimensional ◽  
Physics Literature ◽  
Deformed Algebras

In this paper we review the glq(n)-covariant oscillators algebra and its Fock space representation. We use these results to construct the glq(n)-covariant realization of some infinite dimensional Lie algebras which occurs in the physics literature. Some new explicit realizations of glq(n)-covariant oscillators algebra are also given. At the end, some aspects of the physical behavior of glq(n)-covariant oscillators systems are discussed.

scholarly journals A TWO-PARAMETER DEFORMED SUSY ALGEBRA FOR SUp/q(n)-COVARIANT (p,q)-DEFORMED FERMIONIC OSCILLATORS

2005 ◽  
Vol 20 (08) ◽  
pp. 613-622 ◽  
Author(s):  
ABDULLAH ALGIN ◽  
METIN ARIK
Keyword(s):  
Quantum Group ◽  
Fock Space ◽  
Space Representation ◽  
Oscillator Algebra ◽  
Two Parameter

We construct a two-parameter deformed SUSY algebra by constructing SUSY generators which are bilinears of n (p,q)-deformed fermions covariant under the quantum group SU p/q(n) and n undeformed bosons. The Fock space representation of the algebra constructed is discussed and the total deformed Hamiltonian for such a system is obtained. Some physical applications of the quantum group covariant two-parameter deformed fermionic oscillator algebra are also considered.

scholarly journals Computing nilpotent and unipotent canonical forms: a symmetric approach

2011 ◽  
Vol 152 (1) ◽  
pp. 35-53 ◽  
Author(s):  
MATTHEW C. CLARKE
Keyword(s):  
Algebraic Group ◽  
Lie Algebras ◽  
General Linear Group ◽  
Symmetric Bilinear Form ◽  
Unitary Groups ◽  
Closed Field ◽  
Canonical Forms ◽  
Unified Approach ◽  
Geometric Argument ◽  
Complete Set

AbstractLet k be an algebraically closed field of any characteristic except 2, and let G = GLn(k) be the general linear group, regarded as an algebraic group over k. Using an algebro-geometric argument and Dynkin–Kostant theory for G we begin by obtaining a canonical form for nilpotent Ad(G)-orbits in (k) which is symmetric with respect to the non-main diagonal (i.e. it is fixed by the map f : (xi,j) ↦ (xn+1−j,n+1−i)), with entries in {0,1}. We then show how to modify this form slightly in order to satisfy a non-degenerate symmetric or skew-symmetric bilinear form, assuming that the orbit does not vanish in the presence of such a form. Replacing G by any simple classical algebraic group we thus obtain a unified approach to computing representatives for nilpotent orbits of all classical Lie algebras. By applying Springer morphisms, this also yields representatives for the corresponding unipotent classes in G. As a corollary we obtain a complete set of generic canonical representatives for the unipotent classes in finite general unitary groups GUn(q) for all prime powers q.

ON THE FOCK SPACE REPRESENTATION OF THE WITTEN VERTEX

1994 ◽  
Vol 09 (06) ◽  
pp. 465-477
Author(s):  
RAINER DICK
Keyword(s):  
Closed Form ◽  
Fock Space ◽  
Closed String ◽  
Space Representation ◽  
Operator Representations

The bosonic overlap conditions for operator representations of the Witten vertex and its closed string analog are solved in closed form for arbitrary many external strings. This is accomplished by the use of transformed operator bases of the strings. In particular, the bosonic factor of the Witten vertex for three closed strings is realized in Fock space.

scholarly journals Exactness of the Fock Space Representation of the q-Commutation Relations

2011 ◽  
Vol 308 (1) ◽  
pp. 115-132 ◽  
Author(s):  
Matthew Kennedy ◽  
Alexandru Nica
Keyword(s):  
Fock Space ◽  
Space Representation ◽  
Commutation Relations
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