What is SU*(N)? Definition and Explanation

  • Context: Graduate 
  • Thread starter Thread starter William Nelso
  • Start date Start date
  • Tags Tags
    Groups Lie groups Star
Click For Summary
SUMMARY

The discussion centers on the definition and properties of the Lie group notation SU*(N) and its relation to other groups such as SO*(N) and SO(5,1). It is established that SU*(N) consists of elements X from SL(N,C) satisfying the condition X = B-1X*B, where B is the spinor conjugation operator. The paper referenced, Kugo and Townsend's work in Nuclear Physics B221, provides a foundational understanding of these groups, specifically noting the isomorphism between SO(5,1) and SU*(4). The star notation indicates a projective group, factoring in the center of the group with n-th roots of unity.

PREREQUISITES
  • Understanding of Lie groups and their notation
  • Familiarity with the special linear group SL(N,C)
  • Knowledge of spinor representations in the context of SO(D,1)
  • Basic concepts of projective groups and their properties
NEXT STEPS
  • Research the properties of the special unitary group SU(N) and its applications
  • Study the relationship between SO(N) and SU*(N) in the context of supersymmetry
  • Explore the implications of spinor conjugation operators in quantum field theory
  • Examine the structure and significance of projective groups in mathematics
USEFUL FOR

This discussion is beneficial for theoretical physicists, mathematicians specializing in group theory, and researchers involved in supersymmetry and quantum field theory.

William Nelso
Messages
20
Reaction score
1
I've run across a Lie group notation that I am unfamiliar with and having trouble googling (since google won't seem to search on * characters literally).

Does anyone know the definition of the "star groups" notated e.g. SU*(N), SO*(N) ??
The paper I am reading states for example that SO(5,1) is isomorphic to SU*(4).
(Ref Kugo+Townsend, Nuc. Phys. B221, p. 357, "Supersymmetry and the division algebras")

In fact it has a "definition" of these groups, however I am not able to understand it. It appears to say that
SU*(N) consists of elements X of SL(N,C) such that
X = B-1X* B
where B is the spinor conjugation operator (and N is the dimension of a spinor rep of the SO(D,1) that the paper is talking about)
 
Physics news on Phys.org

Similar threads

Graduate Understanding Simple Lie Groups: Definition and Common Misconceptions
  • · Replies 4 ·
Replies
4
Views
4K
Graduate Explicit construction of isomorphisms of low-rank Lie algebras.
  • · Replies 2 ·
Replies
2
Views
6K
Graduate Differential structure of the group of automorphism of a Lie group
  • · Replies 0 ·
Replies
0
Views
3K
Graduate What Are Simple Lie Algebras and How Are They Classified?
  • · Replies 1 ·
Replies
1
Views
4K
Graduate Definition of Lie Group and its Algebras
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Covering Group of SO(g) & Understanding Spinors on Curved Spacetime
  • · Replies 15 ·
Replies
15
Views
2K
Lie groups,Lie algebras, Physics, Lorentz Group,
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Characterizing the adjoint representation
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Complexifying Lie algebras (footnote to group thread)
  • · Replies 2 ·
Replies
2
Views
6K
Graduate What Is the Exponential Map for a Manifold?
  • · Replies 3 ·
Replies
3
Views
4K