On the relation of the monomial group to other algebraic structures

Igor Bayak

July 4, 2001

Summary: In this paper, we show how the monomial group S_{2} ≀ S_{n} is related to the Abelian group Z^{n} and the general linear group GL_{n}(R). It is shown that any subgroup of the Abelian group Z^{n} gives rise to a subgroup of the monomial group S_{2} ≀ S_{n}, which in turn gives rise to a corresponding subgroup of GL_{n}(R). It is shown that any Clifford algebra can be generated by the cyclic monomial permutations.

1 Introduction

In order to the study of the algebraical theory of the monomial groups it would be better to refer to [1] or [2], then for study of the geometrical theory of the classical groups we refer to [3] or [4], and for study of the applications of group theory to physics we start from [5]. Goal of the present article is to investigate the connections between algebra, geometry and metaphysics with the monomial groups application. The references for this article might be the books [6], [7], [8], [9].

2 The monomial group generates the general linear group

Let I = {1, …, n} and s : I → I be a bijection, and let s′s : I → I : s′s(i) = s′ (s(i)) be the composition of bijections. Then the set of all of the n! bijections S = {s} with the product S × S → S : (s′, s) → s′s is a permutation group (symmetric group) of order n. Moreover, all of the even bijections form an alternating group, which, in the general case, is generated by 3–cycles; i.e., A_{n} := ⟨{(i, j, k)}_{I}⟩ where n ≥ 3 and i ≠ j ≠ k.

In addition, let {±I} := {±1, …, ±n}, and let p : I → {±I} be an injection |p| : I → I which is a bijection. Define a product p′p : I → {±I} by p′p(i) = sgn p(i) · p′(|p(i)|). Then the set P of all 2^{n}n! injections whose absolute value is bijective forms, with the product defined above, the group of monomial permutations of order n (monomial group). It is isomorphic to the wreath product P_{n} := S_{2} ≀ S_{n}. An element of monomial group is a transposition if it is an elementary permutation in the S_{n}–component or a permutation in one S_{2}– component of the wreath product. It is not difficult to show that every element p of P_{n} can be decomposed into a product of transpositions. The length modulo 2 of such a decomposition does not depend on the choice of a decomposition, and therefore is an invariant of p. As with the symmetric group, we can define a subgroup P_{n}^{+} ⊂ P_{n} by taking the set of all even elements in P_{n}. This subgroup is generated by the elements of the form (j, −k) : j → (−k), k → j where j ≠ k. Thus, we have P_{n}^{+} = ⟨{(j, −k)}_{I}⟩.

For any I^{∗} ⊂ I, we restrict the domain of definition of the maps s and p to the subset I^{∗}; i.e., we consider maps s^{∗} := s|I^{∗} and p^{∗} := p|I^{∗} such that the map s^{∗} : I^{∗} → I is an injection and p^{∗} : I^{∗} → {±I} is an injection.

Then s^{∗} is an arrangement and p^{∗} is a monomial arrangement of the set I onto the place I^{∗}. Moreover, a transposition of the arrangement (monomial arrangement) directly corresponds to a transposition of the permutation (monomial permutation); i.e., either to an elementary permutation (monomial permutation) in the subset I^{∗} or to an elementary permutation between subset I^{∗} and I\I^{∗}. Thus, every element s^{∗}(p^{∗}) can be decomposed into product of transpositions of this arrangement. Then the parities of the arrangements s^{∗} and p^{∗} are defined to be the lengths of decompositions σ(s^{∗}) and σ(p^{∗}), respectively, factored at Z_{2}

The notion of arrangement parity allows us to form new combinatorial groups; for example, let I^{1} = {1, …, r} and I^{2} = {r + 1, …, n}, and let s^{1} := s(I^{1}) and s^{2} := s(I^{2}). Then we can present s as a component permutation which consists of the arrangements s^{1} and s^{2}. Next, we define the component parity by σ(s^{1}) + σ(s^{2}) mod (2). The set of all the even–component permutations is a group S_{r,n−r}^{+}, which is generated by 3–cycles which act on the subsets I^{1}, I^{2}, and by an arbitrary 2–cycle, acting between them; that is,

S_{r,n−r}^{+} := ⟨{(i, j, k)}_{I^{1}}, (l, m), {(i, j, k)}_{I^{2}}⟩

where l ∈ I^{1} and m ∈ I^{2}.

