Clifford involutions

An involution is a function that is its own inverse, or equivalently \(f(f(x))=x\). There are several important involutions on Clifford objects; these commute past the grade operator with \(f(\left\langle A\right\rangle_r)=\left\langle f(A)\right\rangle_r\) and are linear: \(f(\alpha A+\beta B)=\alpha f(A)+\beta f(B)\).

The dual is documented here for convenience, even though it is not an involution (applying the dual four times is the identity).

• The reverse \(A^\sim\) is given by rev() (both Perwass and Dorst use a tilde, as in \(\tilde{A}\) or \(A^\sim\). However, both Hestenes and Chisholm use a dagger, as in \(A^\dagger\). This page uses Perwass's notation). The reverse of a term written as a product of basis vectors is simply the product of the same basis vectors but written in reverse order. This changes the sign of the term if the number of basis vectors is 2 or 3 (modulo 4). Thus, for example, \(\left(e_1e_2e_3\right)^\sim=e_3e_2e_1=-e_1e_2e_3\) and \(\left(e_1e_2e_3e_4\right)^\sim=e_4e_3e_2e_1=+e_1e_2e_3e_4\). Formally, if \(X=e_{i_1}\ldots e_{i_k}\), then \(\tilde{X}=e_{i_k}\ldots e_{i_1}\).

  $$\left\langle A^\sim\right\rangle_r=\widetilde{\left\langle A\right\rangle_r}=(-1)^{r(r-1)/2}\left\langle A\right\rangle_r $$

  Perwass shows that \(\left\langle AB\right\rangle_r=(-1)^{r(r-1)/2}\left\langle\tilde{B}\tilde{A}\right\rangle_r \)

• The Conjugate \(A^\dagger\) is given by Conj() (we use Perwass's notation, def 2.9 p59). This depends on the signature of the Clifford algebra; see grade.Rd for notation. Given a basis blade \(e_\mathbb{A}\) with \(\mathbb{A}\subseteq\left\lbrace 1,\ldots,p+q\right\rbrace\), then we have \(e_\mathbb{A}^\dagger = (-1)^m {e_\mathbb{A}}^\sim\), where \(m=\mathrm{gr}_{-}(\mathbb{A})\). Alternatively, we might say $$\left(\left\langle A\right\rangle_r\right)^\dagger=(-1)^m(-1)^{r(r-1)/2}\left\langle A\right\rangle_r $$ where \(m=\mathrm{gr}_{-}(\left\langle A\right\rangle_r)\) [NB I have changed Perwass's notation].

• The main (grade) involution or grade involution \(\widehat{A}\) is given by gradeinv(). This changes the sign of any term with odd grade:

  $$\widehat{\left\langle A\right\rangle_r} =(-1)^r\left\langle A\right\rangle_r$$

  (I don't see this in Perwass or Hestenes; notation follows Hitzer and Sangwine). It is a special case of grade negation.

• The grade \(r\)-negation \(A_{\overline{r}}\) is given by neg(). This changes the sign of the grade \(r\) component of \(A\). It is formally defined as \(A-2\left\langle A\right\rangle_r\) but function neg() uses a more efficient method. It is possible to negate all terms with specified grades, so for example we might have \(\left\langle A\right\rangle_{\overline{\left\lbrace 1,2,5\right\rbrace}} = A-2\left( \left\langle A\right\rangle_1 +\left\langle A\right\rangle_2+\left\langle A\right\rangle_5\right)\) and the R idiom would be neg(A,c(1,2,5)). Note that Hestenes uses “\(A_{\overline{r}}\)” to mean the same as \(\left\langle A\right\rangle_r\).

• The Clifford conjugate \(\overline{A}\) is given by cliffconj(). It is distinct from conjugation \(A^\dagger\), and is defined in Hitzer and Sangwine as

  $$\overline{\left\langle A\right\rangle_r} = (-1)^{r(r+1)/2}\left\langle A\right\rangle_r.$$

• The dual \(C^*\) of a clifford object \(C\) is given by dual(C,n); argument n is the dimension of the underlying vector space. Perwass gives \(C^*=CI^{-1}\)

  where \(I=e_1e_2\ldots e_n\) is the unit pseudoscalar [note that Hestenes uses \(I\) to mean something different]. The dual is sensitive to the signature of the Clifford algebra and the dimension of the underlying vector space.

Usage

# S3 method for class 'clifford'
rev(x)
# S3 method for class 'clifford'
Conj(z)
cliffconj(z)
neg(C,n)
gradeinv(C)

Arguments

C,x,z

Clifford object

n

Integer vector specifying grades to be negated in neg()

Author

Robin K. S. Hankin

See also

grade

Examples

x <- rcliff()
x
#> Element of a Clifford algebra, equal to
#> + 5 - 7e_1 + 6e_2 - 4e_12 - 6e_1234 + 2e_2345 + 1e_356 - 2e_1356 + 4e_3456
rev(x)
#> Element of a Clifford algebra, equal to
#> + 5 - 7e_1 + 6e_2 + 4e_12 - 6e_1234 + 2e_2345 - 1e_356 - 2e_1356 + 4e_3456


A <- rblade(g=3)
B <- rblade(g=4)
rev(A %^% B) == rev(B) %^% rev(A)  # should be TRUE
#> [1] TRUE
rev(A * B) == rev(B) * rev(A)          # should be TRUE
#> [1] TRUE

options(maxdim=8)
a <- rcliff(d=8)
dual(dual(dual(dual(a,8),8),8),8) == a # should be TRUE
#> [1] FALSE
options(maxdim=NULL) # restore default