Mark GOULD

A definition for the covalent and ionic bond index in a molecule

Publisher: Springer Science and Business Media LLC

Date: 12-04-2008

DOI: 10.1007/S00214-007-0282-X

Solution of the graded Yang-Baxter equation associated with the vector representation of

Publisher: Elsevier BV

Date: 03-1991

DOI: 10.1016/0370-2693(91)90870-V

From representations of the braid group to solutions of the Yang-Baxter equation

Publisher: Elsevier BV

Date: 05-1991

DOI: 10.1016/0550-3213(91)90369-9

Unitary representations of basic classical Lie superalgebras

Publisher: Springer Science and Business Media LLC

Date: 10-1990

DOI: 10.1007/BF00398365

QUANTUM SUPERGROUPS, LINK POLYNOMIALS AND REPRESENTATION OF THE BRAID GENERATOR

Publisher: World Scientific Pub Co Pte Lt

Date: 06-1993

DOI: 10.1142/S0129055X93000097

Abstract: Unlike the quantum group case, it is shown that the braid generator σ is not always diagonalizable on V ⊗ V, V an irreducible module for a quantum supergroup. Nevertheless a generalization of the Reshetikhin form of the braid generator, obtained previously for quantum groups, is determined corresponding to every finite dimensional standard cyclic module V of a quantum supergroup. This result is applied to obtain a general closed formula for link polynomials arising from standard cyclic modules of a quantum supergroup belonging to a certain class. As explicit ex les we determine link polynomials corresponding to the rank 2 symmetric tensor representation of U q [gl(m|m)] and the defining representation of U q [osp(2n|2n)].

Fine-grained uncertainty relations under relativistic motion

Publisher: IOP Publishing

Date: 30-07-2018

DOI: 10.1209/0295-5075/122/60001

QUANTUM SUPERGROUPS AND SOLUTIONS OF THE YANG-BAXTER EQUATION

Publisher: World Scientific Pub Co Pte Lt

Date: 10-05-1990

DOI: 10.1142/S0217732390000925

Abstract: A method is developed for systematically constructing trigonometric and rational solutions of the Yang-Baxter equation using the representation theory of quantum supergroups. New quantum R-matrices are obtained by applying the method to the vector representations of quantum osp (1/2) and gl (m/n).

Invariants and reduced matrix elements associated with the Lie superalgebra gl(m|n)

Publisher: AIP Publishing

Date: 2013

DOI: 10.1063/1.4773573

Abstract: We construct explicit formulae for the eigenvalues of certain invariants of the Lie superalgebra gl(m|n) using characteristic identities. We discuss how such eigenvalues are related to reduced Wigner coefficients and the reduced matrix elements of generators, and thus provide a first step to a new algebraic derivation of matrix element formulae for all generators of the algebra.

Generalized Gel’fand invariants and characteristic identities for quantum groups

Publisher: AIP Publishing

Date: 09-1991

DOI: 10.1063/1.529152

Abstract: The generalized Gel’fand invariants for an arbitrary quantum group are explicitly constructed, and their eigenvalues in any irreducible representation are computed. These invariants enable one to develop characteristic identities for the quantum group, and as a natural application, these identities are used to construct projection operators for tensor products of representations. To illustrate the general theory, the quantum group Uq(gl(m)) is studied in detail.

Quantum double construction for graded Hopf algebras

Publisher: Cambridge University Press (CUP)

Date: 06-1993

DOI: 10.1017/S0004972700015197

Abstract: A detailed proof of the quantum double construction is given for Z 2 -graded Hopf algebras, and an explicit formula for the graded universal R −matrix is obtained in a general fashion.

Classification of all star irreps of gl(m‖n)

Publisher: AIP Publishing

Date: 11-1990

DOI: 10.1063/1.529001

Abstract: All the finite-dimensional star irreps of gl(m‖n) are classified in terms of their highest weights, thereby completing the classification of all finite-dimensional star irreps of the basic classical Lie superalgebras. The lowest weights of such irreps are determined explicitly and it is shown that the contravariant and covariant tensor irreps of gl(m‖n) are star irreps of type (1) and (2), respectively. This explains the applicability of the Young diagram method to these two classes of representations. However, it is shown that one-parameter families of typical star irreps occur that are intrinsically different from the tensorial irreps.

Characteristic identities for Lie (super)algebras

Publisher: IOP Publishing

Date: 13-04-2015

DOI: 10.1088/1742-6596/597/1/012045

Matrix elements and duality for type 2 unitary representations of the Lie superalgebra gl(m|n)

Publisher: AIP Publishing

Date: 12-2015

DOI: 10.1063/1.4938076

Abstract: The characteristic identity formalism discussed in our recent articles is further utilized to derive matrix elements of type 2 unitary irreducible gl(m|n) modules. In particular, we give matrix element formulae for all gl(m|n) generators, including the non-elementary generators, together with their phases on finite dimensional type 2 unitary irreducible representations which include the contravariant tensor representations and an additional class of essentially typical representations. Remarkably, we find that the type 2 unitary matrix element equations coincide with the type 1 unitary matrix element equations for non-vanishing matrix elements up to a phase.

