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
Key Terms to Review (18)
Action of a Lie Group: The action of a Lie group on a manifold or vector space is a way in which the elements of the group can transform points in that space, preserving certain structures like the vector space or manifold itself. This concept is crucial for understanding how symmetry and transformation properties interact in different mathematical settings, including the structure of representations and module theory.
Composition of representations: The composition of representations refers to the process of combining two or more representations of a Lie algebra or Lie group to create a new representation. This concept is central in understanding how different representations interact and can be constructed, providing insights into the structure of the algebra or group. It also plays a crucial role in the study of tensor products and dual representations, which allow for the exploration of various ways to build new representations from existing ones.
Direct Sum: The direct sum is an important concept in algebra that combines multiple vector spaces or modules into a larger one, where each component retains its individuality. In the context of Lie algebras, the direct sum allows us to construct new Lie algebras from existing ones, facilitating their analysis and decomposition into simpler parts. This concept is also relevant in the study of representations, particularly when discussing how different representations can be combined to form new ones.
Dual Representation: Dual representation refers to a specific way of associating a representation of a Lie algebra with its dual space, creating a correspondence between vectors in the algebra and linear functionals on that space. This concept is essential in understanding how representations can act on various vector spaces and how these actions relate to dual spaces, which often arise in the context of tensor products and related structures.
Equivalence of Representations: Equivalence of representations refers to the condition where two representations of a Lie algebra or Lie group are considered to be 'the same' in a certain sense. This typically involves finding a linear isomorphism between the vector spaces associated with these representations that intertwines the action of the Lie algebra or group, preserving the structure and relations inherent to those representations. Recognizing when two representations are equivalent is crucial for simplifying problems in representation theory, as it allows one to work with a single representative from each equivalence class.
Finite-dimensional representation: A finite-dimensional representation is a way of describing how a group or algebra acts on a finite-dimensional vector space through linear transformations. This concept is crucial in understanding the structure and properties of Lie groups and Lie algebras, as it connects algebraic structures with geometric and analytical frameworks, enabling the analysis of symmetries and the behavior of functions under group actions.
Homomorphism: A homomorphism is a structure-preserving map between two algebraic structures, such as groups, rings, or Lie algebras, that respects the operations of those structures. This concept connects different algebraic systems, allowing us to understand their relationships and properties through consistent transformations.
Infinitesimal Action: Infinitesimal action refers to the way a Lie group acts on a manifold by describing the behavior of elements in the group that are infinitesimally close to the identity. This concept is crucial for understanding how continuous symmetries can be represented and how these representations relate to the geometry of the underlying space. In particular, infinitesimal actions help establish connections between Lie groups, their corresponding Lie algebras, and representations, especially when considering tensor products and dual representations.
Irreducible Representation: An irreducible representation is a representation of a group or algebra that cannot be decomposed into a direct sum of simpler representations. This means that the only invariant subspaces under the action of the group or algebra are the trivial subspace and the entire space itself, making these representations fundamental building blocks in the study of group actions and linear transformations.
Isomorphism: Isomorphism refers to a structural similarity between two mathematical objects, indicating that there is a one-to-one correspondence between their elements that preserves the operations defined on those objects. This concept highlights the idea that two seemingly different entities can be fundamentally the same in terms of their algebraic structure, allowing us to translate problems and solutions between them.
Peter-Weyl Theorem: The Peter-Weyl theorem states that for a compact Lie group, the space of square-integrable functions on the group can be decomposed into a direct sum of finite-dimensional irreducible representations. This theorem connects harmonic analysis on groups with representation theory, highlighting how each irreducible representation corresponds to a unique character, which plays a crucial role in understanding both representations and their properties.
Schur's Lemma: Schur's Lemma is a fundamental result in representation theory stating that if a representation of a group (or algebra) is irreducible, then any linear operator that commutes with all the operators in that representation must be a scalar multiple of the identity operator. This means that irreducible representations are highly structured, as they cannot be decomposed into smaller representations, which connects to the properties of Lie brackets, tensor products, and examples of representations.
Sophus Lie: Sophus Lie was a Norwegian mathematician best known for his groundbreaking work in the theory of continuous transformation groups, now called Lie groups, and their corresponding algebraic structures, known as Lie algebras. His contributions laid the foundation for much of modern mathematics, particularly in the study of symmetries and their applications across various fields such as physics, geometry, and differential equations.
Standard representation: A standard representation is a specific way in which a Lie group or Lie algebra acts on a vector space, typically chosen for its simplicity and naturalness. It often serves as a fundamental example that can be used to build more complex representations, illustrating how abstract algebraic structures can manifest in more concrete settings, like linear transformations of vector spaces.
Subrepresentation: A subrepresentation is a vector space that is a subset of a larger representation and is invariant under the action of the group or algebra. This means that when the elements of the group or algebra act on the subrepresentation, they produce outputs that remain within the same subrepresentation. This concept is crucial when studying representations because it allows for the decomposition of complex representations into simpler, manageable parts.
Tensor Product of Representations: The tensor product of representations is an operation that combines two representations of a Lie algebra or Lie group into a new representation. This construction helps to create new ways to understand how these algebraic structures act on vector spaces, particularly in terms of their interactions and decompositions. It also plays a crucial role in analyzing the properties of finite-dimensional representations and their characters, providing insights into the underlying symmetries and structures.
Trivial representation: A trivial representation is the simplest form of representation of a group or algebra, where every group element is represented as the identity transformation on a vector space. This means that the action of the group does nothing to the vectors in the space, effectively treating all elements uniformly. It serves as a foundation for understanding more complex representations and plays a crucial role in concepts such as tensor products and dual representations, where it can affect how other representations combine and interact.
Wilhelm Killing: Wilhelm Killing was a prominent mathematician known for his foundational contributions to the theory of Lie algebras, particularly the Killing form, which is a bilinear form used to study the structure and classification of semisimple Lie algebras. His work has significant implications across various areas of mathematics, linking algebraic properties with geometric and topological aspects.