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SemiSpin(16)

Contents

Context

Higher spin geometry

Group Theory

Higher Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

The semi-spin group in dimension 16.

Properties

As a subgroup of E 8E_8

The subgroup of the exceptional Lie group E8 which corresponds to the Lie algebra-inclusion 𝔰𝔬(16)𝔢 8\mathfrak{so}(16) \hookrightarrow \mathfrak{e}_8 is the semi-spin group in that dimension

SemiSpin(16)E 8 SemiSpin(16) \;\subset\; E_8

On the other hand, the special orthogonal group SO(16)SO(16) is not a subgroup of E 8E_8 (e.g. McInnes 99a, p. 11).

In heterotic string theory

In heterotic string theory with gauge group the direct product group E 8×E 8E_8 \times E_8 it is usually only this subgroup Semispin(16)×SemiSpin(16)Semispin(16) \times SemiSpin(16) which is considered (but typically denoted Spin(16)/ 2Spin(16)/\mathbb{Z}_2, see also Distler-Sharpe 10, Sec. 1).

rotation groups in low dimensions:

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
\vdots\vdots
D8SO(16)Spin(16)SemiSpin(16)
\vdots\vdots
D16SO(32)Spin(32)SemiSpin(32)

see also

References

Created on May 8, 2019 at 11:48:51. See the history of this page for a list of all contributions to it.