Glossary

10.1 Definition and basic properties of Lie algebras

2 min read•july 25, 2024

Lie algebras are mathematical structures that combine vector spaces with special operations. They're crucial in physics and math, helping us understand symmetries and transformations in various systems.

These algebras use brackets to capture relationships between elements. We'll look at their key properties, how they connect to , and important concepts like the and .

Fundamental Concepts of Lie Algebras

Definition of Lie algebras

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  • Lie algebra comprises vector space LL over field FF with binary operation called Lie bracket [,]:L×L→L[,]: L \times L \rightarrow L
  • Lie bracket satisfies , ([x,x]=0[x,x] = 0 for all x∈Lx \in L), and Jacobi identity ([x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 for all x,y,z∈Lx,y,z \in L)
  • Vector spaces form backbone of Lie algebras allowing linear combinations of elements
  • Field FF typically real or complex numbers providing scalars for vector operations

Concept of Lie brackets

  • Bilinear operation captures algebraic and geometric properties of Lie algebra
  • Antisymmetry property [x,y]=−[y,x][x,y] = -[y,x] ensures non-commutativity
  • Derivation property [x,[y,z]]=[[x,y],z]+[y,[x,z]][x,[y,z]] = [[x,y],z] + [y,[x,z]] links to differential geometry
  • Geometrically represents infinitesimal version of group element commutators
  • Examples include
    • for matrices [A,B]=AB−BA[A,B] = AB - BA
    • in classical mechanics describing evolution of observables

Lie algebras vs Lie groups

  • Lie groups smooth manifolds with compatible group structure
  • at identity element forms associated Lie algebra
  • connects Lie algebra elements to Lie group elements
  • describes action of Lie group on its Lie algebra
  • Examples demonstrate connection:
    • GL(n,R)GL(n,\mathbb{R}) (general linear group) and gl(n,R)\mathfrak{gl}(n,\mathbb{R}) (general linear algebra)
    • SO(3)SO(3) (special orthogonal group) and so(3)\mathfrak{so}(3) (special orthogonal algebra)

Center and derived series

  • Center Z(L)Z(L) contains elements commuting with all others: Z(L)={x∈L:[x,y]=0 for all y∈L}Z(L) = \{x \in L : [x,y] = 0 \text{ for all } y \in L\}
  • Derived series L(0)=L,L(1)=[L,L],L(2)=[L(1),L(1)],…L^{(0)} = L, L^{(1)} = [L,L], L^{(2)} = [L^{(1)},L^{(1)}], \ldots measures "non-commutativity"
  • Solvable Lie algebras have derived series terminating at zero
  • Computation techniques:
    1. Use to determine commutation relations
    2. Employ matrix representations for concrete calculations
  • Applications include classification of Lie algebras and structural analysis

Key Terms to Review (12)

Adjoint representation: The adjoint representation is a specific way of representing a Lie algebra by using its own elements as linear transformations on itself. In this representation, each element of the Lie algebra acts as an endomorphism, encapsulating the structure and behavior of the algebra through its own commutation relations. This concept is vital for understanding the inner workings of Lie algebras and plays a significant role in their classification and the study of their finite-dimensional representations.
Alternating property: The alternating property refers to a characteristic of bilinear maps where switching the order of two arguments results in a change of sign. In the context of Lie algebras, this property is essential because it ensures that the Lie bracket, which is the operation defining the algebra, is skew-symmetric. This means that for any elements x and y in a Lie algebra, the relation $[x,y] = -[y,x]$ holds, which is a foundational aspect of Lie algebras and their structures.
Bilinearity: Bilinearity refers to a property of a function that is linear in each of its arguments when the other argument is held fixed. This concept is crucial in various mathematical frameworks, as it establishes how certain structures can be analyzed and manipulated. Understanding bilinearity allows for the effective handling of tensor products and their decompositions, as well as the foundational structures found in Lie algebras.
Center: In the context of Lie algebras, the center of a Lie algebra is defined as the set of all elements that commute with every other element in the algebra. This concept is important because the center helps identify substructures within the Lie algebra and plays a crucial role in understanding its representation theory and structure.
Commutator bracket: The commutator bracket is an operation in a Lie algebra defined for two elements, typically denoted as $[x, y] = xy - yx$, capturing the idea of non-commutativity. It measures how much two elements fail to commute and is fundamental in defining the structure of Lie algebras, particularly in understanding their algebraic properties and representations.
Derived series: The derived series is a sequence of subalgebras constructed from a given Lie algebra, starting with the algebra itself and iteratively taking the derived algebra of each subsequent subalgebra. This series is essential for understanding the structure and properties of Lie algebras, particularly in how they relate to solvability and nilpotency. Each term in the derived series gives insight into the behavior of the Lie algebra, allowing for the classification and analysis of its components.
Exponential map: The exponential map is a mathematical function that provides a way to relate the elements of a Lie algebra to the elements of a Lie group. It takes a tangent vector at the identity element of the group and produces an element of the group, effectively capturing the local behavior around the identity. This connection is fundamental in studying the structure of Lie groups and their associated Lie algebras.
Lie Groups: Lie groups are mathematical structures that combine algebraic and geometric properties, serving as a group of differentiable transformations that are continuous and have smooth manifolds. They are fundamental in connecting algebraic operations with smooth geometrical actions, allowing for a deeper understanding of symmetry in various mathematical contexts, such as character theory and the study of Lie algebras.
Poisson Bracket: The Poisson bracket is a mathematical operation used in Hamiltonian mechanics to express the time evolution of dynamical systems. It takes two functions defined on the phase space of a system and produces another function that captures how the two functions change with respect to each other over time. This bracket is deeply connected to the structure of Lie algebras, as it follows properties similar to those of a Lie bracket, thus highlighting the underlying algebraic framework in the study of dynamical systems.
Solvable lie algebra: A solvable Lie algebra is a Lie algebra where its derived series eventually reaches the zero subalgebra, meaning that repeated commutation of elements eventually leads to trivial results. This concept is fundamental in understanding the structure and classification of Lie algebras, particularly in recognizing their behavior in relation to semisimplicity and nilpotency, which are key to grasping their overall organization and properties.
Structure Constants: Structure constants are numerical values that arise in the study of Lie algebras, representing the coefficients in the Lie bracket of two basis elements. They provide a way to encode the algebraic structure of the Lie algebra, helping to define its relationships and properties. The structure constants are essential for understanding how elements of the algebra interact with each other and play a crucial role in determining the behavior of associated groups.
Tangent Space: The tangent space at a point on a manifold is a vector space that consists of all possible directions in which one can tangentially pass through that point. It provides a way to study the local properties of the manifold, allowing us to analyze curves, vector fields, and differential structures in the vicinity of that point.
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