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Pin groups

Readings

• Gregory W. Moore. Quantum Symmetries and Compatible Hamiltonians. 17.3
• Daniel S. Freed, Gregory W. Moore. Twisted Equivariant Matter.
• nLab: Pin(2)
• Math Overflow: Representation theory of pin groups

I am focusing on {$\textrm{Pin}_+(2)\equiv\textrm{Pin}(2,0)$} and {$\textrm{Pin}_-(2)\equiv\textrm{Pin}(0,2)$}.

{$Cl(V,Q)$} has generators {$e_1, e_2$} such that {$e_1^2=e_2^2=1$} or {$e_1^2=e_2^2=-1$}

Consider elements {$v\in V$} such that {$Q(v)=\pm 1$}. {$v=\lambda_1 e_1 + \lambda_2e_2$}.

{$\textrm{Pin}_+(2)$}

The real Clifford algebra {$Cl(V,Q)=Cl_{2,0}$} has generators {$e_1^2=e_2^2=1$}

The generators of the Pin group {$\textrm{Pin}_+(2)$} are vectors {$v=\lambda_1 e_1 + \lambda_2e_2$} such that {$Q(v)=\pm 1$}. We have {$\pm 1=v^2=\lambda_1^2+\lambda_1 \lambda_2(e_1e_2+e_2e_1)+\lambda_2^2$} = {$\lambda_1^2+\lambda_2^2$}. So we can write {$v=\cos\theta\; e_1+\sin\theta\; e_2$}.

The elements of {$\textrm{Pin}_+(2)$} are the products of such vectors, which is to say, {$(\cos\theta_1\; e_1+\sin\theta_1\; e_2)(\cos\theta_2\; e_1+\sin\theta_2\; e_2)\cdots (\cos\theta_k\; e_1+\sin\theta_k\; e_2)$} for angles {$\theta_1, \theta_2,\cdots\theta_k\in [0,2\pi)$} and integers {$k\geq 0$}.

The group {$\textrm{Pin}_+(2)$} is {$\mathbb{Z}_2$}-graded. The odd elements have the form {$\cos\theta\; e_1+\sin\theta\; e_2$}.

A product of an odd element with an odd element yields an even element

{$(\cos\theta_1\; e_1+\sin\theta_1\; e_2)(\cos\theta_2\; e_1+\sin\theta_2\; e_2)=$}

{$\cos\theta_1\cos\theta_2 + \sin\theta_1\sin\theta_2 + (\cos\theta_1\sin\theta_2 - \sin\theta_1\cos\theta_2)e_1e_2 =$}

{$\cos (\theta_2-\theta_1) + \sin(\theta_2-\theta_1)\;e_1e_2$}.

A product of an even element and an odd element is an odd element.

{$(\cos\theta_3 + \sin\theta_3\;e_1e_2)(\cos\theta_4\;e_1 + \sin\theta_4\;e_2)=$}

{$(\cos\theta_3\;\cos\theta_4 + \sin\theta_3\;\sin\theta_4)e_1+(\cos\theta_3\;\sin\theta_4 - \sin\theta_3\;\cos\theta_4)e_2=$}

{$\cos(\theta_4-\theta_3)e_1+\sin(\theta_4-\theta_3)e_2$}

A product of an odd element and an even element is an odd element.

{$(\cos\theta_4\;e_1 + \sin\theta_4\;e_2)(\cos\theta_3 + \sin\theta_3\;e_1e_2)=$}

{$(\cos\theta_4\;\cos\theta_3 - \sin\theta_4\;\sin\theta_3)e_1+(\sin\theta_4\;\cos\theta_3 + \cos\theta_4\;\sin\theta_3)e_2=$}

{$\cos(\theta_3+\theta_4)e_1+\sin(\theta_3+\theta_4)e_2$}

A product of an even element and an even element is an even element.

{$(\cos\theta_1 + \sin\theta_1\;e_1e_2)(\cos\theta_2 + \sin\theta_2\;e_1e_2)=$}

{$(\cos\theta_1\;\cos\theta_2 - \sin\theta_1\;\sin\theta_2)+(\cos\theta_1\;\sin\theta_2 + \sin\theta_1\;\cos\theta_2)e_1e_2=$}

{$\cos(\theta_1+\theta_2)+\sin(\theta_1+\theta_2)e_1e_2$}

The even elements form a subgroup, the rotations of a circle. They are the spinors, the spin group {$\textrm{Spin}(2)$}. Note that {$(e_1e_2)(e_1e_2)=-e_1e_2$} whether {$e_1^2=e_2^2=1$} or {$e_1^2=e_2^2=-1$}. Thus there is only one such spin group whereas there are two pin groups for the distinction arises in multiplying the odd elements.

{$\textrm{Pin}_-(2)$}

The real Clifford algebra {$Cl(V,Q)=Cl_{0,2}$} has generators {$e_1^2=e_2^2=-1$}

The generators of the Pin group {$\textrm{Pin}_-(2)$} are vectors {$v=\lambda_1 e_1 + \lambda_2e_2$} such that {$Q(v)=\pm 1$}. {$v=\lambda_1 e_1 + \lambda_2e_2$}. {$\pm 1=v^2=-\lambda_1^2+\lambda_1 \lambda_2(e_1e_2+e_2e_1)-\lambda_2^2$} = {$-\lambda_1^2-\lambda_2^2$}. Thus {$1=\lambda_1^2+\lambda_2^2$}. We can write {$v=\cos\theta\; e_1+\sin\theta\; e_2$}.

The elements of {$\textrm{Pin}_-(2)$} are the products of such vectors, as for {$\textrm{Pin}_+(2)$}, and likewise, {$\textrm{Pin}_-(2)$} is {$\mathbb{Z}_2$} graded.

A product of an odd element with an odd element yields an even element

{$(\cos\theta_1\; e_1+\sin\theta_1\; e_2)(\cos\theta_2\; e_1+\sin\theta_2\; e_2)=$}

{$-\cos\theta_1\cos\theta_2 - \sin\theta_1\sin\theta_2 + (\cos\theta_1\sin\theta_2 - \sin\theta_1\cos\theta_2)e_1e_2 =$}

{$\cos (\pi - (\theta_2-\theta_1)) + \sin(\pi -(\theta_2-\theta_1))\;e_1e_2$}.

A product of an even element and an odd element is an odd element.

{$(\cos\theta_3 + \sin\theta_3\;e_1e_2)(\cos\theta_4\;e_1 + \sin\theta_4\;e_2)=$}

{$(\cos\theta_3\;\cos\theta_4 - \sin\theta_3\;\sin\theta_4)e_1+(\cos\theta_3\;\sin\theta_4 + \sin\theta_3\;\cos\theta_4)e_2=$}

{$\cos(\theta_3-\theta_4)e_1+\sin(\theta_3-\theta_4)e_2$}

A product of an odd element and an even element is an odd element.

{$(\cos\theta_4\;e_1 + \sin\theta_4\;e_2)(\cos\theta_3 + \sin\theta_3\;e_1e_2)=$}

{$(\cos\theta_4\;\cos\theta_3 + \sin\theta_4\;\sin\theta_3)e_1+(\sin\theta_4\;\cos\theta_3 - \cos\theta_4\;\sin\theta_3)e_2=$}

{$\cos(\theta_4-\theta_3)e_1+\sin(\theta_4-\theta_3)e_2$}

A product of an even element and an even element is an even element, as above.