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- W1966028280 abstract "We consider matrix equations of the form dW/dz = [S, W] where S(z) is a matrix-valued function of z, embedded in a given semisimple Lie algbra, L. If the set Xi is a basis for L, we expand W as wiXi. We obtain general conditions on an n-index form pijk… such that p = (uivjwk…pijk…) is a constant if ui, vj, wk, etc., are solutions of the differential equation. It is shown that these forms are those that determine the vector invariants of the corresponding group. We also ask what transformations will map any solution of the differential equation onto another, regardless of the detailed behavior of S(z). It is shown that the transformations that do this involve these same pijk…. With this identification, we are able to obtain procedures that can be used to calculate at least the two-index forms, pij. The existence of two-index forms, other than the Killing metric gij is shown to depend on the reducibility of the adjoint representation. Finally, we show that these same forms generate Casimir operators—i.e., operators that commute with all elements of the algebra, and hence with the group associated with it. The Casimir operator so obtained is given explicitly by (pijk…XiXjXk …), where the set, Xi, is the basis of the dual algebra." @default.
- W1966028280 created "2016-06-24" @default.
- W1966028280 creator A5075534316 @default.
- W1966028280 date "1965-08-01" @default.
- W1966028280 modified "2023-10-14" @default.
- W1966028280 title "Scalar Invariants of a Rotational System in a Lie Algebra. II. Casimir Operators" @default.
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- W1966028280 doi "https://doi.org/10.1063/1.1704780" @default.
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