In turn, all the even–component monomial permutations form a group P_{r,n−r}^{+}, which is generated by the elements (i, −j) acting in the subsets I^{1}, I^{2}, and by a pair of elements of the form ±(l, m), where +(l, m) : l → m, m → l, −(l, m) : l → (−m), m → (−l) and l ∈ I^{1}, m ∈ I^{2}. Hence, we have the presentation:

P_{r,n−r}^{+} := ⟨{(i, −j)}_{I^{1}}, ±(l, m), {(i, −j)}_{I^{2}}⟩ ,

where l ∈ I^{1} and m ∈ I^{2}.

Before we discuss the matrix realization of the monomial group, we find it convenient to define the determinant of a row subset of a square matrix A = (a_{i,j})_{i,j∈I} (i.e., of the rectangular matrix A(I^{∗}) := (a_{i,j})_{i∈I^{∗},j∈I}) as a sum of signed products of elements indexed by arrangements of columns on rows. In other words,

det A(I^{∗}) := ∑_{{s^{∗}}} ∏_{i∈I^{∗}}(−1)^{σ(s^{∗})} · a_{is^{∗}(i)}

Now let P (I) := {(p_{ij})_{I}} be the set of all square matrices with the property that every column and every row have only one nonzero element and, if p_{ij} ≠ 0, then p_{ij} ∈ {±1}. There is a linear representation P (I) of the monomial group P_{n} given by formula p_{ij} = sgn p(i) for every j = |p(i)|, and p_{ij} = 0 for every j ≠ |p(i)|. Then the matrix groups

{(p_{ij})_{I} | det P (I) = +1}

and

{(p_{ij})_{I} | det P (I^{∗}) · det P (I\I^{∗}) = +1}

are images of linear representations of even and even–component monomial groups respectively. Moreover, whenever we are given a set J = {1, …, m}, where m ≤ n, and a family I_{J} := {I_{j}}_{J} , where ∪_{m}I_{j} = I, ∩_{m}I_{j} = ⊘, card I_{j} = n_{j}, ∑_{m} n_{j} = n, and, in addition, it is true that the subset I_{j+1} is filled by sequential sampling from I after filling I_{j}; i.e., I_{j+1} = {n_{j} + 1, …, n_{j+1}} and I_{1} = {1, …, n_{1}}, then matrix group

{(p_{ij})_{I} | det P (I_{1}) ⋯ det P (I_{m}) = +1}

is the image of a linear representation of an arbitrary subgroup P_{n_{1},...,n_{m}}^{+} of even–component monomial permutations.

Finally, we establish the connection of the monomial group with the general linear group. Let RP_{n} be the algebra over R which is generated as the linear span of the group P_{n} presented by subset of the matrix algebra ML_{n}(R); i.e., RP_{n} = R⟨P (I)⟩. In addition, let GP_{n} be the multiplicative group of the algebra RP_{n}, and let R^{∗} be the multiplicative group of field R. Also, let Aut RP_{n} := Int GP_{n} ≃ GP_{n}/R^{∗} be the group of automorphisms of this algebra. Then, we get the fundamental fact that

GP_{n} = GL_{n}(R)

Indeed, since the group P_{n} is realized by the matrices P (I), from which we can always choose n^{2} linearly independent matrices, it follows that we have a basis of the algebra ML_{n}(R); hence, the linear span of this group coincides with ML_{n}(R); i.e., RP_{n} = ML_{n}(R); therefore, GP_{n} = GL_{n}(R). As a consequence of the fundamental fact noted above, we have Aut RP_{n} = P GL_{n}(R), where P GL_{n}(R) is a total projective group.

