2.1 Lie brackets and their properties

The Lie bracket operation is the cornerstone of Lie algebras, defining their structure and behavior. It's a binary operation that captures the non-commutativity of elements and satisfies key properties like antisymmetry, bilinearity, and the Jacobi identity.

Understanding Lie brackets is crucial for grasping the deeper aspects of Lie algebras. They play a vital role in physics, representing commutators of observables, and in mathematics, where they're essential for studying symmetries and conservation laws in various systems.

Lie bracket operation and Lie algebras

Definition and properties of the Lie bracket

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• The Lie bracket is a binary operation that takes two elements of a Lie algebra and returns another element of the Lie algebra
  • Denoted as $[X, Y]$ for elements $X$ and $Y$
• The Lie bracket operation satisfies certain properties that define the structure of a Lie algebra
  • Antisymmetry: $[X, Y] = -[Y, X]$
  • Bilinearity: $[aX, Y] = a[X, Y]$, $[X, aY] = a[X, Y]$, $[X + Y, Z] = [X, Z] + [Y, Z]$, $[X, Y + Z] = [X, Y] + [X, Z]$ for scalar $a$
  • Jacobi identity: $[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0$
• The Lie bracket captures the non-commutativity of the elements in a Lie algebra
  • $[X, Y]$ is generally not equal to $[Y, X]$
  • Measures the extent to which the elements fail to commute with each other

Interpretation and applications of the Lie bracket

• In physics, the Lie bracket often represents the commutator of two observables
  • Quantifies their incompatibility or the presence of quantum uncertainty
  • Example: the commutator of position and momentum operators in quantum mechanics
• The Lie bracket plays a crucial role in the study of symmetries and conservation laws
  • Lie algebras are associated with continuous symmetry groups (Lie groups)
  • The Lie bracket of the algebra elements corresponds to the infinitesimal generators of the symmetry transformations
  • Example: the angular momentum operators in quantum mechanics form a Lie algebra under the commutator bracket, related to rotational symmetry

Antisymmetry and bilinearity of Lie brackets

Proving the antisymmetry property

• For any elements $X$ and $Y$ in a Lie algebra, $[X, Y] = -[Y, X]$
• Proof:
  • By definition of the Lie bracket and properties of the underlying vector space
  • $[X, Y] = XY - YX$ (for matrix Lie algebras)
  • $[Y, X] = YX - XY = -(XY - YX) = -[X, Y]$

Proving the bilinearity properties

• The Lie bracket is linear in both arguments
  • For any elements $X$, $Y$, and $Z$ in a Lie algebra and any scalar $a$:
    • Linearity in the first argument: $[aX, Y] = a[X, Y]$
    • Linearity in the second argument: $[X, aY] = a[X, Y]$
    • Additivity in the first argument: $[X + Y, Z] = [X, Z] + [Y, Z]$
    • Additivity in the second argument: $[X, Y + Z] = [X, Y] + [X, Z]$
• Proof:
  • Using the definition of the Lie bracket, properties of the underlying vector space, and axioms of a Lie algebra
  • Example for linearity in the first argument: $[aX, Y] = (aX)Y - Y(aX) = a(XY) - a(YX) = a(XY - YX) = a[X, Y]$

Jacobi identity for Lie brackets

Statement and significance of the Jacobi identity

• The Jacobi identity is a fundamental property of Lie brackets
  • For any elements $X$, $Y$, and $Z$ in a Lie algebra: $[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0$
• The Jacobi identity provides a relation between nested Lie brackets
  • Allows simplification and manipulation of Lie bracket expressions
  • Essential for proving various theorems in the theory of Lie algebras

Applying the Jacobi identity to Lie bracket expressions

• To apply the Jacobi identity, cyclically permute the elements within the nested Lie brackets
  • Use the antisymmetry property to rearrange terms
  • Example: $[X, [Y, Z]] = -[Y, [Z, X]] - [Z, [X, Y]]$
• The Jacobi identity can be used to prove the existence of Lie algebra homomorphisms
  • A linear map $\phi: \mathfrak{g} \to \mathfrak{h}$ between Lie algebras is a homomorphism if $\phi([X, Y]) = [\phi(X), \phi(Y)]$ for all $X, Y \in \mathfrak{g}$
  • The Jacobi identity ensures that the homomorphism condition is satisfied
• The Jacobi identity is crucial for the classification of low-dimensional Lie algebras
  • Helps determine the possible structures and isomorphism classes of Lie algebras in dimensions 2 and 3

Constructing Lie algebras

Defining a Lie algebra from a vector space and bracket operation

• A Lie algebra can be constructed by defining:
  • A vector space $V$ over a field $F$ (usually the real or complex numbers)
  • A Lie bracket operation $[·, ·]$ that satisfies the required properties (antisymmetry, bilinearity, and the Jacobi identity)
• Given a vector space, define a Lie bracket operation by specifying its action on the basis elements
  • Extend the operation to all elements using bilinearity
  • Ensure that the defined operation satisfies the Lie algebra axioms

Examples of common Lie algebras

• The space of $n \times n$ matrices with the commutator bracket $[A, B] = AB - BA$
  • The Lie algebra of the general linear group $GL(n, F)$
  • Denoted as $\mathfrak{gl}(n, F)$
• The space of vector fields on a manifold with the Lie bracket defined as the Lie derivative
  • For vector fields $X$ and $Y$, the Lie bracket $[X, Y]$ is the Lie derivative of $Y$ with respect to $X$
  • The Lie algebra of the diffeomorphism group of the manifold
• The space of skew-symmetric $n \times n$ matrices with the commutator bracket
  • The Lie algebra of the special orthogonal group $SO(n)$
  • Denoted as $\mathfrak{so}(n)$

Verifying the Lie algebra axioms

• To verify that a given vector space and bracket operation form a Lie algebra, check:
  • Antisymmetry: $[X, Y] = -[Y, X]$ for all elements $X, Y$ in the vector space
  • Bilinearity: The bracket operation is linear in both arguments
  • Jacobi identity: $[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0$ for all elements $X, Y, Z$ in the vector space
• Constructing Lie algebras is essential for studying the properties and representations of Lie groups
  • Every Lie group has an associated Lie algebra that captures its infinitesimal structure
  • The Lie algebra encodes information about the tangent space at the identity element of the Lie group