Question

Prove the following three properties of the commutator: 1\. \([a, b+\lambda c]=[a, b]+\lambda[a, c]\), for \(\lambda \in R\) (linearity); 2\. \([\boldsymbol{a}, \boldsymbol{b}]+[\boldsymbol{b}, \boldsymbol{a}]=0\) (skew-symmetry); 3\. \([[a, b], c]+[[b, c], a]+[[c, a], b]=0\) (the Jacobi identity).

Short answer

Short Answer: To prove the three properties of the commutator, we substitute the definition \([a,b]=ab-ba\) into each property and simplify. For Linearity, we show that \([a, b+\lambda c]=[a, b]+\lambda[a, c]\) by comparing the expressions for both sides. For Skew-symmetry, we show that \([a, b]+[b, a]=0\) by the cancellation of terms. Finally, to prove the Jacobi identity \([[a, b], c]+[[b, c], a]+[[c, a], b]=0\), we evaluate each term in the sum and show that they add to zero upon simplification.

Step-by-step solution

Proof of Linearity

We aim to show that \([a, b+\lambda c]=[a, b]+\lambda[a, c]\). Using the commutator definition, the left side becomes: \([a, b+\lambda c] = a(b+\lambda c) - (b+\lambda c)a = ab + \lambda ac - ba - \lambda ca\). Now, let's look at the right side of the property: \([a, b]+\lambda[a, c] = (ab - ba) + \lambda(ac-ca) = ab - ba + \lambda ac - \lambda ca\). Comparing both sides, we see they are equal. Therefore, the linearity property is proved.

Proof of Skew-symmetry

We want to show that \([a, b]+[b, a]=0\). Using the commutator definition, the left side becomes: \([a, b]+[b, a] = (ab - ba) + (ba - ab) = ab - ba + ba - ab = 0\). Thus, the skew-symmetry property is proved.

Proof of the Jacobi Identity

We aim to show that \([[a, b], c]+[[b, c], a]+[[c, a], b]=0\). Let's evaluate each term in the sum: \([[a, b], c] = [(ab - ba), c] = (ab - ba)c - c(ab-ba) = abc - bac - cab + cba\), \([[b, c], a] = [(bc - cb), a] = (bc - cb)a - a(bc-cb) = bca - cba - abc + acb\), \([[c, a], b] = [(ca - ac), b] = (ca - ac)b - b(ca-ac) = cab - bac - abc + bca\). Now, we add these three expressions: \((abc - bac - cab + cba) + (bca - cba - abc + acb) + (cab - bac - abc + bca) = 0\). Therefore, the Jacobi identity is proved.

Key concepts

Linearity in Commutators

Linearity is one of the core properties when dealing with commutators. But what does linearity mean in this context? In simple terms, linearity ensures that when we compute a commutator involving a linear combination of elements, it behaves just as expected, breaking down neatly into simpler, shorter forms.

With the commutator \([a, b + \lambda c]\), linearity implies that calculating the result involves two separate terms: one with element \(b\) and another with element \(c\). We arrive at: \([a, b] + \lambda [a, c] \). The delightful thing about linearity is how it simplifies computation by dispersing a complex operation into smaller, manageable parts.

To understand this through a mathematical lens, consider how computation proceeds:

• First, distribute operator \(a\) over \(b + \lambda c\), maintaining consistency with operations specific to \(ab\) and \(ac\).
• The term \(\lambda\) acts as a scalar influencing the second term, producing \(\lambda[a, c]\).
• The glorious result is a clear, combined expression, showcasing that commutators preserve linearity effortlessly.

Understanding Skew-symmetry

Skew-symmetry might sound complex, but it's a fundamental concept in the study of commutators and quite insightful once you grasp it. At its heart, skew-symmetry tells us that commutating two elements and reversing them doesn't affect the outcome – besides possibly swapping the sign!

The equation \([a, b] + [b, a] = 0\) elegantly showcases this. Here, we notice that swapping \(a\) and \(b\) results in their respective operations canceling each other.

Breaking it down:

• The first term, \([a, b]\), computes as \(ab - ba\).
• The second term, \([b, a]\), simplifies to \(ba - ab\).
• Combining these, \(ab - ba + ba - ab = 0\), shows perfect cancellation, where each product and its inverse nullify others.

Don't let the notation intimidate you! The beauty of skew-symmetry is in how it reveals underlying patterns and symmetries, even within complex systems.

Exploring the Jacobi Identity

The Jacobi identity is one of the most profound principles, adding a rich layer to the algebraic behavior of commutators. Despite its dense notation and curious formulas, it holds an underlying simplicity that links to consistency and structure.

Expressed as \([[a, b], c] + [[b, c], a] + [[c, a], b] = 0\), it ensures that the interrelation among three operators follows a balanced symmetry, no matter how they're shifted or permuted. To manifest this balance:

• Note that each commutator in the Jacobi identity involves bracketed pairs, allowing us to define interactions between distinct terms like \(ab\) and \(ba\), in differing orders per cycle.
• When expanded, each commutator term meticulously tracks the resulting sequences, ensuring any deviation is counterbalanced by its counterpart expressions.
• The collective addition reliably results in zero, a testament to the internal homogeneity and integrity of the combined operators.

The Jacobi identity is crucial for systems needing consistent interplay and coherence, frequently showing up in physics and higher algebra's realms.