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 (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
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 as a , 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 ei∗(ej)=δij (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 sl(2,C), the tensor product of two irreducible representations V(m)⊗V(n) decomposes into a 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