Now let GL_{ij} := diag [1, …, GL_{2}(R)_{i,j}^{i,j}, …, 1]_{n} , where i ≠ j. Here the upper indices denote row numbers and lower indices denote column numbers. The intersection is the group GL_{2}(R). Then we can also generate the linear groups as follows:

GL_{n}(R) = ⟨{GL_{i,j}}_{I}⟩ = ⟨{GL_{i,i+1}}_{I}⟩

Indeed, since P_{n} = ⟨{P_{i,j}}_{I}⟩ = ⟨{P_{i,i+1}}_{I}⟩ , where P_{i,j} is the restriction of P to the set {i, j}; i.e., P_{i,j} := P |_{{i,j}→{±i,±j}}; it follows that GP_{n} = ⟨{GP_{i,j}}_{I}⟩. But GP_{i,j} = GL_{i,j} Hence, the statement is proved.

Now suppose that we are given linear representations of two monomial groups

P_{2}^{+} = ⟨( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )⟩ and P_{1,1}^{+} = ⟨± ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )⟩ . Then it is not difficult to

show that

Aut RP_{2}^{+} = \{( \begin{matrix}cos x & sin x \\ − sin x & cos x \end{matrix} ) , x ∈ R\} = SO(2)

and

Aut RP_{1,1}^{+} = \{± ( \begin{matrix}cosh x & sinh x \\ sinh x & cosh x \end{matrix} ) , x ∈ R\} = SO(1, 1).

If we denote

SO(2)_{i,j} := diag [1, …, SO(2)_{i,j}^{i,j}, …, 1]_{n}

and

SO(1, 1)_{i,j} := diag [1, …, SO(1, 1)_{i,j}^{i,j}, …, 1]_{n}

we obtain a presentation of the special orthogonal group

Aut RP_{n}^{+} = ⟨{SO(2)_{i,i+1}}_{I}⟩ = SO(n)

and the special pseudo–orthogonal group

Aut RP_{r,n−r}^{+} = ⟨{SO(2)_{i,i+1}}_{I_{1}} , SO(1, 1)_{r,r+1}, {SO(2)_{i,i+1}}_{I_{2}}⟩ = SO(r, n − r)

However, a generalization of special orthogonal group is possible, including special case of pseudo–orthogonal group, namely,

Aut RP_{n_{1},...,n_{m}}^{+} = SO(I_{J}) := SO(n_{1}, …, n_{m}) =
⟨{{SO(2)_{i,i+1}}_{I_{j}}}_{J} , {SO(1, 1)_{n_{j},n_{j+1}}}_{J}⟩

It is now clear that all possible subgroups of the monomial group generate all possible linear groups; therefore, the problem of the classification and construction of all the linear groups coincides with the corresponding problem for subgroups of the monomial group.

3 The monomial group as a group of automorphisms of the Z module Z^{n}

Let Z be the ring of integers. Then Z^{n} is an n–dimensional Z–module. Let e_{I} := (e_{i})_{I} denote a basis. Let GL_{n}(Z) be the general integer linear group; i.e., the group of nonsingular integral–valued matrices. Let Aut Z^{n} := Int GL_{n}(Z) be the group of automorphisms of Z^{n} Then, since Int GL_{n}(Z) ≃ GL_{n}(Z)/{±1} = P (I) ≃ P_{n}, we have the isomorphism

Aut Z^{n} ≃ P_{n}

We will now attempt to find a group of automorphisms of the module Z^{n} which correspond to the topology of the quotient {−e_{I}, o, e_{I}} / ∼; i.e., to the partition of the set consisting of base elements, inverse base elements and the point of reference at equivalence classes. However, since the partition of this base set must be induced by the factor group Z^{n}/H^{n}, then it should be defined by automorphisms of the module Z^{n} which correspond to the topology of factor group. For example, for the factor group Z/Z we have one class [−e ∼ e ∼ o] and for the factor group Z/mZ, where m ≥ 2, we have a partition into two classes [−e ∼ e], [o], so the topological type of the generalized factor group Z/H may be presented by the groups Z/Z and Z/2Z. It is characteristic also that any factor group Z^{n}/H^{n} induces at least the equivalence relation −e_{i} ∼ e_{i} for every i ∈ I.

