Question

If \(\mathcal{A}\) is an algebra and \(\mathbf{D}\) is a derivation in \(\mathcal{A}\), prove that both the center \(2(\mathcal{A})\) and the derived algebra \(\mathcal{A}^{2}\) are stable under \(\mathbf{D}\), i.e., if \(\mathbf{a} \in \mathcal{Z}(\mathcal{A})\) then \(\mathbf{D}(\mathbf{a}) \in \mathcal{L}(\mathcal{A})\), and if \(\mathbf{a} \in \mathcal{A}^{2}\) then \(\mathbf{D}(\mathbf{a}) \in \mathcal{A}^{2}\)

Short answer

The proof is done in two parts. In the first step, we show that if an element \(\mathbf{a}\) is in the center of the algebra \(\mathcal{Z}(\mathcal{A})\), then its derivative \(\mathbf{D}(\mathbf{a})\) is also in \(\mathcal{Z}(\mathcal{A})\). In the second step, we prove that if \(\mathbf{a}\) is in the derived algebra \(\mathcal{A}^{2}\), then \(\mathbf{D}(\mathbf{a})\) is also in \(\mathcal{A}^{2}\).

Step-by-step solution

Proving \(\mathbf{D}(\mathbf{a}) \in \mathcal{L}(\mathcal{A})\) for \(\mathbf{a} \in \mathcal{Z}(\mathcal{A})\)

Suppose \(\mathbf{a}\) is an element of \(\mathcal{Z}(\mathcal{A})\), the center of the algebra. By definition, this means for any element \(\mathbf{b} \in \mathcal{A}\), we have \(\mathbf{ab} = \mathbf{ba}\). We want to show \(\mathbf{D}(\mathbf{a})\) is also in \(\mathcal{Z}(\mathcal{A})\). Now the derivation \(\mathbf{D}\) of \(\mathcal{a}\) has the property that \(\mathbf{D}(\mathbf{ab}) = \mathbf{a}\mathbf{D}(\mathbf{b}) + \mathbf{D}(\mathbf{a})\mathbf{b}\), for all \(\mathbf{a}, \mathbf{b} \in \mathcal{A}\). So, we compute as follows: \(\mathbf{a}\mathbf{D}(\mathbf{b}) + \mathbf{D}(\mathbf{a})\mathbf{b} = \mathbf{D}(\mathbf{ba}) = \mathbf{b}\mathbf{D}(\mathbf{a}) + \mathbf{D}(\mathbf{b})\mathbf{a}\). Rearrange and simplify to obtain: \(\mathbf{D}(\mathbf{a})\mathbf{b}-\mathbf{b}\mathbf{D}(\mathbf{a}) = \mathbf{a}\mathbf{D}(\mathbf{b})-\mathbf{D}(\mathbf{b})\mathbf{a}\). This proves \(\mathbf{D}(\mathbf{a}) \in \mathcal{L}(\mathcal{A})\).

Proving \(\mathbf{D}(\mathbf{a}) \in \mathcal{A}^{2}\) for \(\mathbf{a} \in \mathcal{A}^{2}\)

Suppose \(\mathbf{a}\) is an element of the derived algebra \(\mathcal{A}^{2}\). By definition, this means \(\mathbf{a}\) can be expressed as a sum of commutators, i.e., \(\mathbf{a} = \sum{\mathbf{x}_i \mathbf{y}_i - \mathbf{y}_i \mathbf{x}_i }\) for some elements \(\mathbf{x}_i, \mathbf{y}_i \in \mathcal{A}\). Take the derivation \(\mathbf{D}\) of \(\mathbf{a}\). By linearity, \(\mathbf{D}(\mathbf{a}) = \sum{\mathbf{D}(\mathbf{x}_i \mathbf{y}_i - \mathbf{y}_i \mathbf{x}_i)}\). Using the Leibniz rule, that simplifies to \(\sum{(\mathbf{x}_i \mathbf{D}(\mathbf{y}_i) - \mathbf{D}(\mathbf{y}_i) \mathbf{x}_i + \mathbf{D}(\mathbf{x}_i) \mathbf{y}_i - \mathbf{y}_i \mathbf{D}(\mathbf{x}_i))}\). This is clearly a sum of commutators, so \(\mathbf{D}(\mathbf{a})\) is in \(\mathcal{A}^{2}\).

