Representation of a Lie algebra

Associated with each operator realization of a Lie algebra we generally have a vector space on which these operators act. For the realization given by L this might be either an abstract space of angular momentum states lm), 1 = 0, 1,... m = —/, — l + 1,..., l or a concrete realization of them as spherical harmonic functions Ylm(6, (j)). We can then consider the matrix elements of the operators with respect to this vector space of states and this leads to the important concept of a matrix representation of a Lie algebra. 

Representable functor 5 Representation, see Linear representation Representation of a Lie algebra 96 Ring of functions on S 30 Root system 98... 

We do not dwell on the details of the exploit construction of the generators of a commutative algebra of functions. If p is a certain representation of a Lie algebra then A p and S p will respectively denote the kth external and the A th symmetric degree of the representation p. 

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... 

There is a natural representation of the Lie algebra so 3 using partial differential operators on We can define the three basic angular momentum operators as linear transformations on as follows ... 

Proposition 8.4 (Schur s Lemma) Suppose (fl, Vi, pf) and (fl, Pf) are irreducible representations of the Lie algebra fl. Suppose that T Vi V2 is a homomorphism of representations. Then there are only two possible cases ... 

The Casimir operator is a useful tool for identifying a representation of the Lie algebra 5m(2). In this section we investigate Casimir operators and apply them to the classification of the finite-dimensional irreducible representations of the Lie algebra so 4 ). 

Exercise 8.10 In this exercise we construct infinite-dimensional irreducible representations of the Lie algebra su (2). Suppose k. is a complex number such that L in for any nonnegative integer n. Consider a countable set S = vo, Vi, 172,... and let V denote the complex vector space of finite linear combinations of elements of S. Show that V can be made into a complex... 

Exercise 8.12 Show that the function pi 0 / + / 0 p from Definition 8.10 satisfies the definition of a Lie algebra representation. 

When the representation space is not precisely specified, we often talk about a realization of a Lie algebra rather than the representation (cf. also... 

Belyaev, A. V. "On the motion of an n-dimensional rigid body with the group of symmetries SO (A ) SO (AT — /) in a field with linear potential. Invariants of coadjoint representation of some Lie algebras. Dokl. Akad. Nauk SSSR, 282 (1985) No. 5, 1038-1042. 

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,... 

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... 

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... 

It follows from relations (15) that the basis elements of the Lie algebra c(l, 3) have the form (6), where the functions c a depend on x e X = Rp only and the functions r j are linear in u. We will prove that owing to these properties of the basis elements of c(l, 3), the ansatzes invariant under subalgebras of the algebra (15) admit linear representation. 

It can be easily checked by direct computation that we have really obtained a realization of the Lie algebra g in a Hilbert (Fock) space, [T a, T fc] = ifabc fc, in accordance with (11), where Ta = T f/aL . For an irreducible representation R, the second-order Casimir operator C2 is proportional to the identity operator I, which, in turn, is equal to the number operator N in our Fock representation, that is, if T" —> Ta, then I /V 5/,/a . Thus we obtain an important for our further considerations constant of motion N ... 

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. 

In a sense that can be made quite precise. Lie groups are global objects and Lie algebras are local objects. To put it another way, Lie algebras are infinitesimal versions of Lie groups. In our main examples, the representation of the Lie group 50(3) on operates by rotations of functions, while the rep-... 

Like the raising and lowering operators, the Casimir operator does not correspond to any particular element of the Lie algebra 5m(2). However, for any vector space V, both squaring and addition are well defined in the algebra gt (V) of linear transformations. Given a representation, we can define the Casimir element of that representation. ... 

We can use the representation theory of the Lie algebra 50(4) along with the stunning fact that there is a representation of 5o(4) on the space of bound states of the Schrodinger operator with the Coulomb potential to make a satisfying prediction about the dimensions of the shells of the hydrogen atom and the energy levels of these shells. 

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