Analyze the structure and classification of low-dimensional Lie algebras based on their multiplication properties | Step-by-Step Solution

Classifying Low-Dimensional Lie Algebras

1. What We're Solving:

We want to understand and classify all possible Lie algebras of dimensions 1 and 2. This means finding all the different ways we can define a Lie bracket operation on 1-dimensional and 2-dimensional vector spaces while satisfying the Lie algebra axioms.

2. The Approach:

We'll use the defining properties of Lie algebras (antisymmetry and the Jacobi identity) as our constraints, then systematically examine what multiplication tables are possible for small dimensions. This classification helps us understand the "building blocks" of more complex Lie algebras.

3. Step-by-Step Solution:

Step 1: Recall the Lie Algebra Axioms A Lie algebra is a vector space with a bracket operation [·,·] that satisfies:

• Antisymmetry: [x,y] = -[y,x] for all x,y
• Jacobi identity: [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 for all x,y,z
• Bilinearity: The bracket is linear in both arguments

Step 2: Dimension 1 Case Let's call our 1-dimensional Lie algebra L₁ with basis {e₁}.

• Any element looks like: ae₁ for some scalar a
• The bracket must satisfy: [e₁,e₁] = ?
• By antisymmetry: [e₁,e₁] = -[e₁,e₁], so 2[e₁,e₁] = 0
• Therefore: [e₁,e₁] = 0

Conclusion for dim 1: There's exactly one 1-dimensional Lie algebra (up to isomorphism) - the abelian one where all brackets are zero!

Step 3: Dimension 2 Case Let L₂ have basis {e₁,e₂}. We need to determine [e₁,e₂].

• By antisymmetry: [e₁,e₁] = [e₂,e₂] = 0
• We only need to find [e₁,e₂] = ae₁ + be₂ for some constants a,b
• By antisymmetry: [e₂,e₁] = -ae₁ - be₂

Step 4: Check the Jacobi Identity For any three elements, the Jacobi identity must hold. Let's check with our basis elements:

• [e₁,[e₂,e₁]] + [e₂,[e₁,e₁]] + [e₁,[e₁,e₂]] = 0
• This simplifies to: [e₁,-ae₁-be₂] + 0 + [e₁,ae₁+be₂] = 0
• Which gives us: 0 = 0 ✓

The Jacobi identity is automatically satisfied!

Step 5: Classification of 2D Cases We have two subcases:

• Case 1: a = b = 0, so [e₁,e₂] = 0

This gives us the 2-dimensional abelian Lie algebra

• Case 2: Not both a,b are zero

By choosing a new basis, we can always make this look like [e₁,e₂] = e₂ This gives us a non-abelian 2-dimensional Lie algebra

4. The Answer:

Classification Result:

• Dimension 1: Exactly one Lie algebra (abelian)
• Dimension 2: Exactly two Lie algebras up to isomorphism:

1. The 2D abelian algebra: [e₁,e₂] = 0 2. The 2D non-abelian algebra: [e₁,e₂] = e₂ (also called the "affine" Lie algebra)

5. Memory Tip:

Think "Small dimensions = Simple choices": In low dimensions, you're either completely abelian (everything commutes) or you have just enough room for one "interesting" bracket relation. The constraints of the Lie algebra axioms severely limit your options when you don't have much space to work with! 🌟