A SIMPLE ALGORITHM FOR COMPUTING THE GLOBAL CRYSTAL BASIS OF AN IRREDUCIBLE Uq(sln)-MODULE

2000 ◽  
Vol 10 (02) ◽  
pp. 191-208 ◽  
Author(s):  
BERNARD LECLERC ◽  
PHILIPPE TOFFIN
Keyword(s):  
Irreducible Representation ◽  
Simple Algorithm ◽  
Crystal Basis ◽  
Finite Dimensional ◽  
Dimensional Irreducible Representation

We describe a simple algorithm for computing Kashiwara's global crystal basis of a finite-dimensional irreducible representation of Uq(sln). Résumé: Nous décrivons un algorithme simple pour calculer la base cristalline globale de Kashiwara d'une représentation irréductible de dimension finie de Uq(sln).

ON SEIFERT CIRCLES AND FUNCTORS FOR TANGLES

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 581-610 ◽  
Author(s):  
H. C. LEE
Keyword(s):  
Irreducible Representation ◽  
Hopf Algebra ◽  
Abstract Algebra ◽  
Finite Dimensional ◽  
Dimensional Irreducible Representation ◽  
Quasitriangular Hopf Algebra

The properties of the Seifert circles in an oriented tangle diagram are exploited to prove a theorem that asserts that every (n,n)-tangle diagram is isotopic to a partially closed braid, and a second one that facilitates the assignment of wrong-way edges, one on each Seifert circle, in a tangle diagram. These result are used to identify the structure of an abstract algebra on which a functor for the isotopy of general tangles may be constructed. Any finite dimensional irreducible representation of a quasitriangular Hopf algebra is a realization of this algebra.

Irreducible representations of finitely generated nilpotent groups

1977 ◽  
Vol 81 (2) ◽  
pp. 201-208 ◽  
Author(s):  
Daniel Segal
Keyword(s):  
Irreducible Representation ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Irreducible Representations ◽  
Closed Field ◽  
Finitely Generated ◽  
Dimensional Representation ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Dimensional Irreducible Representation

1. Introduction. It is well known that every finite-dimensional irreducible representation of a nilpotent group over an algebraically closed field is monomial, that is induced from a 1-dimensional representation of some subgroup. However, even a finitely generated nilpotent group in general has infinite-dimensional irreducible representations, and as a first step towards an understanding of these one wants to discover whether they too are necessarily monomial. The main point of this note is to show how far they can fail to be so.

scholarly journals On tensor operators and characteristic identities for semi-simple Lie algebras

1978 ◽  
Vol 20 (3) ◽  
pp. 290-314 ◽  
Author(s):  
M. D. Gould
Keyword(s):  
Irreducible Representation ◽  
Lie Algebras ◽  
Tensor Operator ◽  
Projection Operators ◽  
Simple Lie Algebras ◽  
Commutation Relations ◽  
Finite Dimensional ◽  
Dimensional Irreducible Representation ◽  
Tensor Operators

AbstractTensor identities for finite dimensional representations of arbitrary semi-simple Lie algebras are derived and are applied to the construction of left-projection operators which project out the shift components of tensor operators from the left. The corresponding adjoint identities are also derived and are used for the construction of right-projection operators. It is also shown that, on a finite dimensional irreducible representation, these identities may be considerably reduced. Commutation relations between the shift tensors of a tensor operator are also computed in terms of the roots appearing in the tensor identities.

scholarly journals Description of unitary representations of the group of infinite 𝑝-adic integer matrices

2021 ◽  
Vol 25 (21) ◽  
pp. 606-643
Author(s):  
Yury Neretin
Keyword(s):  
Irreducible Representation ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Infinite Matrices ◽  
Natural Topology ◽  
Content Type ◽  
Residue Ring ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations

We classify irreducible unitary representations of the group of all infinite matrices over a p p -adic field ( p ≠ 2 p\ne 2 ) with integer elements equipped with a natural topology. Any irreducible representation passes through a group G L GL of infinite matrices over a residue ring modulo p k p^k . Irreducible representations of the latter group are induced from finite-dimensional representations of certain open subgroups.

scholarly journals On the block structure of the quantum ℛ-matrix in the three-strand braids

2018 ◽  
Vol 33 (17) ◽  
pp. 1850105 ◽  
Author(s):  
L. Bishler ◽  
An. Morozov ◽  
Sh. Shakirov ◽  
A. Sleptsov
Keyword(s):  
Block Structure ◽  
Representation Formula ◽  
Building Blocks ◽  
Essential Condition ◽  
The Road ◽  
Finite Dimensional ◽  
Dimensional Irreducible Representation ◽  
On The Road ◽  
Homfly Polynomials ◽  
Block Diagonal

Quantum [Formula: see text]-matrices are the building blocks for the colored HOMFLY polynomials. In the case of three-strand braids with an identical finite-dimensional irreducible representation [Formula: see text] of [Formula: see text] associated with each strand, one needs two matrices: [Formula: see text] and [Formula: see text]. They are related by the Racah matrices [Formula: see text]. Since we can always choose the basis so that [Formula: see text] is diagonal, the problem is reduced to evaluation of [Formula: see text]-matrices. This paper is one more step on the road to simplification of such calculations. We found out and proved for some cases that [Formula: see text]-matrices could be transformed into a block-diagonal ones by the rotation in the sectors of coinciding eigenvalues. The essential condition is that there is a pair of accidentally coinciding eigenvalues among eigenvalues of [Formula: see text] matrix. In this case in order to get a block-diagonal matrix, one should rotate the [Formula: see text] defined by the Racah matrix in the accidental sector by the angle exactly [Formula: see text].

