Cartan map between clifford algebras

Cartan's map isomorphisms from \(\operatorname{Cl}(p,q)\) to \(\operatorname{Cl}(p-4,q+4)\) and \(\operatorname{Cl}(p+4,q-4)\)

Usage

cartan(C, n = 1)
cartan_inverse(C, n = 1)

Arguments

C

Object of class clifford

n

Strictly positive integer

Value

Returns an object of class clifford. The default value n=1 maps \(\operatorname{Cl}(4,q)\) to \(\operatorname{Cl}(0,q+4)\) (cartan()) and \(\operatorname{Cl}(0,q)\) to \(\operatorname{Cl}(4,q-4)\).

References

E. Hitzer and S. Sangwine 2017. “Multivector and multivector matrix inverses in real Clifford algebras”, Applied Mathematics and Computation. 311:3755-89

Author

Robin K. S. Hankin

See also

clifford

Examples

a <- rcliff(d=7)   # Cl(4,3)
b <- rcliff(d=7)   # Cl(4,3)
signature(4,3)     # e1^2 = e2^2 = e3^2 = e4^2 = +1; e5^2 = e6^2=e7^2 = -1
ab <- a*b          # multiplication in Cl(4,3)

signature(0,7)   # e1^2 = ... = e7^2 = -1
cartan(a)*cartan(b) == cartan(ab) # multiplication in Cl(0,7); should be TRUE
#> [1] TRUE

signature(Inf)  # restore default