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In this manuscript, a novel effort has been made to theoretically realize the construction of light tetraquarks made up of two quark-antiquark pairs using Lie group representation. It has been amiably shown that they constitute an isospin multiplet. A five-dimensional representation of the SU(2) group has been projected and these exotic hadrons innately fill this submission in the postulated SU(2) isospin group. Additionally, the attainment of isospin triplets and singlets of tetraquarks has also been done which satisfies the desired requirement of the Kronecker product of four doublets of the SU(2) group. Furthermore, the physical analysis of the states and the weights can be credibly given in terms of these light tetraquarks. [ABSTRACT FROM AUTHOR]

Let g be a complex simple Lie algebra and Z (g) be the center of the universal enveloping algebra U (g). Denote by V λ the finite-dimensional irreducible g -module with highest weight λ. Lehrer and Zhang defined the notion of strongly multiplicity free representations for simple Lie algebras motivated by studying the structure of the endomorphism algebra End U (g) (V λ ⊗ r) in terms of the quotients of the Kohno's infinitesimal braid algebra. Kostant introduced the g -invariant endomorphism algebras R λ (g) = (End V λ ⊗ U (g)) g and R λ , π (g) = (End V λ ⊗ π (U (g))) g. In this paper, we give some other criteria for a multiplicity free representation to be strongly multiplicity free by classifying the pairs (g , V λ) , which are multiplicity free and for such pairs, R λ (g) and R λ , π (g) are generated by generalizations of the quadratic Casimir elements of Z (g). [ABSTRACT FROM AUTHOR]

We investigate the nonlinear algebra W 3 generated by the 9 functionally independent permutation-symmetric operators in the three-particle rational quantum Calogero model. Decoupling the center of mass, we pass to a smaller algebra W 3 ′ generated by 7 operators, which fall into a spin-1 and a spin- 3 2 representation of the conformal sl (2) subalgebra. The commutators of the spin- 3 2 generators with each other are quadratic in the spin-1 generators, with a central term depending on the Calogero coupling. One expects this algebra to feature three Casimir operators, and we construct the lowest one explicitly in terms of Weyl-ordered products of the 7 generators. It is a polynomial of degree 6 in these generators, with coefficients being up to quartic in ℏ and quadratic polynomials in the Calogero coupling ℏ 2 g (g − 1) . Putting back the center of mass, our Casimir operator for W 3 is a degree-9 polynomial in the 9 generators. The computations require the evaluation of nested Weyl orderings. The classical and free-particle limits are also given. Our scheme can be extended to any finite number N of Calogero particles and the corresponding nonlinear algebras WN and W N ′ . [ABSTRACT FROM AUTHOR]

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The eigenvalues and eigenfunctions of Dirac–Pauli equation have been obtained for a neutron with anomalous magnetic moment (AMM) in the presence of a strong magnetic field with cylindrical symmetry. In our calculations, the Nikiforov and Uvarov (NU) method has been used. Using the eigenfunctions and construction of the ladder operators, we show that these generators satisfy su(2) Lie algebra and computed the second-order Casimir operator of the lie algebra. [ABSTRACT FROM AUTHOR]

We show that any adjoint absolutely simple linear algebraic group over a field of characteristic zero is the automorphism group of some projector on a central simple algebra. Projective homogeneous varieties can be described in these terms; in particular, we reproduce quadratic equations by Lichtenstein defining them. [ABSTRACT FROM AUTHOR]

We discuss the quantitative ergodicity of quantum Markov semigroups in terms of the trace distance from the stationary state, providing a general criterion based on the spectral decomposition of the Lindblad generator. We then apply this criterion to the bosonic and fermionic Ornstein-Uhlenbeck semigroups and to a family of quantum Markov semigroups parametrized by semisimple Lie algebras and their irreducible representations, in which the Lindblad generator is given by the adjoint action of the Casimir element. [ABSTRACT FROM AUTHOR]

In this work we we prove that all involutive Hom-Lie triple systems are whether simple or semi-simple. Moreover, we prove that an involutive simple Lie triple system give a rise of Involutive Hom-Lie triple system. [ABSTRACT FROM AUTHOR]

We have considered the Iwasawa and Gauss decompositions for the Lie group SL(2,R). According to these decompositions, the Casimir operators of the group and the Hamiltonians with position-dependent mass were expressed. Then, the unbound state solutions of the Schrödinger equations with position-dependent mass were given.

In this paper, we review classical and quantum field theory of massive non-interacting spin-two fields. We derive the equations of motion and Fierz–Pauli constraints via three different methods: the eigenvalue equations for the Casimir invariants of the Poincaré group, a Lagrangian approach, and a covariant Hamilton formalism. We also present the conserved quantities, the solution of the equations of motion in terms of polarization tensors, and the tree-level propagator. We then discuss canonical quantization by postulating commutation relations for creation and annihilation operators. We express the energy, momentum, and spin operators in terms of the former. As an application, quark–antiquark currents for tensor mesons are presented. In particular, the current for tensor mesons with quantum numbers J P C = 2 − + is, to our knowledge, given here for the first time. [ABSTRACT FROM AUTHOR]

Following the Poincaré algebra, in the Hamiltonian approach, for a free spinning particle and using the Casimirs of the algebra, we construct systematically a set of Lagrangians for the relativistic spinning particle which includes the Lagrangian given in the literature. We analyze the dynamics of this generalized system in the Lagrangian formulation and show that the equations of motion support an oscillatory solution corresponding to the spinning nature of the system. Then we analyze the canonical structure of the system and present the correct gauge suitable for the spinning motion of the system. [ABSTRACT FROM AUTHOR]

This paper focuses on semi-simple Jordan triple systems. We prove that a Jordan triple system is semi-simple if and only if its Casimir operator is nondegenerate. Moreover, we show that a pseudo-Euclidean Jordan triple system is semi-simple (resp. simple) if and only if its index is equal to the number of its simple ideals (resp. equal to one). As an application of these results, we give a new proof of the equivalence between the simplicity of a Jordan triple system and that of its TKK-Lie algebra. [ABSTRACT FROM AUTHOR]