Involutions of Bicomplex Numbers

Let $\mathbb{C}$ be the set of complex numbers together with its usual addition and multiplication making it a commutative field. It is an easy exercise to show that the only real-linear homomorphisms $f:\mathbb{C}\rightarrow\mathbb{C}$ such that $f(f(z))=z$ for any complex number $z$ are $f(z)=z$ and $f(z)=\overline{z}$, the complex conjugate. The functions satisfying these properties (real-linear homomorphism and $f\circ f=\mathrm{Id}$) will be called involutions. In fact, for an algebra $A$ over a commutative ring $R$, a function $f:A\rightarrow A$ is said to be an involution if $f^{2}(a)=a$, that is $f$ is its own inverse, and $f$ is a real-linear homomorphism of $A$. We may ask therefore if there are other characterizations of this type for other algebras of numbers.

In [7], the authors were interested in the (anti)involutions of the algebra of quaternions and its connections to the description of rotations and reflections in $\mathbb{R}^{3}$. The quaternion algebra were introduced by Hamilton in the 19th century. They are defined as the set $\mathbb{H}$ of expressions $q=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$ where $\mathbf{i^{2}}=\mathbf{j^{2}}=\mathbf{k^{2}}=-1$ and $a,b,c,d\in\mathbb{R}$, together with an addition defined component wise and a multiplication of the imaginary units $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ defined by the following rules:

With these operations, the quaternions become a noncommutative field (or a noncommutative algebra over $\mathbb{R}$). A pure quaternion is a quaternion ${\boldsymbol{\mu}}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}$. The authors showed that the functions of the form $q\mapsto-{\boldsymbol{\mu}}q{\boldsymbol{\mu}}$ for a given pure quaternion ${\boldsymbol{\mu}}=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}$, with $a^{2}+b^{2}+c^{2}=1$, are involutions of the quaternions $\mathbb{H}$. In [9], the authors then show that this is in fact the only involutions of the quaternions, completing the characterization of the involutions of the quaternions. The results of the authors of [7] were also generalized to dual quaternions and dual split quaternions in [1, 2].

In this paper, we are interested in the same problem of characterizing involutions but in the algebra of bicomplex numbers, denoted by $\mathbb{B}\mathbb{C}$. The precise definition of these numbers is presented in Section 2. The main characteristic of the bicomplex numbers is that, endowed with an addition and a multiplication, they become a commutative ring with zero divisors. The presence of zero divisors makes the algebra of bicomplex numbers rather different from the algebra of quaternions.

According to the literature on the bicomplex numbers, for example [10, 13] where the involution were first mentionned, only three possible non-trivial involutions on the set of bicomplex numbers are presented: if $s=z_{1}+z_{2}\mathbf{i_{2}}$ is a bicomplex number, where $z_{1},z_{2}$ are complex numbers, then

To the best of our knowledge, no attempt was made to show that these are in fact the only involutions of the bicomplex numbers. We however show that there are more involutions than expected. Our main result is the following.

The involutions $\dagger_{1}$, $\dagger_{2}$, and $\dagger_{3}$ corresponds to the known involutions $\ast$, $\star$, and $\dagger$ respectively. The last two were unknown yet in the literature. We also characterize all real-linear homomorphisms $f:\mathbb{B}\mathbb{C}\rightarrow\mathbb{B}\mathbb{C}$ such that $f^{n}=\mathrm{Id}$ where $\mathrm{Id}$ is the identity mapping and $n\geq 2$ is a positive integer. These functions are called $n$-involutions and our main result shows there are eight $n$-involutions and they only occur for $n=2$ and $n=4$. Our main results give another important difference between the algebra of bicomplex numbers and the algebra of quaternions: in the former, there are finitely many involutions and, in the latter, there are infinitely many involutions! The proofs of our main results are presented in Section 3.

In Section 4, we give a description of the group structures of the $n$-involutions. We use the $n$-involutions to give a new characterization of invertible bicomplex numbers. Our description was based on a result stated in [16]. Finally, in the last section, Section 5, we provide some geometric interpretations of the $n$-involutions of bicomplex numbers.

