Representations of Lie Algebras

Like Lie groups. Lie algebras have representations. In this section we define and discuss these representations. In the examples we develop facility calculating with partial differential operators. Finally, we prove Schur s Lemma along with two propositions used to construct subrepresentations. 

Suppose V is a complex vector space. Let gi (V) denote the vector space of all complex linear transformations from V to V. Then we can define a Lie bracket on gi (V) by [A, B] = AB — BA. 

By analogy with our notation for group representations, we denote a representation by a triple (g, V, p or, when the rest is clear from context, simply by V or p. As for groups, we define homomorphisms and isomorphisms of representations. 

Then we say that T is a homomorphism of (Lie algebra) representations. If in addition T is injective and surjective then we say that T is an isomorphism of (Lie algebra) representations and that p is isomorphic fo p. 

Partial differential operators will play a large role in the examples of Lie algebra representations that concern us. Hence it behooves us to consider partial derivative calculations carefully. Consider a simple example  

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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... 

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

A. 11 Example of representations of Lie algebras, 204 A. 12 Eigenvalues of Casimir operators, 204... 

In this section we develop some preliminary algebraic aspects associated to thermal systems. Our main interest will be the analysis of representations of Lie groups (for a more evolving discussion see (I. Ojima, 1981 A.E. Santana et.al., 1999 A.E. Santana et.al., 2000 T. Kopf et.al., 1997)). 

All of the results of Section 6.1 apply, mutatis mutandis, to irreducible Lie algebra representations. For example, if T is a homomorphism of Lie algebra representations, then the kernel of T and the image of T are both invariant subspaces. This leads to Schur s Lemma for Lie algebra representations. 

The results of the current section, both the lowering operators and the classification, will come in handy in Section 8.4, where we classify the irreducible representations of so(4). One can apply the classification of the irreducible representations of the Lie algebra sm(2) to the study of intrinsic spin, as an alternative to our analysis of spin in Section 10.4. More generally, raising and lowering operators are widely useful in the study of Lie algebra representations. 

This allows the analysis of representations in characteristic zero to be reduced in large part to the theory of Lie algebra representations. The theorems in Chapter 10 closely resemble results for Lie algebras in characteristic zero. 

The next class of Lie algebras, in some sense rather close to the class of semisim-ple algebras, is an extension of semisimple Lie algebras by means of linear representations of minimal dimension. Recall some definitions. 

Theorem 4.2.6 motivates the study on Lie algebras of pairs of Poisson brackets which appear from two different structures of Lie algebras on one vector space. Two such structures will be called compatible if the sum of commutators is again a commutator. The natural class of compatible Poisson brackets is determined by effective symmetric Lie algebras dual in the sense of Cartan. Let the triplet (G, Ky a) be an effective symmetric Lie algebra and G = K Q P its decomposition under an involutive automorphism cr. Suppose that the representation ad iiC — End P is irreducible. 

Holm, D. D., and Kupershmidt, B. A. "Poisson brackets and Clebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity. Physica D. Nonlinear Phenomena 6(3) (1983), 347-363. lYofimov, V. V. "Extensions of Lie algebras and Hamiltonian system. Izvest-iya Akad. Nauk SSSR, ser. matem. 47 (1983), 1303-1321. 

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

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... 

LB. Frenkel, Spinor representations of affine Lie algebras, Proc. Natl. Acad. Sci. USA 77 (1980), 6303-6306. 

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... 

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. 

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