Spin group

The spin group is an object from mathematics and physics , in particular from the areas of spectral geometry and quantum mechanics . A central property of the spin group is that it is a 2-fold superposition of the rotation group . ${\displaystyle \operatorname {Spin} (n)}$ ${\displaystyle SO(n)}$

definition

To a finite-dimensional vector space over a field and a square shape on one defines the Clifford algebra than the algebra over that of , and the identity element created is and their multiplication, the relation ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle Q\colon V\to K}$ ${\displaystyle V}$ ${\displaystyle Cl(V,Q)}$ ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle 1_{Cl}}$

${\displaystyle v\cdot v=-Q(v)1_{Cl}}$

Fulfills. Through this relationship the Clifford algebra is uniquely determined except for isomorphism.

The spin group to this square shape is then defined as the subgroup of the products of an even number of unit vectors

${\displaystyle \operatorname {Spin} (V,Q):=\{v_{1}\dots v_{2k}\in Cl(V,Q):k\in \mathbb {N} ,v_{1},\ldots ,v_{2k}\in V,Q(v_{1})=\ldots =Q(v_{2k})=1\}\subset Cl(V,Q)}$.

The spin group to the square shape

${\displaystyle Q=x_{1}^{2}+\ldots +x_{n}^{2}}$

on- vector space is briefly referred to as . ${\displaystyle \mathbb {R} }$ ${\displaystyle V=\mathbb {R} ^{n}}$ ${\displaystyle \operatorname {Spin} (n)}$

For denotes the spin group to the square shape ${\displaystyle p+q=n}$ ${\displaystyle \operatorname {Spin} (p,q)}$

${\displaystyle Q=x_{1}^{2}+\ldots +x_{p}^{2}-x_{p+1}^{2}-\ldots -x_{n}^{2}}$

on the vector space . ${\displaystyle \mathbb {R} }$ ${\displaystyle V=\mathbb {R} ^{n}}$

Examples

For one has the following isomorphisms to classical Lie groups: ${\displaystyle n\leq 6}$

• ${\displaystyle \operatorname {Spin} (2)\cong S^{1}}$
• ${\displaystyle \operatorname {Spin} (3)\cong SU(2)}$
• ${\displaystyle \operatorname {Spin} (4)\cong SU(2)\times SU(2)}$
• ${\displaystyle \operatorname {Spin} (5)\cong Sp(2)}$
• ${\displaystyle \operatorname {Spin} (6)\cong SU(4)}$

Spin (n) as a 2-fold superposition of the SO (n)

Theorem: is a two-fold superposition of the . ${\displaystyle \operatorname {Spin} (n)}$ ${\displaystyle SO(n)}$

Proof sketch: In Clifford algebra , with applies to all . The image ${\displaystyle Cl(n)}$ ${\displaystyle v^{-1}=-v}$ ${\displaystyle v\in \mathbb {R} ^{n}}$ ${\displaystyle q(v,v)=1}$

${\displaystyle x\to -vxv^{-1}=vxv=x-2q(x,v)v}$

is a reflection of and it is compatible with products, so it defines a representation ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle \operatorname {Spin} (n)\to SO(n)}$.

Because each element is the product of an even number of reflections, one obtains a surjective map that can be shown to be a superposition . The kernel consists only of , because elements in the kernel must commute with all, i.e. belong to the center of the Clifford algebra, which only consists of scalar multiples of . are the only associated scalar multiples of , as can be seen from the formula valid in , from which it follows for multiples of that their square is. ${\displaystyle SO(n)}$ ${\displaystyle \pm 1_{Cl}}$ ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle 1_{Cl}}$ ${\displaystyle \pm 1_{Cl}}$ ${\displaystyle Spin(n)}$ ${\displaystyle 1_{Cl}}$ ${\displaystyle \operatorname {Spin} (n)}$ ${\displaystyle (v_{1}\ldots v_{2k})^{-1}=v_{2k}\ldots v_{1}}$ ${\displaystyle 1_{Cl}}$ ${\displaystyle 1}$

For is simply connected and the universal superposition of . ${\displaystyle n\geq 3}$ ${\displaystyle \operatorname {Spin} (n)}$ ${\displaystyle SO(n)}$

Analog is a double superposition of , the connected component of the one of . For is connected , but has two connected components. ${\displaystyle \operatorname {Spin} (p,q)}$ ${\displaystyle SO_{0}(p,q)}$ ${\displaystyle SO(p,q)}$ ${\displaystyle p+q\geq 3}$ ${\displaystyle \operatorname {Spin} (p,q)}$ ${\displaystyle \operatorname {Spin} (1,1)}$

Lie algebra of spin (n)

The Lie algebra of the from the products with spanned subspace of . ${\displaystyle {\mathfrak {spin}}(n)}$ ${\displaystyle \operatorname {Spin} (n)}$ ${\displaystyle e_{i}e_{j}}$ ${\displaystyle i\not =j}$ ${\displaystyle Cl(n)}$

The superposition induces an isomorphism to the Lie algebra of the skew-symmetric matrices with trace . This corresponds to the skew-symmetric matrix with entries . ${\displaystyle \operatorname {Spin} (n)\to SO(n)}$ ${\displaystyle {\mathfrak {so}}(n)}$ ${\displaystyle 0}$ ${\displaystyle \textstyle {\frac {1}{4}}\sum _{i,j}a_{ij}e_{i}e_{j}}$ ${\displaystyle a_{ij}}$

Representations of spin (n)

Due to the homomorphism , all representations of also become representations of . These are first the standard representation of on and then the induced representations on the outer algebras for ${\displaystyle \operatorname {Spin} (n)\to SO(n)}$ ${\displaystyle SO(n)}$ ${\displaystyle \operatorname {Spin} (n)}$ ${\displaystyle SO(n)}$ ${\displaystyle V=\mathbb {R} ^{n}}$ ${\displaystyle \Lambda ^{k}V}$ ${\displaystyle k=2,3,\ldots }$

In addition, there is also the spinor representation for odd and especially the two half spinor representations of , which cannot be factored as representations of . Together with the above, all of the fundamental representations of . ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \operatorname {Spin} (n)}$ ${\displaystyle SO(n)}$ ${\displaystyle \operatorname {Spin} (n)}$

literature

• Blaine Lawson , Marie-Louise Michelsohn : Spin Geometry. Princeton University Press, ISBN 978-0-691-08542-5 .
• John Roe : Elliptic Operators, Topology, and Asymptotic Methods. Second edition. Chapman & Hall, CRC Research Notes in Mathematics Series, ISBN 978-0-582-32502-9 .
• Thomas Friedrich : Dirac operators in Riemannian geometry. Vieweg Verlag, ISBN 978-3-528-06926-1 .