Derived Series

Definition

The derived series of a Lie algebra is a sequence of subalgebras formed by iteratively taking the derived algebra, which is the commutator of the algebra with itself. This series is crucial for understanding the structure of the Lie algebra, particularly in distinguishing between solvable and non-solvable algebras. Each successive quotient of this series helps to analyze the properties of the algebra, providing insight into its solvability and potential nilpotency.

5 Must Know Facts For Your Next Test

1. The derived series is defined as follows: if L is a Lie algebra, then the first term is L', the derived algebra, and subsequent terms are formed by taking the derived algebra of the previous term.
2. If a Lie algebra has a finite derived series that terminates at zero, it is classified as solvable.
3. The length of the derived series can provide important information about the complexity and structure of the Lie algebra.
4. In contrast, if the derived series does not reach zero, it indicates that the Lie algebra may exhibit non-solvable behavior.
5. The derived series is closely related to the structure theory of Lie algebras, particularly in understanding their representations and modules.

Review Questions

• How does the derived series help classify a Lie algebra as solvable or non-solvable?
  • The derived series provides a systematic way to analyze a Lie algebra's structure by examining its successive derived algebras. If this series eventually reaches zero after a finite number of steps, then the Lie algebra is classified as solvable. Conversely, if the series does not terminate at zero, this indicates that the algebra has more complex relationships between its elements and is thus non-solvable.
• Discuss the relationship between the derived series and nilpotent Lie algebras.
  • Both the derived series and nilpotent Lie algebras relate to how commutation influences structure. While a nilpotent Lie algebra's lower central series reaches zero, indicating stronger conditions on its commutation relations, a solvable Lie algebra's derived series reaching zero signifies simpler hierarchical structure. In fact, every nilpotent Lie algebra is also solvable, meaning that nilpotency implies an even tighter control over commutation than just solvability.
• Evaluate how understanding the derived series contributes to insights about representations of Lie algebras.
  • Understanding the derived series is crucial when evaluating representations of Lie algebras since it reveals the hierarchical structure underlying an algebra. The simpler components represented in a solvable derived series can facilitate constructing representations that reflect these layers. This analysis also informs which representations are feasible based on how tightly knit or complex the relationships within the algebra are, ultimately guiding us in understanding how these structures operate under various transformations.

Related terms

Solvable Lie Algebra: A Lie algebra is solvable if its derived series eventually reaches zero after a finite number of steps, indicating that it can be broken down into simpler components.

Nilpotent Lie Algebra: A nilpotent Lie algebra is one where the lower central series reaches zero after a finite number of steps, often indicating even stronger structural properties than solvable algebras.

Commutator: The commutator of two elements in a Lie algebra measures their non-commutativity and is defined as [x, y] = xy - yx, which plays a key role in forming derived algebras.