Lie algebras

The explicit symplectic integrator can be derived in terms of free Lie algebra in which Hamilton equations (5) are written in the form... 

M. R. Bremner et ai, Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras (1985)... 

H. Strade and R. Farnsteiner, Modular Lie Algebras and Their Representations (1988)... 

Thus unitary operators for the group are associated with anti-Hermitian operators for the Lie algebra. Replacing P — iP, gives P = P ... 

A Hopf algebra emerges by a proper redefinition of the antilinear characteristics of TFD. Consider g = giti = 1,2,3,.. be an associative algebra defined on the field of the complex numbers and let g be equipped with a Lie algebra structure specified by giOgj = C gk, where 0 is the Lie product and Cfj are the structure constants (we are assuming the rule of sum over repeated indeces). Now we take g first realized by C = Ai,i = 1,2,3,.. such that the commutator [Ai,Aj is the Lie product of elements Ai,Aj G C. Consider tp and (p two representations of C, such that ip (A) (linear operators defined on a representation vector space As a consequence,... 

But some algebraic characteristics of the enveloping algebra can be immediately pointed out. U(A) is a Lie algebra and admits a Hopf structure, which is induced by the product A(A), where the maps of the coalgebra are... 

This w -algebra structure can be used to develop a representation theory of symmetry groups, taking H as a representation space for Lie algebras. As before let g be a Lie algebra specified by giOgj = C gu-A unitary representation of g in H is then given by... 

Notice that this Lie algebra, to be denoted by qt, as a vector space is given by gy = g g, where g (g) is a sub-vector space of gr given by... 

This is, incidentally, the first example of a Lie algebra that we encounter. We will return to it later. Also we have set h = 1 in (1.19) to simplify the notation. 

More elementary introductions to the material in the rest of this section can be found in Messiah (1976) or Cohen-Tannoudji, Diu, and Laloe (1977). More detailed discussions are available in Fano and Racah (1959), Edmonds (1960), Brink and Satchler (1968), de Shalit and Talmi (1963), and Judd (1975). Zare (1988) is particularly useful on both the theory and the manner of its application. Special reference to diatomic molecules is made by Judd (1975) and Mizushima (1975). The close connection to Lie algebra is emphasized by Biedenham and Louck (1981). A summary of the results we need is in Appendix B. 

Algebraic theory makes use of an algebraic structure. The structure appropriate to ordinary quantum mechanical problems is that of a Lie algebra. We begin this chapter with a brief review of the essential concepts of Lie algebras.1... 

The binary operation ( multiplication ) in the Lie algebra is that of taking the commutator. As usual, we denote the commutator by square brackets, [A, fi] = AB- BA. A set of operators A is a Lie algebra when it is closed under commutation. That is, for every operator X in the algebra G (which we write as X e G)... 

The constants ccab, which characterize a given algebra, are called the Lie structure constants. A familiar example of a Lie algebra is the angular momentum algebra of Eq. (1.19), which, because of its importance, we repeat it here replacing Iby J,... 

The algebra (2.3) is called the special orthogonal algebra in three dimensions, SO(3). Associated with each Lie algebra there is a group of transformation... 

All admissible Lie algebras were classified by Cartan in 1905. Cartan s classification is given in Appendix A, where many other properties and definitions are provided. 

For each Lie algebra, one can construct a set of operators, called invariant (or Casimir, 1931) operators after the name of the physicist who first introduced them in connection with the rotation group. These operators play a very important role since they are associated with constants of the motion. They are defined as those operators that commute with all the elements of the algebra... 

The next important problem in algebraic theory is the construction of the basis states (the representations) on which the operators X act. A particular role is played by the irreducible representations (Appendix A), which can be labeled by a set of quantum numbers. For each algebra one knows precisely how many quantum numbers there are, and a list is given in Appendix A. The quantum numbers are conveniently arranged in patterns (or tableaux), called Young tableaux. Tensor representations of Lie algebras are characterized by a set of integers... 

The eigenvalues / have been evaluated for any Casimir operator of any Lie algebra, and a summary of the results is given in Appendix A. Using the expressions of the appendix, we find, for example, that the eigenvalues of the Casimir operator of SO(3), J2, in the representation 1/ > is... 

For any admissible Lie algebra, one knows the number of operators in the algebra, denoted by r in Section A.l. This number is called the order of the algebra and is given in Table A.2. In this book, we make extensive use of U(4), with 16 operators, U(3) with 9, U(2) with 4, U(l) with 1 and of the orthogonal algebras SO(4) with 6 operators, SO(3) with 3 and SO(2) with 1. 

For any Lie algebra, one can construct a set of operators, called invariant or Casimir operators, C, such that... 

Table A.4 Number of integers that characterize the tensor representations of Lie algebras...

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