Group algebras

It is customary in mathematical treatments of group theory to develop the representation theory entirely in terms of the group algebra. 9> Our procedure will be to use those aspects of representation theory which we already know by other means to help in developing the theory of the algebra. 

Given a group , the group algebra U is defined as consisting of all possible symbolic sums of group elements multiplied by complex numbers. Thus, an element a of the algebra is written... 

The irreducible representations of may be used to define a set of elements of the group algebra called "projection operators . The projection operator associated with the / th irreducible representation is defined by... 

For each irreducible representation JW there are different projection operators, so in all there are 2 n — g such operators. As we shall see presently, they are linearly independent, and thus form an alternative basis for the group algebra any element of U can be represented as a linear combination of the... 

Since the projection operators themselves are obviously not zero, it follows that all the coefficients ay are zero, so the e(y are indeed linearly independent. It follows that they can be used as a basis for the group algebra. In particular, any group element s may be represented as... 

A mixture of isomers may be represented by an ensemble operator , which is defined simply as a member a of the group algebra U of . 

We need to generalize the idea of a group to that of group algebra. The reader has probably already used these ideas without the terminology. The antisymmetrizer we have used so much in earlier discussions is just such an entity for a symmetric group, S ,... 

The idea of a group algebra is very powerful and allowed Frobenius to show constructively the entire structure of irreducible matrix representations of finite groups. The theory is outlined by Littlewood[37], who gives references to Frobenius s work. 

The matrices of the irreducible representations provide one with a special set of group algebra elements. We define... 

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. 

A matrix basis for group algebras of symmetric groups... 

We now define another operator (group algebra element) using the bi coset generator sums,... 

Just as we saw with the symmetric groups, groups of spatial operations have associated group algebras with a matrix basis for this 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. 

SL 2,C) Group Algebra. The special linear group of 2 x 2 matrices of determinant 1 with complex entries is SL(2,C). 

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. 

Historically, electromagnetic theory was developed for situations described by the U(l) group. The dynamic equations describing the transformations and inter-relationships of the force-field are the well-known Maxwell equations, and the group algebra underlying these equations is U(l). There was a need to... 

---