What is SU*(N)? Definition and Explanation

• William Nelso
In summary, the conversation is discussing the notation and definition of "star groups" such as SU*(N) and SO*(N), as well as their relationship to other groups such as SO(5,1). The original paper being referenced provides a definition for these groups, but it is difficult to understand. It is suggested that the star notation may indicate the projective group, which is a factor group with n-th roots of unity.
William Nelso
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)

1. What is SU*(N)?

SU*(N) is a mathematical term that refers to the special unitary group of degree N. It is a subgroup of the general linear group and is often used in quantum mechanics and other branches of physics.

2. How is SU*(N) different from SU(N)?

SU*(N) is a subgroup of SU(N) and includes only those elements that have a determinant of 1. This means that SU*(N) is a smaller group than SU(N) and has different properties and applications.

3. What does the asterisk (*) indicate in SU*(N)?

The asterisk (*) is used to distinguish SU*(N) from SU(N) and is often used to denote a special or restricted version of a group. In this case, it indicates that the elements of SU*(N) have a determinant of 1.

4. How is SU*(N) related to quantum mechanics?

SU*(N) is a fundamental group in quantum mechanics and is used to represent the symmetries of quantum systems. It is particularly important in the study of quantum entanglement and quantum computing.

5. What are some examples of SU*(N)?

The most well-known example of SU*(N) is SU*(2), which is used to represent the symmetries of particles with spin ½ in quantum mechanics. Other examples include SU*(3), which is used in the study of quarks, and SU*(4), which is used in quantum systems with four energy levels.

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