The bicomplex numbers, or more generally, the multicomplex numbers, were introduced by Segre [14] and Cockle [3, 4, 5, 6] to give another generalization of the complex numbers apart from the quaternions. For a modern treatment of the bicomplex numbers, we refer the reader to [10] or [12]. Here, we follow the presentation from [12].

A bicomplex number is defined by a duplication process of the complex numbers. To fix the notation, we will use the imaginary unit $\mathbf{i_{1}}$ such that $\mathbf{i_{1}^{2}}=-1$ and produce the set

of complex numbers. The set of bicomplex numbers is then defined in the following way.

If we endow the set $\mathbb{B}\mathbb{C}$ with the following operations: if $s=z_{1}+z_{2}\mathbf{i_{2}}$ and $t=w_{1}+w_{2}\mathbf{i_{2}}$, then

• •

  $s+t=(z_{1}+w_{1})+(z_{2}+w_{2})\mathbf{i_{2}}$;

• •

  $s\cdot t=(z_{1}w_{1}-z_{2}w_{2})+(z_{1}w_{2}+z_{2}w_{1})\mathbf{i_{2}}$.

then the triplet $(\mathbb{B}\mathbb{C},+,\cdot)$ is a commutative ring with zero divisors. Two divisors of zero play an important role in the theory of bicomplex numbers. We shall talk about this fact later in this section.

If $s$ is a bicomplex number, then by writing down explicitly the real and imaginary parts of the components of $s$, we then obtain the following representation in terms of real components:

where $\mathbf{j_{1}}=\mathbf{i_{1}}\mathbf{i_{2}}$ is called a hyperbolic unit because $\mathbf{j_{1}^{2}}=1$. Using a hyperbolic unit, we can define the set of hyperbolic numbers:

Equipped with the bicomplex multiplication, the set of hyperbolic numbers becomes a commutative subring of the set of bicomplex numbers. The hyperbolic numbers have a strong connection with the hyperbolic cosine and hyperbolic sine functions similar to the connection the complex numbers have with the cosine and sine functions. For more details on this subject, the reader is referred to Sobczyk’s article [15].

We mentioned that the bicomplex numbers form a commutative ring with non invertible elements. This is one of the major differences with the quaternions. Fortunately, in the case of the bicomplex numbers, we can completely characterize the non-invertible elements. To present the characterization, we introduce the following bicomplex numbers:

These two bicomplex numbers are called idempotent elements because of the following properties:

They also satisfy the following additionnal properties

Multiplying a bicomplex number $s$ respectively by $\mathbf{e_{1}}$ and $\mathbf{e_{2}}$, we obtain

Adding these last two identities and using the property $\mathbf{e_{1}}+\mathbf{e_{2}}=1$, we then see that a bicomplex number $s$ can be rewritten in the following form

This is called the idempotent representation of a bicomplex number. The complex numbers $z_{1}-z_{2}\mathbf{i_{1}}$ and $z_{1}+z_{2}\mathbf{i_{1}}$ are called the idempotent components. To simplify the notation, we denote the idempotent components by $z_{\mathbf{e_{1}}}$ and $z_{\mathbf{e_{2}}}$ respectively. This implies that the idempotent representation of $s$ can be rewritten in the following way:

The idempotent representation is useful because of the following properties: given two bicomplex numbers $s=z_{\mathbf{e_{1}}}\mathbf{e_{1}}+z_{\mathbf{e_{2}}}\mathbf{e_{2}}$ and $t=w_{\mathbf{e_{1}}}\mathbf{e_{1}}+w_{\mathbf{e_{2}}}\mathbf{e_{2}}$, we have

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  $s=0$ if and only if $z_{\mathbf{e_{1}}}=z_{\mathbf{e_{2}}}=0$;

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  $s=t$ if and only if $z_{\mathbf{e_{1}}}=w_{\mathbf{e_{1}}}$ and $z_{\mathbf{e_{2}}}=w_{\mathbf{e_{2}}}$;

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  $s+t=(z_{\mathbf{e_{1}}}+w_{\mathbf{e_{1}}})\mathbf{e_{1}}+(z_{\mathbf{e_{2}}}+w_{\mathbf{e_{2}}})\mathbf{e_{2}}$;

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  $st=(z_{\mathbf{e_{1}}}w_{\mathbf{e_{1}}})\mathbf{e_{1}}+(z_{\mathbf{e_{2}}}w_{\mathbf{e_{2}}})\mathbf{e_{2}}$.

