Algebra G2 (mathematics)

1 algebra

1.1 dynkin diagram , cartan matrix
1.2 roots of g2
1.3 weyl/coxeter group
1.4 special holonomy

algebra
dynkin diagram , cartan matrix

the dynkin diagram g2 given .

its cartan matrix is:

[








2


−
1




−
3





2





]



{\displaystyle \left[{\begin{smallmatrix}\;\,\,2&-1\\-3&\;\,\,2\end{smallmatrix}}\right]}



roots of g2

although span 2-dimensional space, drawn, more symmetric consider them vectors in 2-dimensional subspace of three-dimensional space.

one set of simple roots, is:

(0,1,−1), (1,−2,1)

weyl/coxeter group

its weyl/coxeter group



g
=
w
(

g

2


)


{\displaystyle g=w(g_{2})}

dihedral group,




d

6




{\displaystyle d_{6}}

of order 12. has minimal faithful degree



μ
(
g
)
=
5


{\displaystyle \mu (g)=5}

.

special holonomy

g2 1 of possible special groups can appear holonomy group of riemannian metric. manifolds of g2 holonomy called g2-manifolds.