Question

Prove that \(\mathcal{A}^{2}\), the derived algebra of \(\mathcal{A}\), is indeed an algebra.

Short answer

Yes, given all the described properties, it can indeed be confirmed that \(\mathcal{A}^{2}\) is an algebra.

Step-by-step solution

Establish Properties

First, state the properties that make an object an algebra. For our case, it is sufficient to show that \(\mathcal{A}^{2}\) is closed under addition and scalar multiplication, and that the product in \(\mathcal{A}^{2}\) is bilinear. In particular, we should show that for all \(a, b, c \in \mathcal{A}^{2}\) and any scalar \(r\), the following holds: \(a * (b + c) = a*b + a*c\), \((a + b) * c = a*c + b*c\), and \(r*(a*b) = (r*a)*b = a*(r*b)\).

Show Closure Under Addition and Scalar Multiplication

If \(a, b \in \mathcal{A}^{2}\), then the sum \(a + b\) is also in \(\mathcal{A}^{2}\). Likewise, for any scalar \(r\), the product \(r*a\) is in \(\mathcal{A}^{2}\). These results come directly from the fact that \(\mathcal{A}^{2}\) is a subset of \(\mathcal{A}\), and \(\mathcal{A}\) is an algebra.

Show Bilinearity

Firstly, for \(a, b, c \in \mathcal{A}^{2}\), we have \(a * (b + c) = a*b + a*c\), because both operations are being performed in \(\mathcal{A}\) and so the product is bilinear. Secondly, for any \(a, b, c \in \mathcal{A}^{2}\), we have \((a + b) * c = a*c + b*c\), and so the product is left-distributive in \(\mathcal{A}^{2}\). Lastly, for any scalar \(r\) and any \(a, b \in \(\mathcal{A}^{2}\)), we have \(r*(a*b) = (r*a)*b = a*(r*b)\), and so this is consistent with scalar multiplication in \(\mathcal{A}\) and so \(\mathcal{A}^{2}\) is closed under scalar multiplication.

Key concepts

Closure Under Addition

In mathematics, the concept of closure under addition is fundamental when considering algebraic structures. An object, like a subset of a vector space or an algebra, is said to be closed under addition if the addition of any two elements within it remains inside the set.
This is a crucial property to prove when trying to establish that a set forms an algebra.
In the case of the derived algebra, denoted as \(\mathcal{A}^2\), we must check that if \(a\) and \(b\) are elements of \(\mathcal{A}^2\), then their sum \(a + b\) is also in \(\mathcal{A}^2\).

• This involves showing that the addition operation retains the defined structure and properties of the original set.
• By confirming this property, we ensure that \(\mathcal{A}^2\) indeed preserves the closure under addition, pivotal for algebraic structure.

Understanding this concept helps us recognize why and how mathematical structures maintain their form across operations, ensuring any sum of elements remains within the set.

Scalar Multiplication

Scalar multiplication is an operation on an algebraic set or structure where each element is multiplied by a scalar (a constant value), which could be a real number or other defined unit.
The purpose is to determine whether the set is closed under this operation, meaning any element of the set multiplied by the scalar should still remain within the set.
For \(\mathcal{A}^2\), showing closure under scalar multiplication might look like this: if \(a\) is in \(\mathcal{A}^2\) and \(r\) is any scalar, then \(r * a\) should also be in \(\mathcal{A}^2\).

• This property confirms that the structure adheres to algebra's rules, maintaining consistency even under external scaling.
• Ensuring closure under scalar multiplication upholds scaling transformations which are vital in numerous applications, such as transformations in geometry and physics.

It is a key concept in proving that \(\mathcal{A}^2\) is an algebra, helping to depict its behavorial integrity across different mathematical contexts.

Bilinearity

Bilinearity is a property that involves two operations being linear in each of the variables separately.
In the context of algebra, it's essential for establishing how operations such as multiplication interact within the structure.
Proving bilinearity in \(\mathcal{A}^2\) involves checking conditions like:

• For any elements \(a, b, c\) in \(\mathcal{A}^2\), \(a * (b + c) = a * b + a * c\).
• It also involves demonstrating that \((a + b) * c = a * c + b * c\), a key aspect indicating left-distributive law.
• For a scalar \(r\) and elements \(a, b\), it is required that \(r * (a*b) = (r*a)*b = a*(r*b)\).

Verifying these helps confirm that the product operation distributes over addition and is compatible with scalar multiplication. Bilinearity assures any systemic multiplication or combination within the set \(\mathcal{A}^2\) maintains the set's algebraic integrity, ensuring operations maintain their value expectations and consistent structure.