The generalized Casimir operator and tensor representations of groups (original) (raw)
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Journal of Physics A: Mathematical and General, 1996
The technique for determination of the symmetry adapted bases for finitedimensional representations of the Lie groups is developed, in analogy to the finite group theory. The method uses the corresponding Lie algebra, and relates the group projectors to the Casimir operators. As an application to the semisimple algebras, the general formula for generating function of the values of the Casimir operators is established. §
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2012
This paper uses the structure of the Lie algebras to identify the Casimir invariant functions and Lax operators for matrix Lie groups. A novel mapping is found from the cotangent space to the dual Lie algebra which enables Lax operators to be found. The coordinate equations of motion are given in terms of the structure constants and the Hamiltonian.
Casimir operators of groups of motions of spaces of constant curvature
Theoretical and Mathematical Physics, 1981
Limit transitions are constructed between the generators (Casimir operators) of the center of the universal covering algebra for the Lie algebras of the groups of motions of n-dimensional spaces of constant curvature. A method is proposed for obtaining the Casimir operators of a group of motions of an arbitrary n-dimensional space of constant curvature from the known Casimir operators of the group SO(n + 1). The method is illustrated for the example of the groups of motions of four-dimensional spaces of constant curvature, namely, the Galileo, Poincare, Lobachevskii, de Sitter, Carroll, and other spaces.
2005
In this letter, we recall briefly the generalized Casimir elements of a finite dimensional Lie algebra. We specify those of orders two and three : when the Lie algebra is simple (even semisimple), we begin by normalizing the former (the quadratic), and then we study some actions of the latter (the cubic). In particular, we introduce a graphical formalism, translating rigorously the tensorial calculus. This allows us to prove the main theorem in a graphic theoretic manner. MIRAMARE TRIESTE August 1996 1 E-mail: elhouari@ictp.trieste.it and elhouari@drakkar.ens.fr 1History and Introduction The Casimir elements of a Lie algebra are those in the center of its enveloping algebra. They play a crucial role in representation theory. They are specially useful for both theoretical physicists and mathematicians who work in the frontier of physics, J-P Elliot [?], M. Gell-Mann [?]. In L-C Biedenharn's and Racah's works, [?] and [?] respectively, we notice the use of the Casimir elements...
Split Casimir Operator and Universal Formulation of the Simple Lie Algebras
Symmetry, 2021
We construct characteristic identities for the split (polarized) Casimir operators of the simple Lie algebras in adjoint representation. By means of these characteristic identities, for all simple Lie algebras we derive explicit formulae for invariant projectors onto irreducible subrepresentations in T⊗2 in the case when T is the adjoint representation. These projectors and characteristic identities are considered from the viewpoint of the universal description of the simple Lie algebras in terms of the Vogel parameters.
Documenta Mathematica
Chapter 1. Generalities Definitions of group, isomorphism, representation, vector space and algebra. Biographical notes on Galois, Abel and Jacobi are given. Chapter 2. Lie groups and Lie algebras Lie Groups, infinitesimal generators, structure constants, Cartan's metric tensor, simple and semisimple groups and algebras, compact and non-compact groups. Biographical notes on Euler, Lie and Cartan are given. Chapter 3. Rotations: SO(3) and SU(2) Rotations and reflections, connectivity, center, universal covering group. Chapter 4. Representations of SU(2) Irreducible representations, Casimir operators, addition of angular momenta, Clebsch-Gordan coefficients, the Wigner-Eckart theorem, multiplicity. Biographical notes on Casimir, Weyl, Clebsch, Gordan and Wigner are given. Chapter 5. The so(n) algebra and Clifford numbers Spin(n), spinors and semispinors, Schur's lemma. Biographical notes on Clifford and Schur are given. Chapter 6. Reality properties of spinors Conjugate, orthogonal and symplectic representations. Chapter 7. Clebsch-Gordan series for spinors Antisymmetrie tensors, duality. Chapter 8. The center and outer automorphisms of Spin(n) Inversion, 12, 14 and 12 x 12 centers. A biographical note on Dynkin is given.
Classification and Casimir invariants of Lie–Poisson brackets
Physica D: Nonlinear Phenomena, 2000
We classify Lie-Poisson brackets that are formed from Lie algebra extensions. The problem is relevant because many physical systems owe their Hamiltonian structure to such brackets. A classification involves reducing all brackets to a set of normal forms, and is achieved partially through the use of Lie algebra cohomology. For extensions of order less than five, the number of normal forms is small and they involve no free parameters. We derive a general method of finding Casimir invariants of Lie-Poisson bracket extensions. The Casimir invariants of all low-order brackets are explicitly computed. We treat in detail a four field model of compressible reduced magnetohydrodynamics.
Deformations of inhomogeneous classical Lie algebras to the algebras of the linear groups
Journal of Mathematical Physics, 1973
We study a new class of deformations of algebrll representations, namely, ilso(n) ~ sl(n, R), ilu(n) ~ s l(n.C) '" u(1) and ils p (n) '" s p (I) ~ sl(n, Q) '" sp(l). The new generators are built as commutators between the Casimir invariant of the maximal compact subalgebra and a second-rank mixed tensor. These algebra deformations are related to multiplier representations and manifold mappings of the corresponding Lie groups. Behavior of the representations under Iniinii-Wigner contractions is exhibited. Through the use of these methods we can construct a principal degenerate series of representations of the linear groups and their algebras.
arXiv (Cornell University), 2022
We constructed characteristic identities for the 3-split (polarized) Casimir operators of simple Lie algebras in the adjoint representations ad and deduced a certain class of subrepresentations in ad ⊗3. The projectors onto invariant subspaces for these subrepresentations were directly constructed from the characteristic identities for the 3-split Casimir operators. For all simple Lie algebras, universal expressions for the traces of higher powers of the 3-split Casimir operators were found and dimensions of the subrepresentations in ad ⊗3 were calculated. All our formulas are in agreement with the universal description of (irreducible) subrepresentations in ad ⊗3 for simple Lie algebras in terms of the Vogel parameters.