Unitary representation

Definition

A unitary representation is a way of representing a group by unitary operators on a Hilbert space, ensuring that the group operation corresponds to the composition of these operators. This type of representation preserves the inner product, allowing for the analysis of symmetry and structure in mathematical objects. It plays a crucial role in harmonic analysis, representation theory, and the duality relationships found in Fourier analysis on groups.

5 Must Know Facts For Your Next Test

1. Unitary representations are essential in understanding the harmonic analysis on groups, particularly in how they relate to functions defined on these groups.
2. The Peter-Weyl theorem states that any unitary representation of a compact group can be decomposed into a direct sum of irreducible representations.
3. In the context of Pontryagin duality, unitary representations play a critical role in connecting a locally compact abelian group with its dual group via Fourier transform techniques.
4. A key property of unitary representations is that they preserve the norm of vectors, making them useful in analyzing various physical and mathematical phenomena.
5. The study of unitary representations includes exploring their characters, which are homomorphisms from the group to the complex numbers, providing valuable insights into the structure of the group.

Review Questions

• How do unitary representations facilitate the study of harmonic analysis on locally compact abelian groups?
  • Unitary representations help in harmonic analysis by providing a framework to study functions defined on locally compact abelian groups through operators on Hilbert spaces. They allow us to translate group operations into linear transformations while preserving inner products, which makes analyzing symmetries and structures much more manageable. This connection enables us to apply techniques from functional analysis to understand properties of functions in relation to their corresponding groups.
• Discuss the significance of the Peter-Weyl theorem in relation to unitary representations of compact groups.
  • The Peter-Weyl theorem highlights that every unitary representation of a compact group can be decomposed into irreducible representations, which are simpler and more manageable components. This decomposition is significant because it allows mathematicians to classify and analyze representations systematically, leading to a better understanding of the group's structure. By breaking down complex representations into irreducible parts, one can apply various tools from representation theory to study properties and applications in physics and other fields.
• Evaluate how Pontryagin duality utilizes unitary representations in Fourier analysis on groups.
  • Pontryagin duality establishes a profound relationship between a locally compact abelian group and its dual group through the lens of unitary representations. In this framework, each character of the dual group corresponds to a unitary representation on an associated Hilbert space. This interplay allows for effective Fourier analysis, where functions on the original group can be expressed in terms of their projections onto characters, enabling powerful analytical tools to study harmonic properties and convergence behaviors within the context of abstract algebraic structures.

Related terms

Hilbert Space: A complete inner product space that provides the framework for quantum mechanics and various branches of mathematics, serving as the setting for unitary representations.

Group Homomorphism: A structure-preserving map between two groups that maintains the operation of the groups, which is relevant when studying representations.

Compact Group:

A type of topological group that is both compact and Hausdorff, where every unitary representation can be decomposed into irreducible components.