Homomorphic groups

An element of prostereoisomerism is a partial structure that can be converted into an element of stereoisomerism not otherwise present, by considering one of a pair of homomorphic groups to be different from the other. The groups involved in this operation are necessarily heterotopic. Depending on the character of the element of stereoisomerism thus produced, one can divide the elements of prostereoisomerism into centers, lines, and planes and subdivide them, as appropriate, into those that are (fully) prochiral, only prographochiral, only... 

Levy-Leblond [16] has realized that not only the Lorentz group (or rather the homomorphic group SL(2) [32, 7]), but also the Galilei group has spinor-field representations. While the simplest possible spinor field with s = I and m 0 in the Lorentz framework is described by the Dirac equation, the corresponding field in a Galilei-invariant theory satisfies the Levy-Leblond equation (LLE)... 

At least in alkaline solution, glucosamine may not exist in the pyranose form 278). The rate of oxidation of glucosamine by hypoiodous acid coincides closely with that of the galactose-arabinose homomorphous group of aldoses rather than that of the glucose-xylose group. 

We are particularly concerned with isomorphisms and homomorphisms, in which one of the groups mvolved is a matrix group. In this circumstance the matrix group is said to be a repre.sentation of the other group. The elements of a matrix group are square matrices, all of the same dimension. The successive application of two... 

Note 1. To maintain homomorphic relationships between classes of sugars, the (potential) aldehyde group of a uronic acid is regarded as the principal function for numbering and naming (see 2-Carb-2.2.1 and 2-Carb-22). 

The groups G and G are said to be homomorphic to each other with a one-to-n correspondence, or that there is a homomorphism between G and G. ... 

In an n-dimensional space L, the linear operators of the representation can be described by their matrix representatives. This procedure produces a homomorphic mapping of the group G on a group of n x n matrices D(G), i.e., a matrix representation of the group G. From equations (6) it follows that the matrices are non-singular, and that... 

The group T of matrices is homomorphic to G and the matrices T can be characterized by the same parameters as used for characterizing the elements of G. It is implied that as the values of the parameters change continuously from those for x to those for x in the parameter space, the corresponding matrix r( r) goes continuously to T(a ). 

This result amounts to a 1 to 2 homomorphic mapping of the unitary group SU(2) onto the rotation group. From (28) it follows that the two unitary matrices... 

They are dehned by identifying locally hnite homology groups with relative cohomology groups. For example, the lower homomorphism is induced from... 

The 0(3) group is homomorphic with the SU(2) group, that of 2 x 2 unitary matrices with unit determinant [6]. It is well known that there is a two to one mapping of the elements of SU(2) onto those of 0(3). However, the group space of SU(2) is simply connected in the vacuum, and so it cannot support an Aharonov-Bohm effect or physical potentials. It has to be modified [26] to SU(2)/Z2 SO(3). 

In this section we define and discuss groups and group homomorphisms, including differentiable group homomorphisms, otherwise known as Lie group homomorphisms. 

It is often useful to think of relationships between various groups. To this end we define group homomorphisms and group isomorphisms. 

As an example, consider the detenninant. It is a standard result in linear algebra that if A and B are square matrices of the same size, then det( AB) = (detA)(detB). In other words, for each natural number n, the function det QL (C") C 0 is a group homomorphism. The kernel of the determinant is the set of matrices of determinant one. The kernel is itself a group, in this example and in general. See Exercise 4.4. A composition of... 

Definition 4.4 An injective group homomorphism 4 Gi G2 whose inverse is a group homomorphism from G2 to Gi is a group isomorphism. If there is a group isomorphism from a group G to another group G2, we say that the groups Gi anz/ G2 are isomorphic. 

Definition 4.6 Suppose Gi and G-. are Lie groups. Suppose 4 Gi —> G2 is a group homomorphism. If is differentiable, then T is a Lie group homomorphism. If is a also a group isomorphism and is differentiable then... 

There is an important surjective group homomorphism from 5 U (2) to 50(3). We will find the homomorphism useful in Section 6.6 for deriving the list of irreducible representations of 50(3) from the list of irreducible representations of 50(2). There is no a priori reason to expect such a homomorphism between two arbitrary groups, so the fact that 50(2) and 50(3) are related in this way is quite special. Here is the definition of 4> ... 

Readers should take a Utde time to familiarize themselves with this homomorphism by concrete calculations such as those in Exercise 4.38. Readers who wish to check by brute calculation that 4> is indeed a homomorphism should consult Exercise 4.32. We will take another approach, one that is more appealing geometrically (because we will see how an element of 50(2) can rotate an actual geometric object) and theoretically (because it uses concepts that generalize to other Lie groups). 

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