Question

(a) Show that \(\left[X_{i}, X_{j}\right]=-\left[X_{j}, X_{i}\right]\) implies \(c_{i j}^{k}=-c_{j t}^{k}\). (b) Show that the Jacobi identity $$ \left[\left[X_{i}, X_{j}\right], X_{k}\right]+\left[\left[X_{l}, X_{k}\right], X_{i}\right]+\left[\left[X_{k}, X_{i}\right], X_{j}\right]=0 $$ implies \(\begin{aligned} c_{i j}^{l} c_{i k}^{m}+c_{j k}^{l} c_{L i}^{m}+c_{k i}^{l} c_{l j}^{m}=0 & \text { (summation on repeated indices is } \\ & \text { implied) } \end{aligned}\) [Conditions \((a)\) and \((b)\) turn out to be the only conditions on the structure constants; any set of real numbers \(c_{i j}^{i}\) obeying these two conditions defines a Lie algebra.]

Short answer

a) \ \(c_{ij}^k = -c_{ji}^k\) b) \ \(c_{ij}^l c_{lk}^m + c_{jk}^l c_{li}^m + c_{ki}^l c_{lj}^m = 0\)

Step-by-step solution

- Understanding the commutator

Recall the definition of the commutator between two elements in a Lie algebra: \ \([X_i, X_j] = X_i X_j - X_j X_i\).

- Given property of the commutator

We are given that \([X_i, X_j] = -[X_j, X_i]\), which means the commutator is antisymmetric.

- Writing out commutators in terms of structure constants

In a Lie algebra, the commutation relation can be written as \([X_i, X_j] = c_{ij}^k X_k \).

- Using the antisymmetry property

Substitute the antisymmetric property: \ \([X_i, X_j] = -[X_j, X_i]\) implies \ \(c_{ij}^k X_k = -c_{ji}^k X_k\).

- Equating coefficients

Since the generators \(X_k\) are linearly independent, it follows that the structure constants must satisfy \ \(c_{ij}^k = -c_{ji}^k\).

- Jacobi Identity Introduction

Write down the Jacobi identity: \ \([ [X_i, X_j], X_k] + [ [X_j, X_k], X_i] + [ [X_k, X_i], X_j] = 0\).

- Expand each term using structure constants

Use the structure constants to expand each term in the Jacobi identity: \ \([X_i, X_j] = c_{ij}^l X_l\), so \ \([ [X_i, X_j], X_k] = c_{ij}^l [X_l, X_k] = c_{ij}^l c_{lk}^m X_m\). \ Apply this expansion to all terms.

- Substitution and simplification

Substitute these expansions into the Jacobi identity: \ \(c_{ij}^l c_{lk}^m X_m + c_{jk}^l c_{li}^m X_m + c_{ki}^l c_{lj}^m X_m = 0\).

- Factorization and final result

Since the generators \(X_m\) are linearly independent, the coefficients of each term must sum to zero: \ \(c_{ij}^l c_{lk}^m + c_{jk}^l c_{li}^m + c_{ki}^l c_{lj}^m = 0\). This is the Jacobi identity in terms of structure constants.

Key concepts

Understanding Commutators

In the context of Lie algebras, a commutator is a fundamental operation that measures the non-commutativity of elements. The commutator of two elements, typically denoted as \[ [X_i, X_j] = X_i X_j - X_j X_i \], provides a way to operate on algebraic structures where the order of operations matters.
Given the property \[ [X_i, X_j] = -[X_j, X_i] \], we observe that commutators are antisymmetric. This antisymmetry means that swapping the order of the elements changes the sign of the result.
In practical terms, understanding and computing commutators is crucial for exploring and solving problems involving Lie algebras, as they are widely used to define relationships between different elements within the algebra.

Structure Constants

Structure constants, denoted as \[ c_{ij}^k \], are coefficients that define the specific relationships between elements in a Lie algebra. The commutation relation can be expressed as \[ [X_i, X_j] = c_{ij}^k X_k \].
These constants play a pivotal role in characterizing the Lie algebra's structure.
Given the antisymmetric property of commutators, \[ [X_i, X_j] = -[X_j, X_i] \], we can infer that \[ c_{ij}^k = -c_{ji}^k \]. This relationship ensures that structure constants respect the antisymmetric nature of commutators.
Structure constants are essential for defining the interactions between different generators of the Lie algebra and are foundational in calculations involving these generators.

Jacobi Identity

The Jacobi identity is a key property of Lie algebras that ensures the algebra's consistency and structure. It is expressed as \[ [[X_i, X_j], X_k] + [[X_j, X_k], X_i] + [[X_k, X_i], X_j] = 0 \].
This identity can also be written in terms of structure constants. By expanding each term, we find \[ [X_i, X_j] = c_{ij}^l X_l \], leading to \[ [[X_i, X_j], X_k] = c_{ij}^l [X_l, X_k] = c_{ij}^l c_{lk}^m X_m \].
Applying this expansion, we have \[ c_{ij}^l c_{lk}^m X_m + c_{jk}^l c_{li}^m X_m + c_{ki}^l c_{lj}^m X_m = 0 \].
Because the generators \[ X_m \] are linearly independent, their coefficients must sum to zero, resulting in the relation \[ c_{ij}^l c_{lk}^m + c_{jk}^l c_{li}^m + c_{ki}^l c_{lj}^m = 0 \]. This is the Jacobi identity expressed in terms of structure constants and is critical for maintaining the algebra's integrity.

Antisymmetry

Antisymmetry is a fundamental property in Lie algebras, particularly in the context of commutators. It is defined by the equation \[ [X_i, X_j] = -[X_j, X_i] \], indicating that swapping the order of elements in a commutator changes the sign.
This property extends to the structure constants, leading to the important condition \[ c_{ij}^k = -c_{ji}^k \]. Antisymmetry is key to ensuring that the algebraic structure remains consistent and well-defined.
Understanding antisymmetry helps in verifying the relationships between different elements and ensures that more complex identities, such as the Jacobi identity, hold true.

Linear Independence

Linear independence in Lie algebras refers to the idea that a set of generators \[ X_1, X_2, ..., X_n \] do not express any of these elements as a linear combination of the others. This concept is vital for understanding the structure and dimensional properties of the algebra.
In the step-by-step solution, we used the linear independence of the generators to conclude that the only solution to \[ c_{ij}^l c_{lk}^m X_m + c_{jk}^l c_{li}^m X_m + c_{ki}^l c_{lj}^m X_m = 0 \] is if the coefficients themselves sum to zero: \[ c_{ij}^l c_{lk}^m + c_{jk}^l c_{li}^m + c_{ki}^l c_{lj}^m = 0 \].
The independence of the generators means that if their coefficients sum to zero, it must be true for each individual coefficient. This reasoning helps verify the Jacobi identity and other fundamental properties embedded in the structure of Lie algebras.