Unitary representation theory of the lie superalgebra gl(m|n)

2015 ◽  
Author(s):  
Jason Leonard Werry
Keyword(s):  
Representation Theory ◽  
Unitary Representation ◽  
Lie Superalgebra

scholarly journals Schramm–Loewner evolution with Lie superalgebra symmetry

2018 ◽  
Vol 33 (20) ◽  
pp. 1850117 ◽  
Author(s):  
Shinji Koshida
Keyword(s):  
Differential Equations ◽  
Representation Theory ◽  
Stochastic Differential Equations ◽  
Degrees Of Freedom ◽  
Lie Superalgebra ◽  
Classical Type ◽  
Finite Dimensional ◽  
Loewner Evolution ◽  
Internal Degrees Of Freedom

We propose a generalization of Schramm–Loewner evolution (SLE) that has internal degrees of freedom described by an affine Lie superalgebra. We give a general formulation of SLE corresponding to representation theory of an affine Lie superalgebra whose underlying finite-dimensional Lie superalgebra is basic classical type, and write down stochastic differential equations on internal degrees of freedom in case that the corresponding affine Lie superalgebra is [Formula: see text]. We also demonstrate computation of local martingales associated with the solution from a representation of [Formula: see text].

THE CLASSICAL LIMIT OF REPRESENTATION THEORY OF THE QUANTUM PLANE

2013 ◽  
Vol 24 (04) ◽  
pp. 1350031 ◽  
Author(s):  
IVAN C. H. IP
Keyword(s):  
Fourier Transform ◽  
Representation Theory ◽  
Unitary Representation ◽  
Classical Limit ◽  
Mellin Transform ◽  
Tensor Products ◽  
Quantum Plane ◽  
The Fourier Transform

We showed that there is a complete analogue of a representation of the quantum plane [Formula: see text] where |q| = 1, with the classical ax+b group. We showed that the Fourier transform of the representation of [Formula: see text] on [Formula: see text] has a limit (in the dual corepresentation) toward the Mellin transform of the unitary representation of the ax+b group, and furthermore the intertwiners of the tensor products representation has a limit toward the intertwiners of the Mellin transform of the classical ax+b representation. We also wrote explicitly the multiplicative unitary defining the quantum ax+b semigroup and showed that it defines the corepresentation that is dual to the representation of [Formula: see text] above, and also correspond precisely to the classical family of unitary representation of the ax+b group.

scholarly journals Remarks on the Representation Theory of the Moyal Plane

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
J. M. Isidro ◽  
P. Fernández de Córdoba ◽  
J. M. Rivera-Rebolledo ◽  
J. L. G. Santander
Keyword(s):  
Representation Theory ◽  
Unitary Representation ◽  
Explicit Construction ◽  
Commutator Algebra ◽  
Moyal Plane

We present an explicit construction of a unitary representation of the commutator algebra satisfied by position and momentum operators on the Moyal plane.

scholarly journals COMMENTS ON HAMILTONIAN FORMALISM OF AdS/CFT CORRESPONDENCE

1999 ◽  
Vol 14 (02) ◽  
pp. 147-159 ◽  
Author(s):  
TOSHIO NAKATSU ◽  
NAOTO YOKOI
Keyword(s):  
Representation Theory ◽  
Unitary Representation ◽  
Canonical Quantization ◽  
Hamiltonian Formulation ◽  
Hamiltonian Formalism ◽  
Higher Dimensions ◽  
Irreducible Representations ◽  
Boundary Theory ◽  
Boundary Values ◽  
Simple Mapping

As a toy model to search for Hamiltonian formalism of the AdS/CFT correspondence, we examine a Hamiltonian formulation of the AdS2/CFT1 correspondence emphasizing unitary representation theory of the symmetry. In the course of a canonical quantization of the bulk scalars, a particular isomorphism between the unitary irreducible representations in the bulk and boundary theories is found. This isomorphism defines the correspondence of field operators. It states that field operators of the bulk theory are field operators of the boundary theory by taking their boundary values in a specific way. The Euclidean continuation provides an operator formulation on the hyperbolic coordinates system. The associated Fock vacuum of the bulk theory is located at the boundary, thereby identified with the boundary CFT vacuum. The correspondence is interpreted as a simple mapping of the field operators acting on this unique vacuum. Generalization to higher dimensions is speculated.

The Primitive Spectrum and Category  for the Periplectic Lie Superalgebra

2018 ◽  
Vol 72 (3) ◽  
pp. 625-655 ◽  
Author(s):  
Chih-Whi Chen ◽  
Kevin Coulembier
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Braid Group ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Left Action ◽  
Right Action ◽  
New Type

AbstractWe solve two problems in representation theory for the periplectic Lie superalgebra $\mathfrak{p}\mathfrak{e}(n)$, namely, the description of the primitive spectrum in terms of functorial realisations of the braid group and the decomposition of category ${\mathcal{O}}$ into indecomposable blocks.To solve the first problem, we establish a new type of equivalence between category ${\mathcal{O}}$ for all (not just simple or basic) classical Lie superalgebras and a category of Harish-Chandra bimodules. The latter bimodules have a left action of the Lie superalgebra but a right action of the underlying Lie algebra. To solve the second problem, we establish a BGG reciprocity result for the periplectic Lie superalgebra.

scholarly journals A Basis Theorem for the Degenerate Affine Oriented Brauer–Clifford Supercategory

2019 ◽  
Vol 71 (5) ◽  
pp. 1061-1101 ◽  
Author(s):  
Jonathan Brundan ◽  
Jonathan Comes ◽  
Jonathan Robert Kujawa
Keyword(s):  
Representation Theory ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Basis Theorem

AbstractWe introduce the oriented Brauer–Clifford and degenerate affine oriented Brauer–Clifford supercategories. These are diagrammatically defined monoidal supercategories that provide combinatorial models for certain natural monoidal supercategories of supermodules and endosuperfunctors, respectively, for the Lie superalgebras of type Q. Our main results are basis theorems for these diagram supercategories. We also discuss connections and applications to the representation theory of the Lie superalgebra of type Q.

scholarly journals The unitary representation theory of GL(n) of an infinite discrete field

1989 ◽  
Vol 67 (1) ◽  
pp. 67-81 ◽  
Author(s):  
Roger E. Howe ◽  
Jonathan Rosenberg
Keyword(s):  
Representation Theory ◽  
Unitary Representation
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