Turning to the automorphisms of Z^{n}, we note that the map −I × I → −P (I) × P (I) realizes rearrangements of elememts of the set {−e_{I}, e_{I}} , rearrangements which are coordinated with automorphisms Z^{n}, according to the formula

−e_{I} × e_{I} → −e′_{I} × e′_{I} : e′_{i} = e_{p(i)} := sgn p(i) · e_{|p(i)|}

Let −e_{I} × e_{I} ↔ −e′_{I} × e′_{I} be a reversible map (a map whose arrow can be reversed; i.e., a map that is equal to its inverse map), then the maps

−e_{i} × e_{i} ↔ e_{i} × −e_{i}
−e_{j}, −e_{k} × e_{j}, e_{k} ↔ −e_{k}, −e_{j} × e_{k}, e_{j}
−e_{j}, −e_{k} × e_{j}, e_{k} ↔ e_{k}, e_{j} × −e_{k}, −e_{j}

realize elementary reversible rearrangements of the elements of set {−e_{I}, e_{I}}; i.e., the 2–cycle (e_{i}, −e_{i}) and the two pairs of 2–cycles (e_{j}, e_{k}), (−e_{j}, −e_{k}) and (e_{j}, −e_{k}), (e_{k}, −e_{j}), respectively. If we present now the quotient {−e_{I}, o, e_{I}} / ∼ as a 2n–link graph glued at equivalence points, then it is clear that we cannot reconstruct this factored graph; indeed, it is impossible to realize a rearrangement of two links, the ends of which are glued with each other but not glued to the point of reference, where a 2–link element of the graph should be concidered as 2–dimensional; i.e., a planar graph.

Thus, starting from topological arguments, we postulate the impossibility of a reversible rearrangement of an arbitrary pair elements of the set {−e_{I}, e_{I}} if they are equivalent in some class other than the class of the point of reference. Hence, we get a restriction of the group of automorphisms Z^{n} to some subgroup of the monomial group. So, if the pairs {(−e_{i}, e_{i})}_{I} of base elements are grouped in m equivalence classes, none of which is not equivalent to the point of reference, then group of topological automorphisms Z^{n} corresponds to subgroup P_{n_{1},...,n_{m}}^{+}. Indeed, only the generators of the even–component monomial group exclude the permutations which correspond to all the elementary reversible rearrangements in each set {−e_{I_{j}}, e_{I_{j}}}, but include the permutations which correspond to the pairs of 2–cycles between each {−e_{I_{j}}, e_{I_{j}}} and {−e_{I_{k}}, e_{I_{k}}}.

