Weyl spinor (changes) in nLab

representation theory

geometric representation theory

Ingredients

• linear algebra, algebra, higher algebra

representation, 2-representation, ∞-representation

• group, ∞-group

• group algebra, algebraic group, Lie algebra

• vector space, n-vector space

• affine space, symplectic vector space

• action, ∞-action

• module, equivariant object

• bimodule, Morita equivalence

• induced representation, Frobenius reciprocity

• Hilbert space, Banach space, Fourier transform, functional analysis

• orbit, coadjoint orbit, Killing form

• unitary representation

• geometric quantization, coherent state

• socle, quiver

• module algebra, comodule algebra, Hopf action, measuring

Geometric representation theory

• D-module, perverse sheaf,

• Grothendieck group, lambda-ring, symmetric function, formal group

• principal bundle, torsor, vector bundle, Atiyah Lie algebroid

• geometric function theory, groupoidification

• Eilenberg-Moore category, algebra over an operad, actegory, crossed module

• reconstruction theorems

• Borel-Weil-Bott theorem

• Beilinson-Bernstein localization

• Kazhdan-Lusztig theory

• BBDG decomposition theorem

Representation theory

spin geometry, string geometry, fivebrane geometry …

Ingredients

• supergeometry

• higher differential geometry

Spin geometry

spin geometry

• spin group

  pin group

  semi-spin group

  central product spin group

  spin^c group

• spin representation,

  • spinor, fermion

  • charge conjugation matrix, Fierz identity

  • real spin representation

  • Dirac conjugate

  • Dirac spinor, Weyl spinor, Majorana spinor

• spin structure, spin^c structure

• spinor bundle

• Dirac operator

  • index theory, K-theory

• Dirac equation

• Dirac field

rotation groups in low dimensions:

Dynkin label	sp. orth. group	spin group	pin group	semi-spin group
	SO(2)	Spin(2)	Pin(2)
B1	SO(3)	Spin(3)	Pin(3)
D2	SO(4)	Spin(4)	Pin(4)
B2	SO(5)	Spin(5)	Pin(5)
D3	SO(6)	Spin(6)
B3	SO(7)	Spin(7)
D4	SO(8)	Spin(8)		SO(8)
B4	SO(9)	Spin(9)
D5	SO(10)	Spin(10)
B5	SO(11)	Spin(11)
D6	SO(12)	Spin(12)
	⋮\vdots	⋮\vdots
D8	SO(16)	Spin(16)		SemiSpin(16)
	⋮\vdots	⋮\vdots
D16	SO(32)	Spin(32)		SemiSpin(32)

see also

• Spin(5).Spin(3)

• finite rotation groups

• ADE-classification

String geometry

string geometry

• string 2-group, string^c 2-group

• string structure, string^c structure

Fivebrane geometry

• fivebrane 6-group

• fivebrane structure

Ninebrane geometry

• ninebrane 10-group