The nectarines implemented using Clifford algebra

In clifford: Arbitrary Dimensional Clifford Algebras

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To cite the clifford package in publications, use @hankin2022_clifford. Here I generate the nectarines $\mathbb{H}_\mathrm{nec}$ [@janovska2023] using Clifford package idiom. We need

$$i^2=1\qquad j^2=-1\qquad k^2=1$$

$$ij=k\qquad jk=i\qquad ki=-j$$

OK will try an even subalgebra, but what signature? Will try signature(2,1), giving $e_1^2=e_2^2=1$ and $e_3^2=-1$. Then we will try $i=a_ie_{13}$, $j=a_je_{12}$, $k=a_ke_{23}$ [on the grounds that $j$ is the odd man out and also $e_3$ is the odd man out, and $j$ is unique in having no 3]. Then:

$$i^2=e_{1313}=-e_{1331}=+e_{11}=1$$

$$j^2=e_{1212}=-e_{1221}=-e_{11}=-1$$

$$k^2=e_{2323}=-e_{2233}=+e_{33}=-1$$

(whatever values of $a_i$, $a_j$, $a_k$ are used), and

$$ij=a_ie_{13}\cdot a_je_{12}=a_ia_je_{1312}=-(a_ia_j)e_{1132}=(a_ia_j)e_{23}= \frac{a_ia_j}{a_k}k$$

$$jk=a_je_{12}a_ke_{23}=a_ja_ke_{1223}=a_ja_ke_{13}=\frac{a_ja_k}{a_i}i$$

$$ki=a_ke_{23}a_ie_{13}=a_ka_ie_{2313}=-a_ka_ie_{2133}=a_ka_ie_{21}=-a_ka_ie_{12}=\frac{-a_ka_i}{a_j}j$$

$$\frac{a_ia_j}{a_k}=1\qquad \frac{a_ja_k}{a_i}=1\qquad \frac{-a_ka_i}{a_j}=-1$$

so $a_i=a_j=a_k=1$ will satisfy the requirements [there are other solutions: any two of the $a$'s being $-1$ and the other being $+1$ would also work].

Verification

Package idiom to implement the above would be:

signature(2,1)
options(maxdim=3)
i <- e(c(1,3))
j <- e(c(1,2))
k <- e(c(2,3))

And then the verification would be

c(`i^2`=drop(i^2),`j^2`=drop(j^2),`k^2`=drop(k^2))
c(`ij=k`=i*j==k,`jk=i`=j*k==i,`ki=-j`=k*i==-j)

So we may use direct package idiom to generate some random nectarines and verify some simple identities:

f <- function(...){sample(9,1)}
n1 <- f() + i*f() + j*f() + k*f()
n2 <- f() + i*f() + j*f() + k*f()
n3 <- f() + i*f() + j*f() + k*f()
n1

c(
    `commutative`  = n1*n2 == n2*n1,
    `distributive` = n1*(n2+n3) == n1*n2 + n1*n3,
    `associative`  = n1*(n2*n3) == (n1*n2)*n3
)

References

clifford documentation built on June 8, 2025, 10:56 a.m.