So the bicomplex multiplication can be performed by simply multiplying the idempotent components together. This is a very powerful tool that can be used to extend many results from Complex Analysis to the bicomplex setting.

We can now use the idempotent representation to describe all non-invertible elements (see [12, Corollary 6.7]).

In Section 4, we will give another characterization of invertible elements using our result on bicomplex involutions.

Let recall more precisely the definition of an involution on the set of bicomplex numbers.

The usual definition of an involution involves only the first condition. However, for the quaternions and general algebra over commutative field (see [7] and [9]), the above definition was adopted. Therefore, to compare our result with the one obtained by the authors of [7], we will stick to the above definition. If we take a closer look at our definition of the term “involution”, we require that the function is a real linear homomorphism which is its own inverse.

We will now show our main result presented in the Introduction, that is Theorem 1. Before doing the proof, we will first prove some lemmas.

If we compare this Lemma 5 with what we know on the quaternions, then we find a second major difference between the two algebras. Indeed, any pure quaternion $q=a\mathbf{i}+b\mathbf{j}+c\mathbf{k}$ with $a^{2}+b^{2}+c^{2}=1$ squares to $-1$. In other words, there are infinitely many quaternions $q$ such that $q^{2}=-1$.

The next lemma combined with the previous one explain why we encounter only finitely many involutions of the bicomplex numbers.

We will now need two more lemmas. In the first lemma, by a signed imaginary unit $\mathbf{i}$, we mean any element of the set $\{\mathbf{i_{1}},-\mathbf{i_{1}},\mathbf{i_{2}},-\mathbf{i_{2}}\}$. Also, by a signed hyperbolic unit, we mean any element of the set $\{\mathbf{j_{1}},-\mathbf{j_{1}}\}$.

The proof of our main theorem is then a direct consequence of the last lemma.

It is quite useful to have the expression of the involution in terms of the idempotent components. It is not difficult to accomplish this task with some basic manipulations.

A natural question to ask now would be: how many functions $f:\mathbb{B}\mathbb{C}\rightarrow\mathbb{B}\mathbb{C}$ are there satisfying the conditions (2) and (3) of Definition 4, together with the more general condition $f^{n}(s)=s$ for any bicomplex number $s$, where $n\geq 2$ is an integer? We call the functions satisfying the previous conditions $n$-involutions where $n\geq 2$ is an integer. Note that $n$-involutions are still bijections because the condition $f^{n}=\mathrm{Id}$ means that $f^{n-1}$ is the inverse of $f$.

To answer completely this question, we need several lemmas. We will denote by $S_{4}$ the set of all permutations of $\{\mathbf{i_{1}},-\mathbf{i_{1}},\mathbf{i_{2}},-\mathbf{i_{2}}\}$. We use the same notation as the symmetric group because we are permutating four different symbols. For the symmetric group $S_{4}$, we will use the same term “$n$-involution” to refer to an element $\sigma\in S_{4}$ such that $\sigma^{n}=\sigma$ where $\sigma^{n}:=\sigma\circ\sigma\cdots\circ\sigma$, the $n$-fold composition of $\sigma$.

Lemma 10 has the following consequence. In the sequel, a non-trivial $n$-involution for $S_{4}$ is an $n$-involution that is different from the identity permutation.

Similar to the context of $S_{4}$, the term “non-trivial $n$-involution” for the set $\mathbb{B}\mathbb{C}$ refers to an $n$-involution for $\mathbb{B}\mathbb{C}$ which is different from the identity map.