We also present a convenient method of constructing an arbitrary factor group. Let |z| mod 2 : Z → Z/2Z : z′ ≡ z (mod 2) : |z| mod 2 = r, where r is the remainder the division z: 2, taken without the sign of z. Then we have the field Z_{2} = ⟨{0, 1}; ⊕, ·, 0, 1⟩, where x ⊕ y = |x + y| mod 2. Moreover, if Z_{2}^{n} := ∏_{n} Z_{2}, then Z_{2}^{n} is an Abelian group with component–wise addition. Furthermore, Z_{2}^{n} is a linear space over Z_{2} and e_{I} := (e_{i})_{I}, where e_{i} = (0_{(1)}, …, 1_{(i)}, …, 0_{(n)}), is its proper base. Let H_{2}^{n} denote a generalized subgroup of the group Z_{2}^{n}, and let H^{+}(I) denote the concrete subgroup which consists of all of the possible even sums of elements of its proper base, namely, H^{+}(I) = ⟨{e_{j} ⊕ e_{k}}_{I}⟩, which divides Z_{2}^{n} at the two coset, so that Z_{2}^{n}/H^{+}(I) ≃ Z_{2}. Let us assume that H(I_{J}) := ⟨{H^{+}(I_{j})}_{J}⟩ . Then Z_{2}^{n}/H^{+}(I_{J}) ≃ Z_{2}^{m}. Moreover, since we have the group homomorphism z : Z_{2}^{n} → Z^{n} : (0 → 2Z, 1 → 2Z + 1), it follows that we have the factor group homomorphism Z_{2}^{n}/H_{2}^{n} → Z^{n}/H^{n} and hence, Aut Z^{n}/H^{n} = Aut Z^{n}/z ∗ H_{2}^{n} ⊆ P_{n} is a group of topological automorphisms of Z^{n}; in particular, Aut Z^{n}/z∗H^{+}(I_{J}) = P_{n_{1},...,n_{m}}^{+}, and Aut Z^{n}/z ∗ Z_{2}^{n} = Aut Z^{n}/Z^{n} = P_{n}.

Furthermore, keeping in mind that any quotient {R^{n}/H^{n}} may be identified with some compact space, the topology of which is defined by factor group Z^{n}/H^{n}, we obtain an identification with the n–sphere {R^{n}/z ∗ H^{+}(I)} ≃ S^{n} or, more generally, with the toroidal space {R^{n}/z ∗ H^{+}(I_{J})} ≃ S^{n_{1}} × ⋯ × S^{n_{m}}. Moreover, since RP_{n} = End R^{n}, it follows that End S^{n_{1}} × ⋯ × S^{n_{m}} ≃ RP_{n_{1},...,n_{m}}^{+} and Aut S^{n_{1}} × ⋯ × S^{n_{m}} ≃ SO(n_{1}, …, n_{m}); hence, the rigidity of Euclidean spaces determined by a certain type of functional of the scalar product is the result of a restriction of the algebra of endomorphisms and a group of automorphisms of the space R^{n} arising from its factorization. In particular, the geometry of a proper Euclidean space is associated with a factorization of n–dimensional arithmetical space into the n–dimensional sphere, and therefore, End E(n) ≃ End S^{n} ≃ RP_{n}^{+}. On the other hand, the projective space RP^{n} ≃ {R^{n}/Z^{n}} does not possess topological rigidity, and therefore, End RP^{n} = RP_{n} and Aut RP^{n} = P GL_{n}(R).

Thus, the monomial group, through its own subgroups, provides conformity between the subgroups of the Abelian group Z^{n} and the subgroups of the general linear group GL_{n}(R); moreover, any Abelian subgroup is associated with some topologically compact space, the group of automorphisms of which coincides with the corresponding linear subgroup.

4 The Clifford algebras formed by the cyclic monomial permutations

For the even set I = {1, …, 2n}, we define a reversion Ĩ: I → I : {1, …, 2n} → {2n, …, 1}, an inversion −1 : I → {−I} : {1, …, 2n} → {−1, …, −2n}, a half–inversion Ǐ: I → {±I} : {1, …, 2n} → {−1, …, −n, n + 1, …, 2n}, and a rotation Î:= Ĩ∗Ǐ = −1∗Ǐ∗Ĩ. Next, let I = {1, …, 2^{m}}, J = {1, …, k, …, m}, K = {1, …, 2^{k−1}} and suppose that we have a partition of I at the subset of equal cardinality — from 2 to 2^{m} elements in every subset; i.e., a partition such that for all k ∈ J, we have the family I^{k} := {I_{j}}_{K} , where ∩_{K}I_{j} = ⊘, ∪_{K}I_{j} = I, card I_{j} = 2^{m−k−1} ∀j ∈ K. Suppose also that every subset I_{j+1} is filled by sequential sampling from I after filling I_{j} and I_{1} = {1, …, 2^{m−k−1}}. Then we define a k–reversion k̃ : I → I : I_{j} → Ĩ_{j} ∀j ∈ K, a k–half–inversion Ǐ: I → I : I_{j} → Ǐ_{j} ∀j ∈ K and a k–rotation k̂ := k̃ ∗ ǩ = −1 ∗ ǩ ∗ k̃. Now we can show that the k–rotations satisfy the conditions of generators of classical

