Question

Let \(\mathbf{D}_{1}\) and \(\mathbf{D}_{2}\) be derivations of a Lie algebra \(\mathfrak{v}\). Show that \(\mathbf{D}_{1} \mathbf{D}_{2} \equiv\) \(\mathbf{D}_{1} \circ \mathbf{D}_{2}\) is not a derivation, but \(\left[\mathbf{D}_{1}, \mathbf{D}_{2}\right]\) is.

Short answer

No, the composition of two derivations \(\mathbf{D}_{1} \circ \mathbf{D}_{2}\) is not a derivation of this Lie algebra. However, their commutator \(\left[\mathbf{D}_{1}, \mathbf{D}_{2}\right]\) is a derivation of the Lie algebra.

Step-by-step solution

Exploring D1∘D2

Consider the operation \(\mathbf{D}_{1} \circ \mathbf{D}_{2}\) on the Lie bracket of two elements \(x\) and \(y\) of the Lie algebra. Using the properties of derivations, expand this expression into \( \mathbf{D}_1 (\mathbf{D}_2[x,y]) = \mathbf{D}_1 (\mathbf{D}_2x.y - x.\mathbf{D}_2y). \)

Expand \( \mathbf{D}_1 (\mathbf{D}_2x.y - x.\mathbf{D}_2y) \)

Further expanding the right-hand side using the properties of derivations again, we get \( \mathbf{D}_1 (\mathbf{D}_2x.y) - \mathbf{D}_1 (x.\mathbf{D}_2y) = \mathbf{D}_1\mathbf{D}_2x.y - \mathbf{D}_2x.\mathbf{D}_1y - x.\mathbf{D}_1\mathbf{D}_2y + x.\mathbf{D}_1y \) However, after rearranging, this straightaway implies \( =[(\mathbf{D}_{1} \circ \mathbf{D}_{2}(x),y]+[x,(\mathbf{D}_{1} \circ \mathbf{D}_{2}(y)]-[x,y] \) which definitely did not become \( = \mathbf{D}_{1}\circ\mathbf{D}_{2}([x,y]) \). Consequently, \(\mathbf{D}_{1} \circ \mathbf{D}_{2}\) is not a derivation because its action on \( [x, y] \) doesn't satisfy the derivation property.

Exploring \([D_1, D_2]\)

Next, consider the operation \(\left[\mathbf{D}_{1}, \mathbf{D}_{2}\right] = \mathbf{D}_{1}\mathbf{D}_{2} - \mathbf{D}_{2}\mathbf{D}_{1}\) on the Lie bracket of \(x\) and \(y\). Expand this expression using the properties of derivations and commutators to find: \( \left[\mathbf{D}_{1}, \mathbf{D}_{2}\right][x, y] = (\mathbf{D}_{1}\mathbf{D}_{2} - \mathbf{D}_{2}\mathbf{D}_{1})[x,y] = \mathbf{D}_{1}(\mathbf{D}_{2}[x, y]) - \mathbf{D}_{2}(\mathbf{D}_{1}[x, y]) \) which after rearranging and simplification, we obtain \( = [\mathbf{D}_{1} \mathbf{D}_{2}(x),y]+[x,\mathbf{D}_{1} \mathbf{D}_{2}(y)] \). This expression confirms \(\left[\mathbf{D}_{1}, \mathbf{D}_{2}\right]\) to be a derivation as it satisfies the derivation rule.

Key concepts

Derivation

In mathematics, a derivation on a Lie algebra is a type of linear operator that nicely behaves with the Lie bracket, maintaining certain structural aspects of the algebra. Essentially, a derivation is a function \( \mathbf{D} \) that maps elements from the Lie algebra \( \mathfrak{v} \) into itself. It should satisfy the Leibniz rule:

• \( \mathbf{D}([x, y]) = [\mathbf{D}(x), y] + [x, \mathbf{D}(y)] \) for any \( x, y \in \mathfrak{v} \).

This property is key because it ensures the derivation respects the fundamental structure of the Lie algebra, distributing over the Lie bracket like this.
Derivations can be thought of as defining how the algebra changes infinitesimally. They are linear maps, meaning they respect scalar multiplication and addition. When multiple derivations are applied in sequence, like \( \mathbf{D}_1 \circ \mathbf{D}_2 \), they do not always preserve the derivation property, as shown in the detailed steps of the solution. This makes derivation a somewhat delicate operation, demanding rigorous care to preserve the algebra's structure.

Commutator

The concept of commutators is central in Lie algebras, capturing the idea of measuring the "non-commutativity" of two elements. For two operators, such as derivations \( \mathbf{D}_1 \) and \( \mathbf{D}_2 \), the commutator is defined as:

• \( [\mathbf{D}_1, \mathbf{D}_2] = \mathbf{D}_1 \mathbf{D}_2 - \mathbf{D}_2 \mathbf{D}_1 \)

The commutator tells us how much these operations "miss" each other when reversed.
In the context of derivations, using the commutator bracket retains the essential properties required for derivations. This is because the expression incorporates ideas of both actions but considers their difference, often resulting in an operation which respects the necessary properties of a derivation. Thus, while sequential application (\( \mathbf{D}_1 \circ \mathbf{D}_2 \)) in derivations often fails to maintain derivation properties, the commutator operation can succeed, revealing its structural importance in maintaining algebraic consistency.

Lie Bracket

The Lie bracket is a fundamental operation in the theory of Lie algebras, acting analogously to multiplication but defined in terms of the algebra’s elements. It is denoted by \( [x, y] \) for any two elements \( x \) and \( y \) in the algebra \( \mathfrak{v} \). This operation must satisfy certain properties:

• Antisymmetry: \( [x, y] = -[y, x] \)
• Jacobi Identity: \( [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 \)

Together, these conditions define the heart of Lie algebra structure.
The study of derivations involves these brackets deeply because any derivation \( \mathbf{D} \) must interact with the Lie bracket under the derivation property already discussed. By ensuring properties such as antisymmetry and the Jacobi identity, the Lie bracket maintains the non-abelian nature that’s often crucial in fields like theoretical physics and abstract algebra. This structured, rule-based interaction of elements within an algebra is what keeps the entire framework coherent and consistent.