Universal R matrices and invariants of quantum supergroups

Publisher: AIP Publishing

Date: 12-1991

DOI: 10.1063/1.529487

Abstract: A general method is developed for constructing invariants of quantum supergroups using universal R matrices. Applied to Uq(gl(2/1)), this method yields the invariants of this quantum supergroup in explicit form. The eigenvalues of these invariants in arbitrary irreducible highest weight representations are computed.

Classification of all star and grade star irreps of gl(n‖1)

Publisher: AIP Publishing

Date: 06-1990

DOI: 10.1063/1.528695

Abstract: A method for inducing nondegenerate forms on irreducible gl(m‖n) modules that implies some general results on star and grade star representations is investigated. These results are applied to obtain a complete classification, in terms of highest weights, of the irreducible star and grade star representations of gl(n‖1). It is demonstrated that while gl(n‖1) admits a large class of star representations the irreducible grade star representations are comparatively rare. Moreover, for n≠2 all grade star irreducible representations are also star representations and, for n& , are atypical. The superalgebra gl(2‖1) proves to be a special case and admits a two-parameter family of four-dimensional typical grade star irreducible representations that are not star representations. In particular, typical grade star irreducible representations of gl(n‖1) exist only for n=1,2.

Uncertainty relation in Schwarzschild spacetime

Publisher: Elsevier BV

Date: 04-2015

DOI: 10.1016/J.PHYSLETB.2015.02.058

Invariants and reduced Wigner coefficients for quasi-triangular Hopf superalgebras

Publisher: IOP Publishing

Date: 16-05-2022

DOI: 10.1088/1751-8121/AC6715

Abstract: We develop explicit formulae for the eigenvalues of various invariants for highest weight irreducible representations of the quantum supergroup U q [ gl ( m | n )]. The techniques employed make use of modified characteristic identity methods and allow for the evaluation of generator matrix elements and reduced Wigner coefficients.

Classification of star and grade-star representations of C(n+1)

Publisher: AIP Publishing

Date: 08-1990

DOI: 10.1063/1.528687

Abstract: The two types of * and grade-* representations of the Lie superalgebra C(n+1) are classified. A type (1) irreducible *-representation is characterized by the single condition (Λ,αs)≥0, Λ and αs being the highest weight and the odd simple root, respectively, while the type (2) *-representations are duals of type (1) *-representations. For n& , only the identity and vector representations of C(n+1) are shown to be grade -*. The superalgebra C(2) proves to be an exception and admits several classes of nontrivial irreducible grade -* representations.

Matrix elements for type 1 unitary irreducible representations of the Lie superalgebra gl(m|n)

Publisher: AIP Publishing

Date: 2014

DOI: 10.1063/1.4861706

Abstract: Using our recent results on eigenvalues of invariants associated to the Lie superalgebra gl(m|n), we use characteristic identities to derive explicit matrix element formulae for all gl(m|n) generators, particularly non-elementary generators, on finite dimensional type 1 unitary irreducible representations. We compare our results with existing works that deal with only subsets of the class of type 1 unitary representations, all of which only present explicit matrix elements for elementary generators. Our work therefore provides an important extension to existing methods, and thus highlights the strength of our techniques which exploit the characteristic identities.

Probing Planckian physics in de Sitter space with quantum correlations

Publisher: Elsevier BV

Date: 12-2014

DOI: 10.1016/J.AOP.2014.10.014

Entropic uncertainty relations under the relativistic motion

Publisher: Elsevier BV

Date: 10-2013

DOI: 10.1016/J.PHYSLETB.2013.08.069

Reduced Wigner coefficients for Lie superalgebragl(m|n) corresponding to unitary representations and beyond

Publisher: IOP Publishing

Date: 08-08-2017

DOI: 10.1088/1751-8121/AA8119

Discovery Projects - Grant ID: DP0663005

Start Date: 2006

End Date: 12-2009

Amount: $246,000.00

Funder: Australian Research Council

Discovery Projects - Grant ID: DP140101492

Start Date: 2014

End Date: 12-2017

Amount: $335,000.00

Funder: Australian Research Council

Linkage - International - Grant ID: LX0346913

Start Date: 09-2002

End Date: 12-2006

Amount: $66,000.00

Funder: Australian Research Council

Discovery Projects - Grant ID: DP220102530

Start Date: 07-2022

End Date: 06-2025

Amount: $313,000.00

Funder: Australian Research Council

Discovery Projects - Grant ID: DP0208808

Start Date: 2002

End Date: 12-2005

Amount: $457,836.00

Funder: Australian Research Council

Discovery Projects - Grant ID: DP0663773

Start Date: 2006

End Date: 01-2009

Amount: $501,000.00

Funder: Australian Research Council

Discovery Projects - Grant ID: DP110103434

Start Date: 2011

End Date: 12-2014

Amount: $300,000.00

Funder: Australian Research Council

Discovery Projects - Grant ID: DP160101376

Start Date: 2016

End Date: 12-2020

Amount: $305,500.00

Funder: Australian Research Council

Discovery Projects - Grant ID: DP0344455

Start Date: 2003

End Date: 12-2006

Amount: $245,000.00

Funder: Australian Research Council

Discovery Projects - Grant ID: DP180102185

Start Date: 2018

End Date: 12-2022

Amount: $312,438.00

Funder: Australian Research Council

Special Research Initiatives - Grant ID: SR0354741

Start Date: 2004

End Date: 12-2004

Amount: $10,000.00

Funder: Australian Research Council