Clifford algebras Cl (\overbrace{−1, …, −1}^{m}), namely, e_{i}e_{j} + e_{j}e_{i} = −2δ_{ij} · 1. Indeed, it is not difficult to show that k̂ ∗ k̂ = −1 = −1 · 1 ∀k ∈ J and k̂ ∗ k̂′ = −1 ∗ k̂′ ∗ k̂ = −1 · k̂′ ∗ k̂ ∀k′ ≠ k, where 1 is the identity map. Hence, the group of cyclic monomial permutations ⟨{k̂}_{J}⟩ contains a base of classical Clifford algebra,

and since the decomposition −1 = −1 · 1 defines a splitting of ⟨{k̂}_{J}⟩ in two subsets, it follows that by choosing a base of desired algebra, we can exclude the linearly dependent elements e′_{i} = −1 ∗ e_{i} = −e_{i}.

Thus, if we denote the newly formed algebra in the same way as the group algebra over field F, then we have isomorphism

F ⟨\{k̂\}_{J}⟩ ≃ Cl (\overbrace{−1, …, −1}^{m}).

Note also that

F⟨{˜1, ˇ1}⟩ ≃ Cl (1, 1)

and

F ⟨ˇ1, \{k̂\}_{J}⟩ ≃ F ⟨˜1, \{k̂\}_{J}⟩ ≃ Cl (+1, \overbrace{−1, …, −1}^{m})

If we define a reversion of pairs (and analogously of the 2^{k} elements of I, where k ≠ m) by the map ^{2}Ĩ : I → I : {(1, 2), …, (2n − 1, 2n)} → {(2n − 1, 2n), …, (1, 2)}, and if for all k ≠ m we define the maps ^{2}k̃ : I → I : I_{j} →^{2} Ĩ_{j} ∀j ∈ K and ^{2}k̂ :=^{2} k̃ ∗ ǩ, then

F ⟨\{^{2}k̂ ∗ m̂\}_{J\m}⟩ ≃ Cl (\overbrace{1, …, 1}^{m−1})

Moreover, let m ∉ J^{∗} ⊂ J and card J^{∗} = r. Then

F ⟨\{^{2}k̂\}_{J^{∗}}, \{^{2}k̂ ∗ m̂\}_{J\m\J^{∗}}⟩ ≃ Cl (\overbrace{−1, …, −1}^{r}, \overbrace{1, …, 1}^{m−r−1})

If we use the linear representation of the monomial group, then it is not difficult to show that

C ≃ R⟨ˆ1⟩ where m = 1,
H ≃ R⟨ˆ1, ˆ2⟩ where m = 2,
σ̂ ≃ C⟨ˇ1,^{2} ˜1,^{2} ˆ1 ∗ ˆ2⟩, m = 2,

where σ̂ is the Pauli algebra,

γ̂ ≃ C⟨ˇ1,^{4} ˆ1 ∗ ˇ2,^{4} ˆ1 ∗^{2} ˜2,^{4} ˆ1 ∗^{2} ˆ2 ∗ ˆ3⟩, m = 3,

where γ̂ is the Dirac algebra, and

STA ≃ R⟨ˇ1, ˆ1, ˆ2, ˆ3⟩, m = 3,

where STA is the space–time algebra.

