Unitary algebras

Expansions including terms of the type bfab and bab are sometimes used. The corresponding algebraic structures are more complicated than those of the unitary algebras U n + 1) of Eq. 2.21. The algebras constructed from bah y b b, b bp are the sym-plectic algebras Sp (2n + 2). 

A Casimir operator containing p operators X, is called of order p. Only unitary algebras U(n) have linear Casimir operators. All other algebras have Casimir... 

The irreducible representations of unitary algebras, U( ), are characterized by a set of n integers, corresponding to all possible partitions of an integer s,... 

Kellman, M. E. (1983), Dynamical Symmetries in a Unitary Algebraic Model of Coupled Local Modes of Benzene, Chem. Phys. Lett. 103,40. 

This is the equivalent of stating that the G s give the unitary algebra U(n +1). Moreover, it is possible to write the Hamiltonian operator (2.36) directly in terms of the generators (2.37),... 

We start our discussion with the construction of a proper, spectrumgenerating algebra for two interacting oscillators by attaching a unitary algebraic structure to each of them. To be more precise, we need two unitary algebras in four dimensions, Ui(4) and 113(4), whose boson operators are a replacement of the vector coordinates, and (as explained in Section II.C.2). Thus we have... 

The transformation U(it) which maps the operator algebra /(x),An x) onto the operator algebra of the time reversed operators is fundamentally different from the unitary mappings previously considered. This can most easily be seen as follows ... 

Thus unitary operators for the group are associated with anti-Hermitian operators for the Lie algebra. Replacing P — iP, gives P = P ... 

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... 

We shall omit from here on the letter S in the orthogonal algebras since there is no difference in the algebraic structure of SO(n) and 0(n). However, we will keep the letter S, if appropriate, in the unitary groups, since there is a difference in the algebraic structure of SU( ) and U(n). 

As mentioned before in connection with one-dimensional problems, the states (2.101) or (2.96) provide bases in which all algebraic calculations can be done. These bases are orthogonal bases for three-dimensional problems. They can be converted one into the other by unitary transformations that have been (Frank and Lemus, 1986) written down explicitly. 

These considerations make the elements of a group embedded in the algebra behave like a basis for a vector space, and, indeed, this is a normed vector space. Let X be any element of the algebra, and let [x] stand for the coefficient of / in x. Also, for all of the groups we consider in quantum mechanics it is necessary that the group elements (not algebra elements) are assumed to be unitary. There will be more on this below in Section 5.4 This gives the relation pt = p h Thus we have... 

Permutations are unitary operators as seen in Eq. (5.27). This tells us how to take the Hermitian conjugate of an element of the group algebra. 

SU(n) Group Algebra. Unitary transformations, U( ), leave the modulus squared of a complex wavefunction invariant. The elements of a U( ) group are represented by n x n unitary matrices with a determinant equal to 1. Special unitary matrices are elements of unitary matrices that leave the determinant equal to +1. There are n2 — 1 independent parameters. SU( ) is a subgroup of U(n) for which the determinant equals +1. 

IJ(n) Group Algebra. Unitary matrices, U, have a determinant equal to 1. The elements of U(n) are represented by n x n unitary matrices. 

Four decades ago, Bell [3] introduced a particle-hole conjugation operator CB into nuclear shell theory. Its operator algebra is essentially isomorphic to that of Cq (for example, CB is unitary), the filled Dirac sea now corresponding to systems with half-filled shells. This was later extended to other areas of physics. For example,... 

The main idea of the proof of Theorem 12.4.6 is to apply Theorem 9.1.7(ii) to an algebraically closed field C of characteristic 0. According to Lemma 9.2.5, the left hand side of the equation in Theorem 9.1.7(ii) is algebraic over the smallest unitary subring Z of C and the right hand side of that equation is in the smallest subfield of C. Thus, according to Lemma 8.2.5, both sides must be in Z, and one obtains an integral divisibility condition. 

Proof. Let C be an algebraically closed field of characteristic 0. Let us write Q to denote the smallest subfield of C, and let Z denote the smallest unitary subring of C. 

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