Lie subalgebra

A Lie subalgebra is a subset G of operators of G, which, by itself, is closed with respect to commutation. In other words, the commutator of two elements is a linear combination of the same elements. In mathematical terms,... 

This is the exact meaning of relations such as (2.6) and (2.7) from the viewpoint of the definition of a Lie subalgebra. 

Note that the Lie subalgebras H M) and H oc M) depend on the choice of the form u) on M. With a change of their symplectic structure, these subalgebras, in general, will change in the Lie algebra of all functions. 

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. 

Another definition of interest here involves the subalgebra G of a given Lie algebra G. One has to consider a subset G CG that is closed with respect to the same commutation laws of G,... 

Before providing more detail on this procedure, we prefer to outline the strategy specifically adopted in the U(4) problem. By analogy with the U(2) case, we need to perform two distinct steps (1) to identify the subalgebra chains of U(4) closing in SO(3), and (2) to construct the Hamiltonian operator in a dynamical symmetry sense. The first step is a strictly Lie algebraic question, which can be solved by making explicit use... 

Up to this moment we have dealt with complex semisimple Lie algebras. But an important role is also played by various real subalgebras contained in complex algebras. One of them is especially remarkable, since the corresponding Lie group is compact. 

Let Go be a certain real form of a complex semisimple Lie algebra G. Then any element of the algebra G can be uniquely represented in the form X - -iYj where X,Y 6 Go. This decomposition of the algebra G gives rise to a natural involution 6 which maps the algebra G into itself. Namely a X- -iY) = X—iY. This involution depends on the subalgebra Go and possesses the following obvious properties ... 

Lemma 1.4.1. On a Lie algebra G let an involution a be given which possesses the properties speciSed above. Then this involution de nes a certain subalgebra Go in the algebra G, which is the real form. 

First we consider our model example sl(n,C). Consider an involution a A = A on G, that is, an operation of complex conjugation. The set of fixed points of this involution coincides, obviously, with the subalgebra of real matrices, which is the Lie algebra sl(n,R). It is clear that the Killing form on the algebra sl(n, R) is a... 

Now consider an adjoint action of the Lie algebra Gc on itself, that is, examine the action of transformation of the form ad/ Gc — Gc, where h iTq. Since the element h lies in the Cart an subalgebra, it follows that the transformation adh carries into itself the plane orthogonal to the plane tTo. We make use of the fact that the operators ad are skew-symmetric with respect to the Killing form and therefore preserve the orthogonal complement by carrying it into itself. 

Lemma 1.4.4. The standard embedding of the subalgebra su(n) into the Lie algebra sl(n, C) coincides with the canonical embedding of the compact form Gc into G,... 

Definition 1.4.7 The real subalgebra EatE aiH is called a normal noncompact form of the Lie algebra G,... 

The compact real form Gc admits another invariant characteristic. This subalgebra is a maximal compact subalgebra in the complex Lie algebra G. 

A linear (infinite-dimensional) space of all smooth functions on a sympletic manifold will be denoted by (7 (A/). As we already know from Ch. 1, this space naturally transforms into an infinite-dimensional Lie algebra with respect to the Poisson bracket /, g, where /, y 6 Different subalgebras (both... 

Definition 3.1.1 We will say that on a sympletic manifold a maximal linear commutative subalgebra of functions Go is given (in the Lie algebra C (M) with respect to the Poisson bracket) if dimGo = n and if in Go one can choose an additive basis consisting of n functions /i,..., /n functionally independent on (almost everywhere). Such an algebra of functions will be sometimes called a complete involutive (commutative) set of functions. 

It turns out that a Hamiltonian system can be integrated not only in case its Hamiltonian is included in the MLC subalgebra, but also in case it is included in a noncommutative Lie algebra G of functions on Ai which has the property dimCr + rank G = dim Af ". This question is considered in the next subsection. 

Definition 3.1.2 Let G be a finite-dimensional subalgebra in a Lie algebra C [M) (with respect to the Poisson bracket). The subalgebra G will be called a maximal linear (ML) subalgebra on a sympletic manifold if dimG + indG = dimM and if we can choose in G an additive basis consisting of functionally independent (almost everywhere) functions on the manifold M. 

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