invariant bilinear form exists if and only if is self-dual

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21-01-2025

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An important concept in linear algebra is the relationship between a vector space and its dual. This article explores the crucial connection between the existence of an invariant bilinear form and the property of a vector space being self-dual. We'll demonstrate that a finite-dimensional vector space V possesses an invariant bilinear form if and only if it is self-dual.

Understanding the Key Concepts

Before diving into the proof, let's clarify the core terms:

1. Bilinear Form: A bilinear form on a vector space V over a field F is a map B: V x V → F that is linear in both arguments. This means:

• B(au + bv, w) = aB(u, w) + bB(v, w)
• B(u, av + bw) = aB(u, v) + bB(u, w)

for all u, v, w ∈ V and a, b ∈ F.

2. Invariant Bilinear Form: Let V be a vector space and G a group of linear transformations acting on V. A bilinear form B on V is said to be G-invariant if B(gu, gv) = B(u, v) for all u, v ∈ V and all g ∈ G. In simpler terms, the bilinear form remains unchanged under the action of the group G.

3. Dual Space: The dual space V* of a vector space V is the vector space of all linear functionals on V. A linear functional is a linear map from V to the underlying field F.

4. Self-Dual Vector Space: A vector space V is self-dual if it is isomorphic to its dual space V*. This means there exists a linear isomorphism φ: V → V*.

The Main Theorem: Invariance and Self-Duality

Theorem: A finite-dimensional vector space V is self-dual if and only if there exists a non-degenerate invariant bilinear form on V.

Proof:

Part 1: If V is self-dual, then there exists a non-degenerate invariant bilinear form on V.

Assume V is self-dual. This means there's an isomorphism φ: V → V*. We can define a bilinear form B on V as follows:

B(u, v) = φ(u)(v)

This means we apply the linear functional φ(u) to the vector v. It's straightforward to verify that B is bilinear. The non-degeneracy of B follows from the fact that φ is an isomorphism. If B(u,v) = 0 for all v, then φ(u) = 0, implying u = 0 (since φ is injective). Similarly, if B(u,v) = 0 for all u, then v = 0.

The invariance of B depends on the specific group G acting on V. If the action of G on V induces a corresponding action on V* such that φ(gu) = gφ(u), then B is G-invariant. This is because:

B(gu, gv) = φ(gu)(gv) = (gφ(u))(gv) = φ(u)(g⁻¹gv) = φ(u)(v) = B(u, v)

This condition highlights the crucial role of the group action's compatibility between V and V*.

Part 2: If there exists a non-degenerate invariant bilinear form on V, then V is self-dual.

Suppose there exists a non-degenerate invariant bilinear form B on V. We can define a map φ: V → V* by:

φ(u)(v) = B(u, v)

It's clear that φ(u) is a linear functional for each u ∈ V. The non-degeneracy of B ensures that φ is injective (if φ(u) = 0, then B(u, v) = 0 for all v, implying u = 0). Since V is finite-dimensional, injectivity implies surjectivity, making φ an isomorphism. Therefore, V is self-dual.

Implications and Applications

The existence of an invariant bilinear form has significant consequences in various areas of mathematics and physics, including:

• Representation Theory: Invariant bilinear forms are crucial in the study of group representations.
• Differential Geometry: They play a vital role in defining metrics and inner products on manifolds.
• Physics: Invariant bilinear forms appear in numerous physical theories, such as classical mechanics and quantum field theory.

This theorem provides a fundamental link between the algebraic structure of a vector space and its geometric properties through the lens of bilinear forms. Understanding this relationship is essential for deeper explorations in these fields.

Further Exploration

This article provides a foundational understanding of the connection between invariant bilinear forms and self-dual vector spaces. Further research could explore:

• Specific examples of invariant bilinear forms in different contexts.
• The role of the group action in determining the invariance of the bilinear form.
• Extensions of this theorem to infinite-dimensional vector spaces.

This connection between invariant bilinear forms and self-duality is a cornerstone result in linear algebra with far-reaching implications. Understanding this theorem provides a solid foundation for more advanced study in various mathematical and physical disciplines.