Is the spin group spin(n) a double cover for O(n)?

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In summary, the pin group of a definite form is a double cover of the orthogonal group and each component is simply connected. However, for semidefinite forms, the pin group is still a double cover but each element is not necessarily simply connected to the orthogonal group.
  • #1
redtree
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The elements of the orthogonal group O(n) may not be continuously connected as in SO(n), but it seems that they should bijectively map to spin(n) for ##n\geq 3##
Every rotation in O(n) seems like it should map to spin(n) even if some rotations are not continuously connected to the identity.
 
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##Spin\left(n\right)## is the double cover of ##SO\left(n\right)##, while ##Pin\left(n\right)## is the double cover of ##O\left(n\right)##. The name is a pun. See (and references therein)

https://en.wikipedia.org/wiki/Pin_group
 
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  • #3
Thanks! I read the link (and much else).

From the Wikipedia page: "The pin group of a definite form maps onto the orthogonal group, and each component is simply connected: it double covers the orthogonal group. The pin groups for a positive definite quadratic form Q and for its negative −Q are not isomorphic, but the orthogonal groups are."

Why doesn't the same hold true for semidefinite quadratic forms?
 
  • #4
redtree said:
From the Wikipedia page: "The pin group of a definite form maps onto the orthogonal group, and each component is simply connected: it double covers the orthogonal group. The pin groups for a positive definite quadratic form Q and for its negative −Q are not isomorphic, but the orthogonal groups are."

Why doesn't the same hold true for semidefinite quadratic forms?
##\mathrm{Pin}\left(p.q\right)## is a double cover of ##\mathrm{O}\left(p.q\right)##. Is this what you mean?
 
  • #5
I apologize. My question was poorly worded.

Pin##(p,q)## is the double cover of ##O(p,q)##. For a definite form, each element in Pin##(p,q)## is simply connected to ##O(p,q)##.

My question: for a semidefinite form, Pin##(p,q)## is still the double cover of ##O(p,q)##, but each element in Pin##(p,q)## is NOT simply connected to ##O(p,q)##. Is my understanding correct?
 

1. What is the spin group?

The spin group, denoted as Spin(n), is a mathematical group that is closely related to the special orthogonal group O(n). It is a double cover for O(n), meaning that for every element in O(n), there are two corresponding elements in Spin(n).

2. What is the relationship between the spin group and O(n)?

The spin group is a double cover for O(n), which means that every element in O(n) has two corresponding elements in Spin(n). This relationship is important in mathematics and physics, particularly in the study of rotations and symmetries in higher dimensions.

3. How is the spin group defined?

The spin group is defined as a special type of matrix group, where each element is a 2x2 matrix with complex entries. These matrices have special properties that make them useful in studying rotations and symmetries in higher dimensions.

4. Why is the spin group important?

The spin group is important in mathematics and physics because it provides a way to study rotations and symmetries in higher dimensions. It also has applications in quantum mechanics, where it is used to describe the spin of particles.

5. What is the significance of the spin group being a double cover for O(n)?

The fact that the spin group is a double cover for O(n) has significant implications in mathematics and physics. It allows for a deeper understanding of rotations and symmetries in higher dimensions, and also has applications in quantum mechanics and other fields of science.

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