Pin & Spin Groups: Double Covers of Orthogonal & SO Groups

In summary, Pin & Spin groups are mathematical structures used to describe double covers of orthogonal and special orthogonal (SO) groups, denoted as Pin(n) and Spin(n). These double covers are significant in understanding spinorial representations of these groups, which have applications in mathematics and physics. Pin & Spin groups are closely related to Clifford algebras and can be constructed using them. An example of a Pin & Spin group is Pin(3), the double cover of SO(3). These groups are used in various areas of physics, such as in the study of symmetries and quantum mechanics.
  • #1
redtree
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TL;DR Summary
Do both sets of the double cover contain an identity element?
Pin Groups are the double cover of the Orthogonal Group and Spin Groups are the double cover of the Special Orthogonal Group. Both sets of the double cover are considered to be groups, but it seems that only one of the sets of the double cover actually contains the identity element, which means that both are not groups. Am I missing something?
 
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  • #2
$$
\underbrace{\{\ldots \det =\pm 1\}}_{\text{Group}}=\underbrace{\{\ldots \det =-1\}}_{\text{ no Group}}\cup \underbrace{\{\ldots \det =1\}}_{\text{Group}}
$$
 
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Likes redtree
  • #3
Got it; thanks!
 
  • #4
redtree said:
Got it; thanks!
Just to be clear, both Pin and Spin are groups.
 

FAQ: Pin & Spin Groups: Double Covers of Orthogonal & SO Groups

What are Pin & Spin groups?

Pin & Spin groups are mathematical structures that are used to describe the double covers of orthogonal groups and special orthogonal groups. They are important in the study of spinors and their applications in physics and geometry.

What is the difference between Pin and Spin groups?

Pin groups are the double covers of orthogonal groups, while Spin groups are the double covers of special orthogonal groups. This means that Pin groups have twice the dimension of their corresponding orthogonal groups, while Spin groups have twice the dimension of their corresponding special orthogonal groups.

What are the applications of Pin & Spin groups?

Pin & Spin groups have various applications in mathematics, physics, and engineering. They are used in the study of spinors, which are mathematical objects that describe the intrinsic angular momentum of particles. They also have applications in gauge theory, string theory, and quantum field theory.

How are Pin & Spin groups related to Lie groups?

Pin & Spin groups are special types of Lie groups, which are mathematical structures that describe continuous symmetries. Pin groups are non-compact Lie groups, while Spin groups are compact Lie groups. This means that they have different properties and behave differently under certain operations.

Are there any open problems related to Pin & Spin groups?

Yes, there are still many open problems related to Pin & Spin groups. Some of the current research topics include the classification of Pin & Spin groups, their representations, and their applications in physics and geometry. There is also ongoing research on the relationship between Pin & Spin groups and other mathematical structures, such as Lie algebras and Clifford algebras.

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