Polynomial Invariant Theory and Taylor Series

1991 ◽  
Vol 43 (6) ◽  
pp. 1243-1262 ◽  
Author(s):  
John E. Gilbert
Keyword(s):  
Irreducible Representation ◽  
Taylor Series ◽  
Invariant Theory ◽  
Regular Representation ◽  
Polynomial Functions ◽  
Polynomial Invariant ◽  
Finite Dimensional ◽  
Isotypic Component ◽  
Image Position ◽  
The Right

For any group K and finite-dimensional (right) K-module V let be the right regular representation of K on the algebra of polynomial functions on V. An Isotypic Component of is the sum of all k-submodules of on which π restricts to an irreducible representation can then be written as f = ΣƬ ƒƬ with ƒƬ in .

Note on Weight Spaces of Irreducible Linear Representations

1968 ◽  
Vol 11 (3) ◽  
pp. 399-403 ◽  
Author(s):  
F. W. Lemire
Keyword(s):  
Irreducible Representation ◽  
Weight Function ◽  
Wide Class ◽  
Irreducible Representations ◽  
Weight Space ◽  
Closed Field ◽  
One Dimensional ◽  
Highest Weight ◽  
Linear Representations ◽  
Finite Dimensional

Let L denote a finite dimensional, simple Lie algebra over an algebraically closed field F of characteristic zero. It is well known that every weight space of an irreducible representation (ρ, V) admitting a highest weight function is finite dimensional. In a previous paper [2], we have established the existence of a wide class of irreducible representations which admit a one-dimensional weight space but no highest weight function. In this paper we show that the weight spaces of all such representations are finite dimensional.

scholarly journals On the Structure of Finite Groupoids and Their Representations

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 414 ◽  
Author(s):  
Alberto Ibort ◽  
Miguel Rodríguez
Keyword(s):  
Irreducible Representation ◽  
Regular Representation ◽  
Irreducible Representations ◽  
Quantum Mechanical ◽  
Linear Representations ◽  
Finite Dimensional ◽  
Completely Reducible ◽  
Quantum Mechanical Systems ◽  
Maschke’S Theorem ◽  
Totally Disconnected

In this paper, both the structure and the theory of representations of finite groupoids are discussed. A finite connected groupoid turns out to be an extension of the groupoids of pairs of its set of units by its canonical totally disconnected isotropy subgroupoid. An extension of Maschke’s theorem for groups is proved showing that the algebra of a finite groupoid is semisimple and all finite-dimensional linear representations of finite groupoids are completely reducible. The theory of characters for finite-dimensional representations of finite groupoids is developed and it is shown that irreducible representations of the groupoid are in one-to-one correspondence with irreducible representation of its isotropy groups, with an extension of Burnside’s theorem describing the decomposition of the regular representation of a finite groupoid. Some simple examples illustrating these results are exhibited with emphasis on the groupoids interpretation of Schwinger’s description of quantum mechanical systems.

Infinitesimal Generators on the Quantum Group SUq(2)

1998 ◽  
Vol 01 (04) ◽  
pp. 573-598 ◽  
Author(s):  
Michael Schürmann ◽  
Michael Skeide
Keyword(s):  
Irreducible Representation ◽  
Quantum Group ◽  
Lie Group ◽  
Lévy Processes ◽  
Levy Processes ◽  
Infinite Dimensional ◽  
Infinitesimal Generators ◽  
Dimensional Irreducible Representation

Quantum Lévy processes on a quantum group are, like classical Lévy processes with values in a Lie group, classified by their infinitesimal generators. We derive a formula for the infinitesimal generators on the quantum group SU q(2) and decompose them in terms of an infinite-dimensional irreducible representation and of characters. Thus we obtain a quantum Lévy–Khintchine formula.

scholarly journals Analog of selfduality in dimension nine

2015 ◽  
Vol 2015 (699) ◽  
Author(s):  
Anna Fino ◽  
Paweł Nurowski
Keyword(s):  
Irreducible Representation ◽  
Riemannian Geometry ◽  
Four Dimensions ◽  
Dimensional Irreducible Representation

AbstractWe introduce a type of Riemannian geometry in nine dimensions, which can be viewed as the counterpart of selfduality in four dimensions. This geometry is related to a 9-dimensional irreducible representation of

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