What we need to do now is to find all involutions of order $2^{m}$ where $m\geq 2$ is an integer, namely the $2^{m}$-involutions. From Lemma 8, however, we simply have to check if the two cases excluded in the proof of Theorem 1 are $4$-involutions.

Here are the expressions of the two remaining $4$-involutions:

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  $s^{\ddagger_{6}}=\operatorname{Re}(z_{1})-\operatorname{Re}(z_{2})\mathbf{i_{1}}+\operatorname{Im}(z_{1})\mathbf{i_{2}}-\operatorname{Im}(z_{2})\mathbf{j_{1}}=\overline{z}_{\mathbf{e_{2}}}\mathbf{e_{1}}+z_{\mathbf{e_{1}}}\mathbf{e_{2}}$.

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  $s^{\ddagger_{7}}=\operatorname{Re}(z_{1})+\operatorname{Re}(z_{2})\mathbf{i_{1}}-\operatorname{Im}(z_{1})\mathbf{i_{2}}-\operatorname{Im}(z_{2})\mathbf{j_{1}}=z_{\mathbf{e_{2}}}\mathbf{e_{1}}+\overline{z}_{\mathbf{e_{1}}}\mathbf{e_{2}}$.

We will adopt the following terminology from now on. Any $2$-involution on $\mathbb{B}\mathbb{C}$ is called a conjugate and any $4$-involution which is not a $2$-involution is called a pseudo-conjugate. The reason for this terminology is the following. The $2$-involutions satisfy the same properties as the complex conjugate and it is natural to call them conjugates. To be more precise, they are bicomplex conjugates. The $4$-involutions which are not $2$-involutions do not quite satisfy the property of the complex conjugate; they do not satisfy the condition $f(f(s))=s$. However, we just need to apply them two other times to obtain the identity map. For this reason, they almost satisfy the property of being a conjugate.

From the work of Rochon and Shapiro [13], the four conjugates $\dagger_{0}$, $\dagger_{1}$, $\dagger_{2}$, and $\dagger_{3}$ form a group under the operation of composition of functions. The Cayley table of the relations between each conjugates is presented in Table 1 (see the Appendix).

However, if we try to join $\dagger_{4}$ and $\dagger_{5}$ to the group, then we no longer have a group. A way to see it is to compose one of the conjugates, $\dagger_{4}$ or $\dagger_{5}$, with the first four conjugates and see that we obtain an element outside of the set of $\{\dagger_{0},\dagger_{1},\dagger_{2},\dagger_{3},\dagger_{4},\dagger_{5}\}$. We can compose $\dagger_{4}$ and $\dagger_{1}$. We obtain

We know that $\ddagger_{6}$ is not a conjugate from the proof of Corollary 13. We therefore conclude that the set $\{\dagger_{0},\dagger_{1},\dagger_{2},\dagger_{3},\dagger_{4},\dagger_{5}\}$ is not closed under the composition and is therefore not a group.

If we change the point of view and consider the set of all $4$-involutions, we then obtain a group under the composition which is of order $8$. Simple calculations show that the Cayley table for the set $\{\dagger_{0},\dagger_{1},\dagger_{2},\dagger_{3},\dagger_{4},\dagger_{5},\ddagger_{6},\ddagger_{7}\}$ equiped with the composition is the Table 2 (see the Appendix). From this table, we see that we obtain a group under the composition and the order of the group is $8$ which is a power of $2$. We denote the group of conjugates and pseudoconjugates by ${\boldsymbol{\dagger}}$. The following summary of the six conjugations and the two $4$-involutions helped to construct the Cayley table:

1. $\dagger_{0}$:

   $f(s)=s$ (identity).

2. $\dagger_{1}$:

   $f(s)=z_{\mathbf{e_{2}}}\mathbf{e_{1}}+z_{\mathbf{e_{1}}}\mathbf{e_{2}}$.

