Infinite-dimensional representations

Definition

Infinite-dimensional representations refer to representations of groups or algebras that act on vector spaces with infinite dimensions. These representations extend the concept of finite-dimensional representations, allowing for a richer structure that can model more complex systems. They are crucial in areas such as quantum mechanics and functional analysis, where infinite-dimensional spaces naturally arise.

5 Must Know Facts For Your Next Test

1. Infinite-dimensional representations can be represented using sequences or function spaces, which differ from the finite-dimensional cases where matrices suffice.
2. In the context of linear representations, infinite-dimensional representations allow groups to act on spaces like $L^2$ spaces, which are crucial in quantum physics.
3. Not every finite-dimensional representation can be extended to an infinite-dimensional one, making the study of these representations rich with complexity.
4. Maschke's theorem does not apply to infinite-dimensional representations, as it requires finite-dimensionality to ensure complete reducibility.
5. Examples of infinite-dimensional representations include the action of Lie groups on function spaces and representation theory in statistical mechanics.

Review Questions

• How do infinite-dimensional representations differ from finite-dimensional representations in their structure and applications?
  • Infinite-dimensional representations differ fundamentally from finite-dimensional ones by operating on vector spaces with infinitely many dimensions, allowing for the modeling of more complex phenomena. While finite-dimensional representations are often represented using matrices, infinite-dimensional cases typically involve sequences or functions. This difference opens the door to applications in areas like quantum mechanics and functional analysis, where such structures are essential.
• Discuss the implications of Maschke's theorem for the study of infinite-dimensional representations.
  • Maschke's theorem states that a finite group representation over a field of characteristic zero is completely reducible. However, this theorem does not hold for infinite-dimensional representations. As a result, when dealing with these types of representations, one cannot guarantee that every representation can be decomposed into a direct sum of irreducible ones. This leads to unique challenges and considerations in the analysis and application of infinite-dimensional structures.
• Evaluate the significance of Hilbert spaces in the context of infinite-dimensional representations and their applications in physics and mathematics.
  • Hilbert spaces serve as a fundamental framework for infinite-dimensional representations, particularly in quantum mechanics. They provide a rigorous mathematical structure that supports concepts like orthogonality and completeness, essential for formulating physical theories. The ability to use operators on Hilbert spaces allows physicists to describe quantum states and observables, highlighting how infinite-dimensional representations are integral to our understanding of complex systems in both mathematics and physics.

Related terms

Hilbert Space: A complete inner product space that generalizes the notion of Euclidean space to infinite dimensions, providing a framework for quantum mechanics and various mathematical fields.

Compact Operators: Operators on Hilbert spaces that map bounded sets to relatively compact sets, playing a key role in the study of infinite-dimensional representations.

Unitary Representation: A representation where group elements are represented by unitary operators on a Hilbert space, preserving the inner product and often used in quantum mechanics.