3.4 Tensor products and dual representations

Tensor products and dual representations are powerful tools in Lie algebra theory. They let us build new representations from existing ones, revealing the structure of Lie algebras and their representations. This is key for understanding complex algebraic systems.

These concepts are crucial for studying semisimple Lie algebras and Lie groups. They help us construct invariant bilinear forms, classify representations, and uncover important structural properties. This knowledge is essential for advanced topics in Lie theory.

Tensor product of representations

Definition and properties

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• Defines the tensor product of two representations $(π₁, V₁)$ and $(π₂, V₂)$ of a Lie algebra $𝔤$ as a new representation $(π₁ ⊗ π₂, V₁ ⊗ V₂)$ on the tensor product space $V₁ ⊗ V₂$
• Specifies the action of the tensor product representation on the basis elements of $V₁ ⊗ V₂$ given by $(π₁ ⊗ π₂)(X)(v₁ ⊗ v₂) = π₁(X)v₁ ⊗ v₂ + v₁ ⊗ π₂(X)v₂$ for $X ∈ 𝔤$, $v₁ ∈ V₁$, and $v₂ ∈ V₂$
• Possesses associative and bilinear properties, but not commutative ($(V₁ ⊗ V₂) ⊗ V₃ ≅ V₁ ⊗ (V₂ ⊗ V₃)$, but $V₁ ⊗ V₂ ≇ V₂ ⊗ V₁$)
• Generally results in a reducible representation when taking the tensor product of irreducible representations
• Preserves finite-dimensionality, with the dimension of the tensor product space equal to the product of the dimensions of the individual spaces ($\dim(V₁ ⊗ V₂) = \dim(V₁) · \dim(V₂)$)

Importance in Lie algebra representation theory

• Allows for the construction of new representations from existing ones, aiding in understanding the representation theory of Lie algebras
• Provides information about the structure of the Lie algebra and its representations through the decomposition of tensor products of irreducible representations into irreducible components
• Plays a crucial role in the study of semisimple Lie algebras and their associated Lie groups by revealing important structural properties

Constructing tensor products

Forming the tensor product space

• Form the tensor product space $V₁ ⊗ V₂$ by taking the tensor product of basis elements from the individual representation spaces $V₁$ and $V₂$
• Example: If $V₁$ has basis $\{e₁, e₂\}$ and $V₂$ has basis $\{f₁, f₂, f₃\}$, then $V₁ ⊗ V₂$ has basis $\{e₁ ⊗ f₁, e₁ ⊗ f₂, e₁ ⊗ f₃, e₂ ⊗ f₁, e₂ ⊗ f₂, e₂ ⊗ f₃\}$

Defining the action of the tensor product representation

• Define the action of the tensor product representation $π₁ ⊗ π₂$ on the basis elements of $V₁ ⊗ V₂$ using the formula $(π₁ ⊗ π₂)(X)(v₁ ⊗ v₂) = π₁(X)v₁ ⊗ v₂ + v₁ ⊗ π₂(X)v₂$ for $X ∈ 𝔤$, $v₁ ∈ V₁$, and $v₂ ∈ V₂$
• Extend the action of $π₁ ⊗ π₂$ linearly to all elements of $V₁ ⊗ V₂$
• Example: If $π₁(X)e₁ = ae₁ + be₂$ and $π₂(X)f₁ = cf₁ + df₂$, then $(π₁ ⊗ π₂)(X)(e₁ ⊗ f₁) = ae₁ ⊗ f₁ + be₂ ⊗ f₁ + ce₁ ⊗ f₁ + de₁ ⊗ f₂$

Verifying the representation properties

• Verify that $π₁ ⊗ π₂$ satisfies the properties of a Lie algebra representation
  • $(π₁ ⊗ π₂)(aX + bY) = a(π₁ ⊗ π₂)(X) + b(π₁ ⊗ π₂)(Y)$ for $a, b ∈ ℂ$ and $X, Y ∈ 𝔤$ (linearity)
  • $(π₁ ⊗ π₂)([X, Y]) = [(π₁ ⊗ π₂)(X), (π₁ ⊗ π₂)(Y)]$ for $X, Y ∈ 𝔤$ (compatibility with the Lie bracket)

