14,391 research outputs found

    Fermionic realisations of simple Lie algebras

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    We study the representation D{\cal D} of a simple compact Lie algebra \g of rank l constructed with the aid of the hermitian Dirac matrices of a ({\rm dim} \g)-dimensional euclidean space. The irreducible representations of \g contained in D{\cal D} are found by providing a general construction on suitable fermionic Fock spaces. We give full details not only for the simplest odd and even cases, namely su(2) and su(3), but also for the next ({dim} \g)-even case of su(5). Our results are far reaching: they apply to any \g-invariant quantum mechanical system containing {\rm dim} \g fermions. Another reason for undertaking this study is to examine the role of the \g-invariant fermionic operators that naturally arise. These are given in terms of products of an odd number of gamma matrices, and include, besides a cubic operator, (l-1) fermionic scalars of higher order. The latter are constructed from the Lie algebra cohomology cocycles, and must be considered to be of theoretical significance similar to the cubic operator. In the ({\rm dim} \g)-even case, the product of all l operators turns out to be the chirality operator \gamma_q, q=({{\rm dim} \g+1}).Comment: Plain latex file, 17

    Optimally defined Racah-Casimir operators for su(n) and their eigenvalues for various classes of representations

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    This paper deals with the striking fact that there is an essentially canonical path from the ii-th Lie algebra cohomology cocycle, i=1,2,...li=1,2,... l, of a simple compact Lie algebra \g of rank ll to the definition of its primitive Casimir operators C(i)C^{(i)} of order mim_i. Thus one obtains a complete set of Racah-Casimir operators C(i)C^{(i)} for each \g and nothing else. The paper then goes on to develop a general formula for the eigenvalue c(i)c^{(i)} of each C(i)C^{(i)} valid for any representation of \g, and thereby to relate c(i)c^{(i)} to a suitably defined generalised Dynkin index. The form of the formula for c(i)c^{(i)} for su(n)su(n) is known sufficiently explicitly to make clear some interesting and important features. For the purposes of illustration, detailed results are displayed for some classes of representation of su(n)su(n), including all the fundamental ones and the adjoint representation.Comment: Latex, 16 page
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