C_n

Definition

In the context of simple Lie algebras, $c_n$ refers to the dual Coxeter number associated with a given root system of type $A_n$, $B_n$, $C_n$, or $D_n$. This number plays a critical role in the classification and representation theory of simple Lie algebras, influencing properties such as the structure constants and dimensionality of representations.

5 Must Know Facts For Your Next Test

1. $c_n$ is computed using the formula $c_n = n + 1$ for types $A_n$, $B_n$, $C_n$, and $D_n$.
2. The dual Coxeter number $c_n$ is significant because it helps determine the highest weight representations of the corresponding simple Lie algebra.
3. In terms of representation theory, $c_n$ influences various invariants like dimensions and weights associated with representations.
4. For classical types, the values of $c_n$ are 2, 3, 4, and 5 for types $A_1$, $B_2$, $C_3$, and $D_4$, respectively.
5. Understanding $c_n$ is essential for studying fusion rules in conformal field theory, linking algebraic structures to physical theories.

Review Questions

• How does the value of c_n relate to the highest weight representations in simple Lie algebras?
  • $c_n$, as the dual Coxeter number, directly influences the structure of highest weight representations in simple Lie algebras. These representations can be characterized by their weights and dimensions, which are determined in part by this value. Thus, understanding how $c_n$ affects these aspects helps grasp the overall representation theory associated with a particular Lie algebra.
• Discuss the significance of c_n when classifying different types of simple Lie algebras.
  • $c_n$ serves as a key parameter when classifying simple Lie algebras because it encapsulates vital information about their structure and symmetry. Each type of root system (like $A_n$, $B_n$, etc.) has its own specific value for $c_n$, influencing how these algebras are grouped and analyzed within broader classification schemes. This categorization facilitates deeper insights into their algebraic properties and applications.
• Evaluate how understanding c_n enhances our comprehension of fusion rules in conformal field theory.
  • Grasping the concept of c_n allows us to better understand fusion rules in conformal field theory, which describe how different representations combine under certain operations. The value of c_n can determine multiplicities and constraints on these combinations, highlighting connections between algebraic structures and physical interpretations in quantum field theory. This interplay showcases the profound impact that algebraic concepts have on theoretical physics, enabling richer explorations into fundamental interactions.

Related terms

Coxeter Number: The Coxeter number is a vital integer that arises in the study of root systems and reflects the symmetries of the associated Lie algebra.

Simple Lie Algebra: A simple Lie algebra is a non-abelian algebra that cannot be decomposed into smaller algebras and serves as a building block for more complex structures.

Root System: A root system is a collection of vectors in a Euclidean space that encodes the symmetries and structure of a Lie algebra.