Quaternion Algebra Construction
The compact Klein quartic can be constructed as the quotient of the hyperbolic plane by the action of a suitable Fuchsian group Γ(I) which is the principal congruence subgroup associated with the ideal in the ring of integers of the field where . Note the identity
exhibiting as a prime factor of 7 in the ring of integers.
The group Γ(I) is a subgroup of the (2,3,7) hyperbolic triangle group. Namely, Γ(I) is a subgroup of the group of norm 1 elements in the quaternion algebra generated as an associative algebra by the generators i,j and relations . One chooses a suitable Hurwitz quaternion order in the quaternion algebra, Γ(I) is then the group of norm 1 elements in . The least absolute value of a trace of a hyperbolic element in Γ(I) is, corresponding the value 3.936 for the systole of the Klein quartic, one of the highest in this genus.
Read more about this topic: Klein Quartic
Famous quotes containing the words algebra and/or construction:
“Poetry has become the higher algebra of metaphors.”
—José Ortega Y Gasset (1883–1955)
“When the leaders choose to make themselves bidders at an auction of popularity, their talents, in the construction of the state, will be of no service. They will become flatterers instead of legislators; the instruments, not the guides, of the people.”
—Edmund Burke (1729–1797)