By Jouko Mickelsson

Let M be a tender manifold and G a Lie workforce. during this publication we will research infinite-dimensional Lie algebras linked either to the crowd Map(M, G) of soft mappings from M to G and to the gang of dif feomorphisms of M. within the former case the Lie algebra of the gang is the algebra Mg of gentle mappings from M to the Lie algebra gof G. within the latter case the Lie algebra is the algebra Vect M of delicate vector fields on M. although, it seems that during many functions to box idea and statistical physics one needs to take care of definite extensions of the above pointed out Lie algebras. within the least difficult case M is the unit circle SI, G is a straightforward finite dimensional Lie team and the crucial extension of Map( SI, g) is an affine Kac-Moody algebra. the top weight concept of finite dimensional Lie algebras might be prolonged to the case of an affine Lie algebra. the real element is that Map(Sl, g) will be break up to optimistic and unfavorable Fourier modes and the finite-dimensional piece g corre sponding to the 0 mode.

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**Sample text**

Taking commutators of vectors belonging to simple roots or to minus simple roots one can generate the whole algebra a. Thus [z, D] = 0 for all z E a. Next we evaluate D by applying it to the highest weight vector v in a highest weight representation. We get The coefficient in front of v is easily seen to be equal to I>. + pl2 _ Ip12. A Hermitian form H on a a-module V is contravariant if H(xu,v) = H(u,B(x)v), Vu,v E V, xE a. We use the convention that a Hermitian form is linear in the first and antilinear in the second argument.

Note first that a semisimple Lie algebra is always spanned by subalgebras of the type Al' N amely, let h C g be a Cartan subalgebra and ep the system of nonzero roots. If a E ep then also -a E ep (just look at the various root systems listed earlier). Choose 0 i- x", E g", alld 0 =1= y", E g-"" remembering that dimg", = 1 Ya E ep. Set k", = [x"" y",J. If hE h then [h,k",J = [h,[x""y",ll = [x"" [y""hll - [y""[h,x,,,ll = -[x"" a(h)y",J - [y"" a(h)x",l = O. Since the Cartan subalgebra is a maximal commutative subalgebra of g, we have k", E h.

I- 0) is spanned by the vectors hi, where {h}, . ,h1} is an orthonormal basis of h. ,O,n) has multiplicity =1 and each ofthe nonzero roots (O,O,n) has multiplicity =1. , 0, 0) 10'. E ~} U {( -7jJ, 0, 1)}, where 7jJ is the highest root of (g, h), that is, 7jJ is the highest weight of the adjoint representation ad(x)y = [x,y] of g. ,O,n) and ~bras. 10'. ,O,O) I 0'. 1. Let g = Az. 2. The highest weight vector in the adjoint representation is e1,/+1 since [x a ;j,e1,/+1] = [eij,e1,z+1] = 0 for i < j.