We see that physical algebras formed by the cyclic monomial permutations could be considered as geometric algebras; i.e., as the subalgebras of endomorphisms of the corresponding Euclidean spaces, and therefore it would be interesting to find these spaces.

A Physical symmetries and Euclidean geometry

We make use of the mathematical formalism which we developed above, and we discuss certain algebras and groups which we associate with symmetries of physical vacuum and cosmos.

First, we turn to the Pauli and Dirac algebras, and embed the first in the algebra of endomorphisms of an oriented linear space of minimal dimensionality, and the second — in the algebra of endomorphisms of the direct product of oriented linear spaces of minimal dimensionality; that is,

σ̂ ≃ Cl (−1, −1) ≃ R⟨ˆ1, ˆ2⟩ ⊂ RP_{4}^{+} ≃ End E(4)

γ̂ ≃ Cl (+1, −1, −1, −1) ≃ R⟨ˇ1, ˆ1, ˆ2, ˆ3⟩ ⊂ RP_{4}^{+} × RP_{4}^{+} ≃ End E(4) × E(4)

Thus, the Pauli algebra is embedded in End E(4), and the Dirac algebra is embedded in End E(4) × E(4). Moreover, the observed component of vacuum space is 3–dimensional Euclidean space; hence, we have the inclusion E(3) ⊂ E(4) ⊂ E(4) × E(4), which defines the three degrees of physical perception of vacuum space.

We next turn to the group of ideal symmetries SU(3) × SU(2) × U(1) and SU(5), and to the group of violated symmetries SU(3) × U(1); that is, to the groups of invariance of the standard and grand unification models of guage interactions. Note that the inclusion SU(n) ⊆ SO(2n) and the isomorphism R^{2n}/SO(2n) ≃ S^{2n−1}, where R^{2n}/SO(2n) is the quotient of the orbits, point to the geometric origin (subject) of the unitary symmetries, so that we have the logical correspondences S^{9} ⇒ SU(5), S^{5} ⇒ SU(3), S^{3} ⇒ SU(2), S^{1} ⇒ U(1); moreover, SU(2) ≃ S^{3} and U(1) ≃ S^{1}. A Euclidean space, then, is the result of the factorization of an arithmetical space by a sphere; therefore, we have the correspondences E(9) ⇒ SU(5) and E(5) ⊕ E(3) ⊕ E(1) ⇒ SU(3) × SU(2) × U(1). Thus, if f is the endomorphism of E(9) defined by the matrix diag [α, α, α, α, α, β, β, β, 1], then E(9) can be decomposed as the direct sum E(5)⊕E(3)⊕E(1) of f–invariant subspaces, and the similarity ratios α, β, 1 can be compared with the constraint force of nuclear, weak and electromagnetic interaction respectively. If, now, we take into account the violation of symmetries, we need to use the linear transformation f defined by the quasidiagonal matrix diag [A, B], where A is diag [α, α, α, α, α] and B is a matrix with a null corner; i.e.,

( \begin{matrix}β & 0 & 0 & 0 \\ 0 & β & 0 & 0 \\ 0 & 0 & β & 0 \\ χ & χ & χ & 1 \end{matrix} )

Then we have the f–invariant subspaces E(5), E(1) and the potencial–invariant (if χ = 0) subspace E(3). Hence, the space f ∗E(9) can be considered as cosmic space and the space f ∗ S^{9} can be considered as a natural compactification of this space.

In conclusion, we note that the observed Euclidean space E(3) can be factored by S^{3}, from whence it follows that this space can be considered as a 3–dimensional foliation of cosmic space f ∗ E(9), a foliation which covers the sphere S^{3} in every layer. Moreover, since E(4, 4) ⊂ E(4)×E(4) and E(4, 4) can be factored by S^{4} × S^{4}, it follows that the oriented vacuum space E(4, 4) can be considered as an 8–dimensional foliation of cosmic space f ∗ E(9), covering the toroid S^{4} × S^{4} in every layer.

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