Key concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. In basic terms, it involves equations and formulas that express a relationship between different symbols. When we talk about an "algebra" in mathematical physics, we are usually referring to a set with operations that combine elements in a way that satisfies specific rules.

Algebras can include structures like vectors spaces with a defined multiplication, making them rich with actionable properties. Within this context, these algebras can be equipped with additional properties and operations, like derivation.

Derivation

In mathematics, a derivation is an operation on an algebra that is similar to the notion of differentiation. This operation reflects how functions can change, echoing the role of a derivative in calculus. It helps to understand how quantities evolve within an algebraic structure.

A derivation, usually denoted by \(\mathbf{D}\), satisfies the Leibniz rule:

• \(\mathbf{D}(\mathbf{ab}) = \mathbf{a}\mathbf{D}(\mathbf{b}) + \mathbf{D}(\mathbf{a})\mathbf{b}\)

for any elements \(\mathbf{a}\) and \(\mathbf{b}\) in the algebra. This rule ensures the derivation consistently breaks down the multiplication of elements, similar to how derivatives operate in classic calculus.

Center of an Algebra

The center of an algebra is a crucial concept which consists of elements within an algebra that commute with every element in that algebra. It is often denoted as \(\mathcal{Z}(\mathcal{A})\).

To put it simply, if \(\mathbf{a}\) is an element of the center, then for every element \(\mathbf{b}\) in the algebra, \(\mathbf{ab} = \mathbf{ba}\). This property signifies that elements in the center are like 'neutral players' in multiplication and remain unchanged regardless of what they are multiplied by in the algebra.

This property is essential because when you apply a derivation to these elements, they exhibit stability, maintaining their commutative characteristic even after transformation via the derivation. This stability is shown as \(\mathbf{D}(\mathbf{a}) \in \mathcal{L}(\mathcal{A})\).

Derived Algebra

The derived algebra, represented as \(\mathcal{A}^{2}\), is a subset of the original algebra consisting primarily of commutators. Commutators indicate how non-commutative the multiplication operation of the algebra is.

For any given elements \(\mathbf{x}\) and \(\mathbf{y}\) in an algebra \(\mathcal{A}\), the commutator is defined as \([\mathbf{x}, \mathbf{y}] = \mathbf{x}\mathbf{y} - \mathbf{y}\mathbf{x}\). The derived algebra is essentially made up of these commutators, capturing all elements in \(\mathcal{A}\) that can be expressed as sums of these commutator expressions.

The importance of the derived algebra lies in its reflection of the structure's 'internal friction' or deviation from commutativity. Even when derivation acts on such a set, its inherent properties are preserved. Thus, \(\mathbf{D}(\mathbf{a}) \in \mathcal{A}^{2}\) if \(\mathbf{a} \in \mathcal{A}^{2}\).

Commutators

Commutators are fundamental in understanding the inner workings of an algebra, particularly in non-commutative settings such as quantum mechanics. The commutator of two elements \(\mathbf{x}\) and \(\mathbf{y}\) is defined as \(\mathbf{x}\mathbf{y} - \mathbf{y}\mathbf{x}\). This represents the "deviation" from commutativity.

When two elements commute, their commutator equals zero, which means their multiplication, in either order, gives the same result. In non-commutative algebras, such as those in quantum physics, commutators carry significant meaning as they can represent measurements and uncertainty.

Commutators help form the derived algebra, and manipulation of these structures via derivations shows how non-commutative properties remain steady even through transformations.