3.2 Irreducible and completely reducible representations

Irreducible representations are the building blocks of Lie algebra theory. They're representations with no proper, non-zero invariant subspaces. Understanding them is key to grasping the structure of Lie algebras and their applications.

Completely reducible representations can be broken down into irreducible parts. This concept is crucial for analyzing complex representations and their properties. It helps us understand how simpler representations combine to form more intricate ones.

Irreducible vs Completely Reducible Representations

Definitions and Properties

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• An irreducible representation is a representation of a Lie algebra on a vector space $V$ that has no proper, non-zero invariant subspaces under the action of the Lie algebra
• A completely reducible representation is a representation that can be decomposed as a direct sum of irreducible subrepresentations
• Every irreducible representation is completely reducible, but the converse is not true (a completely reducible representation may not be irreducible)
• The study of irreducible representations is crucial in the representation theory of Lie algebras, as they serve as the building blocks for constructing more complex representations

Significance in Lie Algebra Theory

• Irreducible representations are the fundamental building blocks of the representation theory of Lie algebras
• Every representation of a Lie algebra can be decomposed into a direct sum of irreducible representations, making the study of irreducible representations essential for understanding the structure of Lie algebras
• The classification of irreducible representations of a Lie algebra provides insight into the structure and properties of the Lie algebra itself
• Irreducible representations are used to construct character tables, which encode important information about the Lie algebra and its representations ($\mathfrak{sl}(2,\mathbb{C})$ character table)

Identifying Reducible Representations

Checking for Invariant Subspaces

• To determine if a representation is irreducible, one must check if there exist any proper, non-zero invariant subspaces under the action of the Lie algebra
• If no such subspaces exist, the representation is irreducible
• Finding invariant subspaces involves solving a system of linear equations derived from the action of the Lie algebra generators on the basis vectors of the representation space ($\mathfrak{su}(2)$ representation on $\mathbb{C}^2$)

Schur's Lemma

• Schur's lemma provides a powerful tool for determining the irreducibility of a representation
• Schur's lemma states that if a linear map between two irreducible representations commutes with the action of the Lie algebra, then it is either zero or an isomorphism
• In practice, Schur's lemma can be used to show that a representation is irreducible by proving that any linear map commuting with the Lie algebra action is a scalar multiple of the identity (irreducible representation of $\mathfrak{sl}(2,\mathbb{C})$ on a finite-dimensional vector space)

Decomposing Reducible Representations

Finding Invariant Subspaces

• If a representation is reducible, it can be decomposed into a direct sum of irreducible subrepresentations
• The process of decomposition involves finding the invariant subspaces of the representation and expressing the representation as a direct sum of these subspaces
• The invariant subspaces are themselves irreducible representations of the Lie algebra (decomposition of the adjoint representation of $\mathfrak{sl}(2,\mathbb{C})$)

Uniqueness and Isomorphism Classes

• The decomposition of a reducible representation is not always unique, but the number of times each irreducible representation appears in the decomposition (up to isomorphism) is unique
• Representations that have the same decomposition into irreducible subrepresentations (up to isomorphism) are said to be in the same isomorphism class
• The decomposition of a representation into irreducible subrepresentations is an important tool in studying the structure of Lie algebras and their representations (decomposition of tensor product representations)

Importance of Irreducible Representations in Lie Algebras

Building Blocks of Representation Theory

• Irreducible representations are the fundamental building blocks of the representation theory of Lie algebras
• Every representation of a Lie algebra can be decomposed into a direct sum of irreducible representations, making the study of irreducible representations essential for understanding the structure of Lie algebras
• The classification of irreducible representations of a Lie algebra provides insight into the structure and properties of the Lie algebra itself (classification of irreducible representations of semisimple Lie algebras)

Applications in Mathematics and Physics

• The study of irreducible representations has applications in various areas of mathematics and physics, such as quantum mechanics, where they are used to describe the states of a quantum system (irreducible representations of the Poincaré group in relativistic quantum mechanics)
• Irreducible representations play a crucial role in the study of symmetries and invariants in physical systems (irreducible representations of the rotation group $SO(3)$ in the study of angular momentum)
• The representation theory of Lie algebras has deep connections to other areas of mathematics, such as algebraic geometry and number theory (Langlands program)