INTEGRABILITY OF CLASSICAL AFFINE W-ALGEBRAS by SOLE, ALBERTO D, KAC, VICTOR G, JIBLADZE, MAMUKA, VALERI, DANIELE

“…Abstract We prove that all classical affine W-algebras 𝒲(𝔤; f), where g is a simple Lie algebra and f is its non-zero nilpotent element, admit an integrable hierarchy of bi-Hamiltonian PDEs, except possibly for one nilpotent conjugacy class in G2, one in F4, and five in E8.…”
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Description of some classes of Leibniz algebras by Rakhimov, Isamiddin Sattarovich, Rikhsiboev, Ikrom Mirzarustamovich, Khudoyberdiyev, Abror Khakimovich, Karimjanov, Ikboljon Abdulazizovich

“…In this paper we describe the isomorphism classes of finite-dimensional complex Leibniz algebras whose quotient algebra with respect to the ideal generated by squares is isomorphic to the direct sum of three-dimensional simple Lie algebra sl2 and a three-dimensional solvable ideal. …”
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Canonical groups for quantization on the two-dimensional sphere and one-dimensional complex projective space by Ahamad Sumadi, Ahmad Hazazi, Zainuddin, Hishamuddin

“…We explicitly show that the Lie algebra structures of both canonical groups are locally homomorphic. …”
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On the geometry of Riemannian manifolds with a Lie structure at infinity by Bernd Ammann, Robert Lauter, Victor Nistor

“…We also prove that the geometric operators are generated by the given Lie algebra of vector fields. This is the first one in a series of papers devoted to the study of the analysis of geometric differential operators on manifolds with Lie structure at infinity.…”
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Bi-Integrable Couplings of a New Soliton Hierarchy Associated with SO(4) by Yan Cao, Liangyun Chen, Baiying He

“…Based on the six-dimensional real special orthogonal Lie algebra SO(4), a new Lax integrable hierarchy is obtained by constructing an isospectral problem. …”
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On Faithful Matrix Representations of q-Deformed Models in Quantum Optics by Latif A -M. Hanna, Abdullah Alazemi, Anwar Al-Dhafeeri

“…Consider the q-deformed Lie algebra, tq:K^1,K^2q=1−qK^1K^2,K^3,K^1q=sK^3, K^1,K^4q=sK^4,K^3,K^2q=tK^3,K^2,K^4q=tK^4, and K^4,K^3q=rK^1, where r,s,t∈ℝ−0, subject to the physical properties: K^1 and K^2 are real diagonal operators, and K^3=K^4†, († is for Hermitian conjugation). …”
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Two arbitrary-order constraint-preserving schemes for the Yang–Mills equations on polyhedral meshes by Jérôme Droniou, Jia Jia Qian

“…Two numerical schemes are proposed and investigated for the Yang–Mills equations, which can be seen as a nonlinear generalisation of the Maxwell equations set on Lie algebra-valued functions, with similarities to certain formulations of General Relativity. …”
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The SCFT/VOA correspondence for twisted class S by Nair, SS

“…The construction takes as input a choice of simple Lie algebra g, and applies equally well regardless of whether g is simply laced or not. …”

Bi-Integrable and Tri-Integrable Couplings of a Soliton Hierarchy Associated with SO(3) by Jian Zhang, Chiping Zhang, Yunan Cui

“…Based on the three-dimensional real special orthogonal Lie algebra SO(3), by zero curvature equation, we present bi-integrable and tri-integrable couplings associated with SO(3) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. …”
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Erratum to: Classical W-Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents by De Sole, Alberto, Kac, Victor, Valeri, Daniele

“…There is an error in Sect. 6.2 of the original article in the computation of equations of the generalized Drinfeld–Sokolov hierarchy associated to the minimal nilpotent element f of the Lie algebra g = sln, n ≥ 3, corresponding to the unique (up to a constant factor) non-zero central element c ∈ g f0 (such c exists for minimal f only in the case g = sln, n ≥ 3).…”
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Yangians versus minimal W-algebras: A surprising coincidence by Kac, Victor G, Frajria, Pierluigi Möseneder, Papi, Paolo

“…We prove that the singularities of the R-matrix R(k) of the minimal quantization of the adjoint representation of the Yangian Y of a finite dimensional simple Lie algebra are the opposite of the roots of the monic polynomial p(k) entering in the OPE expansions of quantum fields of conformal weight 3/2 of the universal minimal affine W-algebra at level k attached to .…”
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Global quantum differential operators on quantum flag manifolds, theorems of Duflo and Kostant by Backelin, E, Kremnizer, K

“…We also describe the center of the ad-integrable part of the quantum group and the adjoint Lie algebra action on it.…”

Space-Time Diffeomorphisms in Noncommutative Gauge Theories by L. Román Juarez, J. David Vergara, Marcos Rosenbaum

“…Making use of a combination of all of these ideas, we are therefore able to apply our canonical reparametrization approach in order to derive the deformed Lie algebra of the noncommutative space-time diffeomorphisms as well as to consider how gauge transformations act on the twisted algebras of gauge and particle fields. …”
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Unitary Howe dualities in fermionic and bosonic algebras and related Dirac operators by Guner Muarem

“…Moreover, we prove that the algebra of these Dirac operators is isomorphic to the Lie algebra $\mathfrak{su}(1,2)$ which leads to the Howe dual pair $(U(n),\mathfrak{su}(1,2))$.…”
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Geometric Methods to Investigate Prolongation Structures for Differential Systems with Applications to Integrable Systems by Paul Bracken

“…They are based on a set of one forms for which the underlying structure group of the integrability condition corresponds to the Lie algebra of , , and . Each will be considered in turn and the latter two systems represent larger cases. …”
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Lie Symmetry and the Bethe Ansatz Solution of a New Quasi-Exactly Solvable Double-Well Potential by M. Baradaran, H. Panahi

“…Exact expressions for the energies, the corresponding wave functions, and the allowed values of the potential parameters are obtained using two different methods, the Bethe ansatz method and the Lie algebraic approach. Some numerical results are reported and it is shown that the results are in good agreement with each other and with those obtained previously via a different method.…”
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Pseudo-Sasakian manifolds endowed with a contact conformal connection by Vladislav V. Goldberg, Radu Rosca

“…It is proved that such manifolds are space forms M˜(K), K<0, and some remarkable properties of the Lie algebra of infinitesimal transformations of the principal vector field U˜ on M˜ are discussed. …”
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On dimensional reduction of magical supergravity theories by Naoto Kan, Shun'ya Mizoguchi

“…This serves as a basis for the solution generating technique in this supergravity as well as allows to give the Lie algebraic characterizations to some of the parameters and functions in the original D=5 Lagrangian. …”
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The Zakharov–Shabat Spectral Problem for Complexification and Perturbation of the Korteweg–de Vries Equation by Tatyana V. Redkina, Arthur R. Zakinyan, Robert G. Zakinyan

“…In this case, we use the Lie algebra of 4-dimensional quadratic nilpotent matrices. …”
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Classical Lie Point Symmetry Analysis of a Steady Nonlinear One-Dimensional Fin Problem by R. J. Moitsheki, M. D. Mhlongo

“…We perform preliminary group classification to determine forms of the arbitrary functions appearing in the considered equation for which the principal Lie algebra is extended. Some invariant solutions are constructed. …”
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