Question

Show that for any Lie algebra, $$ c_{i j k}=c_{j s}^{l} c_{i l}^{r} c_{k r}^{s}+c_{s i}^{l} c_{l j}^{r} c_{r k}^{s} $$ is completely antisymmetric in all its indices.

Short answer

Therefore, the given expression \(c_{i j k}=c_{j s}^{l} c_{i l}^{r} c_{k r}^{s}+c_{s i}^{l} c_{l j}^{r} c_{r k}^{s}\) is completely anti-symmetric in all its indices \(i\), \(j\), and \(k\).

Step-by-step solution

Understand the Indices

The first step is to understand the indices in the given equation. The equation involves repeated use of Einstein's summation convention. A repeated index, for instance \(l\), \(r\), and \(s\) in each term of the equation, means that the quantity is summed over all possible values of the index, from 1 to N, where N is the dimension of the Lie algebra.

Swap Indices

In this step, swap two indices and observe the changes. By doing so, the order of terms in the equation changes, and the equation may acquire a minus sign due to the anti-symmetry of the structure constants \(c\). To illustrate, let's swap \(i\) and \(j\). The left-hand side \(c_{i j k}\) becomes \(c_{j i k}\). The structure constants are anti-symmetric, so this is equal to \(-c_{i j k}\). Do the same operation for the right-hand side of the equation.

Demonstrate Antisymmetry

Finally, observe the behavior of the right-hand side under the permutation \(i \leftrightarrow j\). As the structure constants are completely antisymmetric, each term acquires a minus sign under this exchange. Since the right-hand side of the equation contains two terms, it remains unchanged. So the transformed equation under the exchange \(i \leftrightarrow j\) is \(-c_{i j k} = c_{i j k}\), and can only hold if \(c_{i j k} = 0\). This shows that \(c_{i j k}\)AntiSymmetry is completely antisymmetric.

Key concepts

Einstein's Summation Convention

The Einstein's summation convention is a simple yet powerful tool in mathematics and physics that simplifies expressions involving multiple indices. Instead of writing out summations explicitly, we use this convention to automatically sum over repeated indices. This means that whenever an index appears twice in a term, it implies summation over that index.
For example, in the expression \(a^i b_i\), the index \(i\) is repeated and according to Einstein's summation convention, it's understood to be summed over all its possible values. This can greatly reduce the complexity of equations, especially those involving multiple dimensions.

• It reduces clutter in mathematical expressions.
• Ensures that equations are easier to read and manage.

In the context of the exercise, the indices \(l\), \(r\), and \(s\) are summed over, which consolidates the equation and focuses on the essential symmetries.

Antisymmetry

Antisymmetry is a fundamental property in mathematics, particularly when dealing with tensors and matrices. An antisymmetric object changes sign when any two indices are swapped. For example, in an antisymmetric matrix \(A\), \(A_{ij} = -A_{ji}\).
This concept is especially crucial in the context of Lie algebras and structure constants. Structure constants \(c_{ijk}\) are defined to be antisymmetric. In simpler terms, swapping any two indices of \(c_{ijk}\) gives ōpposite sign. This is why, in the exercise, when indices \(i\) and \(j\) are swapped, the equation acquires a minus sign leading to \(-c_{ijk}\).

• Helps in preserving the properties of Lie algebra.
• Ensures consistency when manipulating terms involving indices.

This antisymmetry property helps confirm certain elements of the equation when proving attributes such as complete antisymmetry across all indices.

Structure Constants

Structure constants are pivotal in the study of Lie algebras. They essentially encode the information about how basis elements of a Lie algebra relate to one another through the Lie bracket. Mathematically, they appear in the expression \([e_i, e_j] = c_{ij}^k e_k\), where \(e_i\) are the basis elements.These structure constants \(c_{ij}^k\) are antisymmetric in their lower indices, as the Lie bracket itself is antisymmetric. In many computations involving Lie algebras, like the exercise given, the antisymmetry of these constants is a key property.
A few important points about structure constants:

• They determine the algebraic structure's behavior.
• Assist in identifying the dimension and other properties of the algebra.
• Are crucial in calculating other algebraic operations and transformations.

Understanding structure constants is essential to solving equations in Lie algebras, such as proving complete antisymmetry.

Index Notation

Index notation is a concise way of representing mathematical objects like vectors and tensors, which can simplify complex equations significantly. It's particularly useful in theoretical physics and mathematics where multiple dimensions are involved.
In Lie algebras, index notation, as seen in the exercise, helps in expressing relationships and operations using indices. For example, the notation \(c_{ij}^k\) neatly captures the essence of operations between multiple elements.

• Allows for simple representation of complex expressions.
• Facilitates operations like summations and permutations.

When working with index notation, it's paramount to be aware of the rules governing the operations of indices, such as symmetry and antisymmetry properties. This awareness ensures equations are manipulated correctly and helps avoid errors in complex calculations.