3. $\dagger_{2}$:

   $f(s)=\overline{z}_{\mathbf{e_{2}}}\mathbf{e_{1}}+\overline{z}_{\mathbf{e_{1}}}\mathbf{e_{2}}$.

4. $\dagger_{3}$:

   $f(s)=\overline{z}_{\mathbf{e_{1}}}\mathbf{e_{1}}+\overline{z}_{\mathbf{e_{2}}}\mathbf{e_{2}}$.

5. $\dagger_{4}$:

   $f(s)=z_{\mathbf{e_{1}}}\mathbf{e_{1}}+\overline{z}_{\mathbf{e_{2}}}\mathbf{e_{2}}$.

6. $\dagger_{5}$:

   $f(s)=\overline{z}_{\mathbf{e_{1}}}\mathbf{e_{1}}+z_{\mathbf{e_{2}}}\mathbf{e_{2}}$.

7. $\ddagger_{6}$:

   $f(s)=\overline{z}_{\mathbf{e_{2}}}\mathbf{e_{1}}+z_{\mathbf{e_{1}}}\mathbf{e_{2}}$.

8. $\ddagger_{7}$:

   $f(s)=z_{\mathbf{e_{2}}}\mathbf{e_{1}}+\overline{z}_{\mathbf{e_{1}}}\mathbf{e_{2}}$.

Furthermore, we can extract some subgroups from the whole group. Here are the possible subgroups. Straightforward calculations show that

• •

  $\{\dagger_{0},\dagger_{1},\dagger_{2},\dagger_{3}\}$, $\{\dagger_{0},\dagger_{3},\dagger_{4},\dagger_{5}\}$, and $\{\dagger_{0},\dagger_{3},\ddagger_{6},\ddagger_{7}\}$ are subgroups of order $4$.

• •

  $\{\dagger_{0},\dagger_{i}\}$ for $i=1,2,\ldots,5$ are subgroups of order $2$.

• •

  $\{\dagger_{0}\}$ is a subgroup of order $1$.

These are the only subgroups of the group ${\boldsymbol{\dagger}}$. This is a consequence of the following theorem. We let $D_{8}$ be the dihedral group of the symmetries of the square. It is generated by two elements: a rotation of $90$ degrees counterclockwise, denoted by $a$ and a reflection about one of the lines joining midpoints of opposite sides, denoted by $x$. The operation is the composition of functions. More explicitly, we have

where $\mathrm{Id}$ is the identity mapping (doing nothing on the square). The Cayley table is presented in Table 3 (see the Appendix).

Define the map $\rho:{\boldsymbol{\dagger}}\rightarrow D_{8}$ by

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  $\rho(\dagger_{0}):=\mathrm{Id}$.

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  $\rho(\dagger_{1}):=x$.

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  $\rho(\dagger_{2}):=a^{2}x$.

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  $\rho(\dagger_{3}):=a^{2}$.

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  $\rho(\dagger_{4}):=a^{3}x$.

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  $\rho(\dagger_{5}):=ax$.

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  $\rho(\ddagger_{6}):=a$.

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  $\rho(\ddagger_{7}):=a^{3}$.

Comparing the Cayley table of the group ${\boldsymbol{\dagger}}$ (see Table 2) and the group $D_{8}$ (see Table 3), we see that $\rho$ is a group isomorphism. All the proper subgroups of $D_{8}$ are known. There are exactly nine of them:

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  $\{\mathrm{Id},x,a^{2},a^{2}x\}$, $\{\mathrm{Id},ax,a^{2},a^{3}x\}$, $\{\mathrm{Id},a,a^{2},a^{3}\}$ (subgroups with four elements).

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  $\{\mathrm{Id},x\}$, $\{\mathrm{Id},a^{2}x\}$, $\{\mathrm{Id},a^{2}\}$, $\{\mathrm{Id},a^{3}x\}$, $\{\mathrm{Id},ax\}$ (subgroups with two elements).

• •

  $\{\mathrm{Id}\}$ (trivial subgroup).

Applying $\rho^{-1}$ to the above subgroups, we obtain the list of subgroups of ${\boldsymbol{\dagger}}$ enumerated above.

