Unitary group, definition

Definition 4.2 Suppose V is a complex scalar product space. The group of unitary operators on V is called the unitary group (of V) and denoted U (V). IfV is finite dimensional, then yve also define... 

In all the important quantum mechanical applications, the representations are unitary. Recall the unitary group U (V) from Definition 4.2. 

We start by defining the projective unitary representations. Recall the unitary group ZT (V) of a complex scalar product space V from Definition 4.2. The following definition is an analog of Definition 4.11. 

Definition 10.6 Suppose V is a complex scalar product space. projective unitary group of V is... 

Successful model building is at the very heart of modern science. It has been most successful in physics but, with the advent of quantum mechanics, great inroads have been made in the modelling of various chemical properties and phenomena as well, even though it may be difficult, if not impossible, to provide a precise definition of certain qualitative chemical concepts, often very useful ones, such as electronegativity, aromaticity and the like. Nonetheless, all successful models are invariably based on the atomic hypothesis and quantum mechanics. The majority, be they of the ah initio or semiempirical type, is defined via an appropriate non-relativistic, Born-Oppenheimer electronic Hamiltonian on some finite-dimensional subspace of the pertinent Hilbert or Fock space. Consequently, they are most appropriately expressed in terms of the second quantization formalism, or even unitary group formalism (see, e.g. [33]). 

Exercise 4.3 Show that the set ofly. diagonal special unitary matrices is a group and that it is isomorphic to the group T x T. (See Exercise 4.1 for the definition of the Cartesian product of groups.)... 

Definition 10.7 Suppose G is a group and V is a complex scalar product space. Then the triple G, V, p) is called a projective unitary representation if and only if p is a group homomorphism from G to PIT (V). 

Definition 10.8 Suppose G is a group, V is a complex scalar product space and p. G PU (V) is a projective unitary representation. We say that p is irreducible if the only subspace W of V such that [VT] is invariant under p is V itself. 

We begin by reiterating the definition of a PR and listing some conventions regarding PFs. A projective unitary representation of a group G = g, of dimension g is a set of matrices that satisfy the relations... 

Sharpe (2) gave an algebraic definition of the even-dimensional relative L-groups I<2j(f) (which however only applies to the simple L-groups, since it is based on the unitary Steinberg group relations of Sharpe 111). 

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