Generalized symmetries and deformations of the direct sums of Lie algebras

This paper classifies irreducible, integrable highest weight modules for “current Kac–Moody Algebras” with finite-dimensional weight spaces. We prove that these modules turn out to be modules of appropriate direct sums of finitely many copies of Kac–Moody Lie algebras.

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ON THE CONSTRUCTION AND THE STRUCTURE OF OFF-SHELL SUPERMULTIPLET QUOTIENTS

International Journal of Modern Physics A ◽

10.1142/s0217751x12501734 ◽

2012 ◽

Vol 27 (29) ◽

pp. 1250173 ◽

Cited By ~ 3

Author(s):

TRISTAN HÜBSCH ◽

GREGORY A. KATONA

Keyword(s):

Lie Algebras ◽

Central Charge ◽

Unitary Representations ◽

Direct Sums ◽

Connected Network ◽

Finite Dimensional ◽

Component Field ◽

Math Phys

Recent efforts to classify representations of supersymmetry with no central charge [C. F. Doran et al., Adv. Theor. Math. Phys.15, 1909 (2011)] have focused on supermultiplets that are aptly depicted by Adinkras, wherein every supersymmetry generator transforms each component field into precisely one other component field or its derivative. Herein, we study gauge-quotients of direct sums of Adinkras by a supersymmetric image of another Adinkra and thus solve a puzzle in the paper by Doran et al., Int. J. Mod. Phys. A22, 869 (2007): such (gauge-)quotients are not Adinkras but more general types of supermultiplets, each depicted as a connected network of Adinkras. Iterating this gauge-quotient construction then yields an indefinite sequence of ever larger supermultiplets, reminiscent of Weyl's construction that is known to produce all finite-dimensional unitary representations in Lie algebras.

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SOME PROPERTIES ON ISOCLINISM OF LIE ALGEBRAS AND COVERS

Journal of Algebra and Its Applications ◽

10.1142/s0219498808002965 ◽

2008 ◽

Vol 07 (04) ◽

pp. 507-516 ◽

Cited By ~ 13

Author(s):

ALI REZA SALEMKAR ◽

HADI BIGDELY ◽

VAHID ALAMIAN

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Schur Multiplier ◽

Direct Sums ◽

Schur Multipliers ◽

Finite Dimensional ◽

Equivalent Conditions

In this paper, we give some equivalent conditions for Lie algebras to be isoclinic. In particular, it is shown that if two Lie algebras L and K are isoclinic then L can be constructed from K and vice versa using the operations of forming direct sums, taking subalgebras, and factoring Lie algebras. We also study connection between isoclinic and the Schur multiplier of Lie algebras. In addition, we deal with some properties of covers of Lie algebras whose Schur multipliers are finite dimensional and prove that all covers of any abelian Lie algebra have Hopfian property. Finally, we indicate that if a Lie algebra L belongs to some certain classes of Lie algebras then so does its cover.

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INTEGRABLE COUPLING SYSTEMS OF HAMILTONIAN LATTICE EQUATIONS BY SEMI-DIRECT SUMS OF LIE ALGEBRAS

Modern Physics Letters B ◽

10.1142/s0217984910022755 ◽

2010 ◽

Vol 24 (07) ◽

pp. 681-694

Author(s):

LI-LI ZHU ◽

JUN DU ◽

XIAO-YAN MA ◽

SHENG-JU SANG

Keyword(s):

Eigenvalue Problem ◽

Lie Algebras ◽

Hamiltonian Structure ◽

Liouville Integrability ◽

Direct Sums ◽

Direct Sum ◽

Hamiltonian Lattice ◽

Integrable Couplings ◽

Type Lattice ◽

Integrable Coupling

By considering a discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations are derived. The relation to the Toda type lattice is achieved by variable transformation. With the help of Tu scheme, the Hamiltonian structure of the resulting lattice hierarchy is constructed. The Liouville integrability is then demonstrated. Semi-direct sum of Lie algebras is proposed to construct discrete integrable couplings. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.

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THE MULTIPLIER AND THE COVER OF DIRECT SUMS OF LIE ALGEBRAS

Asian-European Journal of Mathematics ◽

10.1142/s179355711250026x ◽

2012 ◽

Vol 05 (02) ◽

pp. 1250026 ◽

Cited By ~ 3

Author(s):

Ali Reza Salemkar ◽

Behrouz Edalatzadeh

Keyword(s):

Tensor Product ◽

Lie Algebras ◽

Schur Multiplier ◽

Direct Sums ◽

Schur Multipliers ◽

Direct Sum ◽

Group Case

In this paper, we prove that the Schur multiplier of the direct sum of two arbitrary Lie algebras is isomorphic to the direct sum of the Schur multipliers of the direct factors and the usual tensor product of the Lie algebras, which is similar to the work of Miller (1952) in the group case. Also, a cover for the direct sum of two Lie algebras in terms of given covers of them will be constructed.

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A discrete variational identity on semi-direct sums of Lie algebras

Journal of Physics A Mathematical and Theoretical ◽

10.1088/1751-8113/40/50/010 ◽

2007 ◽

Vol 40 (50) ◽

pp. 15055-15069 ◽

Cited By ~ 115

Author(s):

Wen-Xiu Ma

Keyword(s):

Lie Algebras ◽

Direct Sums

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AN ALGEBRAIC STRUCTURE OF ZERO CURVATURE REPRESENTATIONS ASSOCIATED WITH COUPLED INTEGRABLE COUPLINGS AND APPLICATIONS TO τ-SYMMETRY ALGEBRAS

International Journal of Modern Physics B ◽

10.1142/s0217979211101351 ◽

2011 ◽

Vol 25 (23n24) ◽

pp. 3237-3252 ◽

Cited By ~ 2

Author(s):

LIN LUO ◽

WEN-XIU MA ◽

ENGUI FAN

Keyword(s):

Lie Algebras ◽

Algebraic Structure ◽

Algebraic Structures ◽

Direct Sums ◽

Zero Curvature ◽

Integrable Couplings ◽

Symmetry Algebras

We establish an algebraic structure for zero curvature representations of coupled integrable couplings. The adopted zero curvature representations are associated with Lie algebras possessing two sub-Lie algebras in form of semi-direct sums of Lie algebras. By applying the presented algebraic structures to the AKNS systems, we give an approach for generating τ-symmetry algebras of coupled integrable couplings.

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