Index Concordia

Group theory – Week 01

Posted in Group theory by Index Guy on September 6, 2007

During this week professor van Nieuwenhuizen started the semester by mentioning the fact that group theory was very important for physics and the importance of the study of transformation groups acting on a carrier space. He mentioned some examples, such as the group of rotations in Euclidean space that keep the distance fixed and the rotations of a solid cube. He then moved forward by defining a group in terms of a left-inverse and left-identity as follows:

• A group $G$ is a set of elements and a multiplication rule $*$ such that
• if $a,b \in G$ then $a*b \in G$ (closure),
• $a*(b*c) = (a*b)*c$ (associativity),
• for every $a \in G$ we also have $a_{L}^{-1} \in G$ such that $a_{L}^{-1}* a = e_{L}$,
• for every $a \in G$ we have $e_{L} \in G$ with the property $e_{L} *a = a.$

When questioned (by himself) why not have item 4 before item 3 the professor stated that from a physical point of view it is easier to understand the existence of the identity as the result of doing and un-doing an operation on an element, instead of just postulating its existence. Then because of item 1 one has the identity as part of the group. Rigor will sit on the back of the room.

He then proceeded to prove that the existence of a left- inverse and identity implies the existence of a right- inverse and identity and that furthermore the left and right inverses are one and the same element and the same goes for the left and right identities. After this subtlety he proceeded to list examples of what is a group and what is not.

Some of the examples include:

1. The set of $n \times n$ matrices with real entries and non-vanishing determinant, with the operation given by matrix multiplication is a group. We also have a group with matrix addition as the operation.
2. The set of all real matrices $M$ with the operation specified as $M_{1}\star M_{2} = [M_{1}, M_{2}]$ is not a group. This is because of failure of associativity.
3. All numbers of the form $a + b\sqrt{2}$ with $a,b$ being rational numbers form a group under traditional multiplication.
4. Polynomials do not form a group since they fail to have an inverse.
5. The set $\left\{ \pm 1 , \pm i \right\}$ is a group under multiplication called the Klein group. An extension of this group is the Quaternion group with elements $\left\{ \pm 1 , \pm \tilde{i} , \pm j , \pm k\right\}$.

After these examples professor van Nieuwenhuizen introduced the concept of a finite group’s multiplication table as a matrix with the entries in the ith row and jth column as $g_{i}*g_{j}$. If there is no element repeated on column or row, then we call this array a Latin square. It turns out that not all Latin squares are represent groups. If two groups have the same multiplication table, then we say that they are isomorphic to each other.

We calculated the entries of the multiplication table for the Klein Group. Then we considered the group formed by rotations by $\pi$ of a three dimensional cube. This group has four elements, but its multiplication table is slightly different from that of the Klein group.

After all these we defined the concept of an algebra as some set of objects that have two operations (addition and multiplication) defined and linear independence over some field (either the complex or the real numbers usually). For example, the complex numbers is an algebra of elements $z = x + iy$ with the basis formed by the elements

$\left\{\pm 1, \pm i \right\}.$

Next we turned to normed division Algebras. Some definitions:

• A normed algebra is one where the norm of elements satisfy $\left|a_{1}a_{2}\right| = \left|a_{1}\right|\left|a_{2}\right|$.
• A division algebra is one where if for any two elements we have $a_{1}a_{2} = 0$ then either $a_{1} = 0$ or $a_{2} = 0$.

We then turned our attention to the four normed division algebras over the real numbers: real numbers, complex numbers, quaternions and octonions. The octonions are the black sheep of the family since the set of basis elements does not form a group, associativity fails.

Finally we mentioned briefly Dirac’s approach to octonions using, well, Dirac matrices. Mr. Dirac considered $n$-dimensional objects of the form

$\displaystyle a_{0} + a_{1}\gamma_{1} + ... + a_{n}\gamma_{n},$

with the gammas satisfying $\left\{\gamma_{i}, \gamma_{j}\right\} = 2\delta_{ij}$. An important step was to expect the product of two gammas to be a new element, not some linear combination of the gammas (like in the case of the octonions). The set of all these objects forms the Dirac group in $n$-dimensions.