Dual representation

Definition and relation to the original representation

• Defines the dual representation $(π^*, V^*)$ of a representation $(π, V)$ as a representation of the Lie algebra $𝔤$ on the dual space $V^*$ of linear functionals on $V$
• Specifies the action of the dual representation $π^*$ on an element $φ ∈ V^*$ given by $(π^*(X)φ)(v) = -φ(π(X)v)$ for $X ∈ 𝔤$ and $v ∈ V$
• Satisfies the properties of a Lie algebra representation, i.e., $π^*([X, Y]) = [π^*(X), π^*(Y)]$ for $X, Y ∈ 𝔤$
• Relates the double dual representation $(π^{**}, V^{**})$ to the original representation $(π, V)$ through a canonical isomorphism

Importance in Lie algebra representation theory

• Allows for the construction of invariant bilinear forms on the Lie algebra, which are essential for the classification of Lie algebra representations
• Provides a way to study the structure of the Lie algebra and its representations by examining the relationship between a representation and its dual
• Plays a crucial role in the study of semisimple Lie algebras, as the tensor product of a representation with its dual contains the trivial representation as a subrepresentation, related to the existence of invariant bilinear forms on the Lie algebra

Computing dual representations

Determining the dual space basis

• Determine a basis for the dual space $V^*$ in terms of the dual basis elements of the original representation space $V$
• Example: If $V$ has basis $\{e₁, e₂, e₃\}$, then $V^*$ has the dual basis $\{e₁^*, e₂^*, e₃^*\}$, where $e_i^*(e_j) = δ_{ij}$ (Kronecker delta)

Defining the action of the dual representation

• Define the action of the dual representation $π^*$ on the basis elements of $V^*$ using the formula $(π^*(X)φ)(v) = -φ(π(X)v)$ for $X ∈ 𝔤$, $φ ∈ V^*$, and $v ∈ V$
• Express the action of $π^*$ on the basis elements of $V^*$ in matrix form, using the structure constants of the Lie algebra and the matrix representation of $π$
• Example: If $π(X)e₁ = ae₁ + be₂$, then $(π^*(X)e₁^*)(e₁) = -e₁^*(π(X)e₁) = -e₁^*(ae₁ + be₂) = -a$, and similarly for other basis elements

Extending the action to the entire dual space

• Extend the action of $π^*$ linearly to all elements of $V^*$
• Example: If $φ = αe₁^* + βe₂^*$, then $π^*(X)φ = απ^*(X)e₁^* + βπ^*(X)e₂^*$

Tensor products and duals in Lie algebras

Constructing new representations

• Tensor products of representations allow for the construction of new representations from existing ones
• The decomposition of tensor products of irreducible representations into irreducible components provides information about the structure of the Lie algebra and its representations
• Example: In the case of $\mathfrak{sl}(2, ℂ)$, the tensor product of two irreducible representations $V(m) ⊗ V(n)$ decomposes into a direct sum of irreducible representations $V(m+n) ⊕ V(m+n-2) ⊕ ... ⊕ V(|m-n|)$

Invariant bilinear forms and classification of representations

• Dual representations are important in the study of Lie algebras because they allow for the construction of invariant bilinear forms
• The tensor product of a representation with its dual representation contains the trivial representation as a subrepresentation, which is related to the existence of invariant bilinear forms on the Lie algebra
• Invariant bilinear forms play a crucial role in the classification of Lie algebra representations, particularly for semisimple Lie algebras

Semisimple Lie algebras and Lie groups

• The study of tensor products and dual representations is crucial for understanding the representation theory of semisimple Lie algebras and their associated Lie groups
• In the case of semisimple Lie algebras, the existence of a non-degenerate invariant bilinear form (the Killing form) allows for a complete classification of finite-dimensional irreducible representations
• The representation theory of semisimple Lie algebras is closely tied to the structure of their associated Lie groups, with the irreducible representations of the Lie algebra corresponding to the irreducible representations of the Lie group