The group structure of the conjugates and pseudoconjugates gives rise to surprising facts on the invertibility of a bicomplex number. If we multiply a bicomplex number with all of its possible conjugates and pseudoconjugates, we obtain

This is a real number. If we instead multiply all the conjugates from the subgroup $\{\dagger_{0},\dagger_{1},\dagger_{2},\dagger_{3}\}$ together, we obtain

Again, we obtain a real number. Similarly, if we multiply the conjugates in the subgroups $\{\dagger_{0},\dagger_{3},\dagger_{4},\dagger_{5}\}$ or $\{\dagger_{0},\dagger_{3},\ddagger_{6},\ddagger_{7}\}$ together, we obtain

Again we obtain a real number. This leads to the following characterization of the invertible elements in $\mathbb{B}\mathbb{C}$.

This last result gives interesting expressions of the inverse of a bicomplex number in terms of the product of its conjugates and/or pseudoconjugates. These expressions are similar to the usual expression of the inverse of a complex number in terms of its complex conjugate. Also, the last identities were discovered by Vallière and Rochon [16] for the tricomplex numbers in a very specific situation. What is suprising in our last result is that the same property can be extended to the other groups of conjugates and pseudoconjugates.

We end this paper by stating geometric interpretations for each conjugate and pseudoconjugate. We seek a geometric description similar to the geometric meaning the complex conjugate has. To do so, we will use the representations of the conjugates and pseudoconjugates in terms of four real coefficients. We will now think of a bicomplex number $s=x_{1}+x_{\mathbf{i_{1}}}\mathbf{i_{1}}+x_{\mathbf{i_{2}}}\mathbf{i_{2}}+x_{\mathbf{j_{1}}}\mathbf{j_{1}}$ as a quadruplet $\mathbf{s}=(x_{1},x_{\mathbf{i_{1}}},x_{\mathbf{i_{2}}},x_{\mathbf{j_{1}}})\in\mathbb{R}^{4}$.

We introduce three reflections of a vector $\mathbf{s}$:

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  $R_{\mathbf{i_{1}}}(\mathbf{s})=(x_{1},-x_{\mathbf{i_{1}}},x_{\mathbf{i_{2}}},x_{\mathbf{j_{1}}})$ (reflection about the plane $x_{\mathbf{i_{1}}}=0$).

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  $R_{\mathbf{i_{2}}}(\mathbf{s})=(x_{1},x_{\mathbf{i_{1}}},-x_{\mathbf{i_{2}}},x_{\mathbf{j_{1}}})$ (reflection about the plane $x_{\mathbf{i_{2}}}=0$).

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  $R_{\mathbf{j_{1}}}(\mathbf{s})=(x_{1},x_{\mathbf{i_{1}}},x_{\mathbf{i_{2}}},-x_{\mathbf{j_{1}}})$ (reflection about the plane $x_{\mathbf{j_{1}}}=0$).

The vector form of the expression for the first, second, and third conjugates become

• $\dagger_{1}$:

  $(x_{1},x_{\mathbf{i_{1}}},x_{\mathbf{i_{2}}},x_{\mathbf{j_{1}}})\mapsto(x_{1},x_{\mathbf{i_{1}}},-x_{\mathbf{i_{2}}},-x_{\mathbf{j_{1}}})$;

• $\dagger_{2}$:

  $(x_{1},x_{\mathbf{i_{1}}},x_{\mathbf{i_{2}}},x_{\mathbf{j_{1}}})\mapsto(x_{1},-x_{\mathbf{i_{1}}},x_{\mathbf{i_{2}}},-x_{\mathbf{j_{1}}})$;

• $\dagger_{3}$:

  $(x_{1},x_{\mathbf{i_{1}}},x_{\mathbf{i_{2}}},x_{\mathbf{j_{1}}})\mapsto(x_{1},-x_{\mathbf{i_{1}}},-x_{\mathbf{i_{2}}},x_{\mathbf{j_{1}}})$.