Commutative Lie algebra

The latter form a commutator Lie algebra, which is mathematically equivalent to the vectorial Lie algebra ... 

Consequently, we may take these functions as additive generators of the commutative algebra Go, which is obviously the maximal linear commutative Lie algebra of functions on M, We have reached the desired conclusion. 

There exist circumstances that sometimes facilitate the search. In fact the Hamiltonians appearing in mechanics and physics are usually quadratic (bilinear) functions. Therefore one should first examine the linear subspace of quadratic functions contained in a commutative Lie algebra V. As a rule, this examination is rather simple, and we will demonstrate it on concrete examples. 

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,... 

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)... 

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... 

In these formulas,. SM v are constant q x q matrices, which realize a representation of the Lie algebra o(l, 3) of the pseudoorthogonal group 0(1,3) and satisfy the commutation relations... 

Representahon theory encompasses more than just group representahons. Because we can add, compose and take commutators T T2 — of linear transformahons, we can represent any algebraic structure whose operahons are limited to these operahons. We will see an important example in Chapter 8, where we introduce and use the representahon theory of Lie algebras to hnd more symmetry in and make hner predictions for the hydrogen atom. 

To define a Lie algebra homomorphism, it suffices to define it on basis elements of 01 and check that the commutation relations are satisfied. Because the homomorphism is linear, it is defined uifiquely by its value at basis elements. Because the bracket is linear, if the brackets of basis elements satisfy the equality in Definition 8.7, then any linear combination of basis elements will satisfy equality in Definition 8.7. 

When two or more Lie algebras are isomorphic, it is common practice to call them equal or the same. For example, we will refer to 0q, 5o(3) and 5m(2) as the same algebra and use the shorthand i for 72(1) or ri(i), etc., when the context precludes confusion. This Lie algebra shows up in yet another guise in many physics texts, where one encounters triples of operators, say,, Ja, , Ji, J , satisfying commutation relations... 

Proposition 8.5 Suppose g is a Lie algebra and (g, V, p) is a Lie algebra representation. Suppose T. V V commutes with p. Then each eigenspace ofT is an invariant space of the representation p. 

On the 0(3) level, particular solutions of the E2> Lie algebra (796) give a total of six commutator relations. Three of these form the B cyclic theorem (B(0) = 1 units) ... 

It should be emphasized that a detailed knowledge of Lie algebras is not essential to the understanding of the applications we shall consider, since all of our results will be presented in a simple pedagogical manner using only the familiar concepts of vector spaces, operators, matrices, and commutators (see, e.g., Hamermesh, 1962 Saletan and Cromer, 1971). 

Lie algebras can often be constructed from associative algebras of operators or matrices. In fact, the Lie algebras we shall consider for physical applications can all be constructed in this manner. Thus, given an associative algebra with multiplication defined by AB we can define the Lie product by the commutator, or Lie bracket of A and B... 

It is now easily verified that the general properties (2)—(5) of a Lie algebra are formally satisfied by this commutator multiplication, so that only the closure property (1) needs to be verified in each particular case. It should also be noted that commutators arise naturally in quantum mechanics. 

Since a Lie algebra has an underlying vector space structure we can choose a basis set ,- i = 1,..., N for the Lie algebra. Furthermore, because of the bilinearity properties Eq. (1), the Lie algebra is completely defined by specifying the commutators of these basis elements ... 

These relations are called the defining commutation relations for the Lie algebra and the coefficients ciJk are called the structure constants. A different choice of basis leads to a different but equivalent set of structure constants. We also say that the are generators of the Lie algebra. The rather difficult problem of classifying different Lie algebras will not be considered here (see, e.g., Wybourne, 1974 Gilmore, 1974), since those we shall use arise quite naturally from physical considerations. 

Finally, we would like to mention that the set of operators which commute with all elements of the Lie algebra is very useful in the study of Lie algebras (cf. Section III). In general these operators do not belong to the Lie algebra and are called Casimir operators. 

Once a Lie algebra has been defined in the abstract sense via the defining commutation relations Eq. (3), it is of practical interest to find physical realizations of the generators in terms of position and momentum operators, which also satisfy these defining commutation relations. We shall call such a set of concrete operators a realization of the Lie algebra. In practice, as we shall see, we often work backwards by starting with a set of concrete operators... 

As an example consider the important three-dimensional Lie algebra often denoted as so(3) or su(2), whose defining commutation relations can be cast in the form... 

Well-known realizations of the generators of this Lie algebra are given by the three components of the orbital angular momentum vector L = r x p, the three components of the spin S = a realized in terms of the Pauli spin matrices (Schiff, 1968), or the total one-electron angular momentum J = L + S. The components of each of these vector operators satisfy the defining commutation relations Eq. (4) if we use atomic units. We should also note that the vector cross-product example mentioned earlier also satisfies Eq. (4) if we define E = iey, j = 1, 2, 3. 

Given a Lie algebra with defining commutation relations Eq. (3), we can consider the generators , as operators acting on some n-dimensional vector space W. If j> i = 1,..., n is a basis set for W then... 

This is as far as we can go with the representation of vector operators without requiring further properties of V in order to obtain the explicit form of the coefficients a and c. The additional properties we shall use are the commutation relations needed to make the six components of J and V into the Lie algebra so(4). 

To make /, V into generators of a Lie algebra we require that the commutators [Vh Vj] be expressible in terms of Jt and Vi. The possibilities we shall consider are given by... 

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