# Some Algebraic Properties of the Wilson Loop

Authors: Han Li; M. Nadeem; M. A. Ali; H. Mutee ur Rehman
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this article, some algebraic properties of the Wilson loop have been investigated in a broad manner. These properties include identities, autotopisms, and implications. We use some equivalent conditions to study the behavior of holomorphism of this loop. Under the shadow of this holomorphism, we are able to observe coincident loops.


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## Body

## 1. Introduction and Definitions

A nonempty set is defined to be a groupoid relative to the operation⋅ if and only if to each ordered pair α,β of elements of L, there exists a uniquely defined element α⋅β∈L (or simply αβ∈L). If μ∈L is a fixed element, then there exist two single-valued mappings Lμ, Rμ of L into itself by(1)αLμ=μαandαRμ=αμfor allα∈L.An element1∈L is said to be unit of L if for each α∈L, we have the identity law α1 = 1α = α. A unit if exists is unique. An element μ∈L is said to be left nonsingular if Lμ is a permutation of L and similarly an element μ of L is said to be right nonsingular if Rμ is a permutation of L. A system L,⋅,\,/ is called a quasigroup provided each of L,⋅, L,\, and L,/ is a groupoid and(2)ααβ=β,ααβ=β,αββ=α,αββ=α.For all,α,β∈L. For a quasigroup L, we can associate a permutation Tα:L⟶L defined by βTα=αβ/α,∀α,β∈L. A quasigroup which admits identity law is called a loop. For each α∈L, the elements αρ = αJρ, αλ = αJλ such that ααρ = eρ, αλα = eλ are called the right and left inverses of α under right Jρ:L⟶L and left Jλ:L⟶L inverse permutations on L, respectively. A loop satisfying Wilson identity in [1] is called Wilson loop. From Table 1 of Wilson loop, we have 11⋅10=0 which shows 10 is the right inverse of 11 with right multiplicative identity 0. For a loop L, the right and left inverses of an element of L are different in general but right eρ and left eλ identities are always same. It is not easy to control this identity algebraically due to the fact that there are three variables appearing at the right and two at the left side. A loop L with weak associativity αβγα=αβγα is called Moufang loop for all α,β,γ∈L. Nucleus of the loop L is the intersection of the left ℵλ=α∈L:αβγ=αβγ,∀β,γ∈L, middle ℵχ=β∈L:αβγ=αβγ,∀α,γ∈L, and right ℵρ=γ∈L:αβγ=αβγ,∀α,β∈L nuclei of L denoted by ℵL. An ordered triple Φ,Ψ,ξ of permutations of loop L is in the group of autotopisms, AUTL, of L if and only if αΦβΨ=αβξ,∀α,β∈L. If these bijections are coincident, then Φ is automorphism in group AUML. To every loop L with AUML, there corresponds another set HL. If we define ⋄ on HL such that Φ,α⋄Ψ,β=ΦΨ,αΨβ for all Φ,α,Ψ,β∈HL, Φ,Ψ⊆AUML, and α,β∈L, then HL is a loop and called the holomorph of L.Table 1
An example of a noncommutative and nonassociative Wilson loop of order 12.
c0123456789101100123456789101111204537861011922015348671191033450121011986744531201191067855342019101178666781191001245377869101112053488671011920134599101178653401210101198673451201111910678453201Inspired by the work on Moufang loop in [1], Goodaire and Robinson [2] introduced a new understanding of Wilson loop with the help of weak inverse property loop and conjugacy closed loop. It is shown in [1] that a Moufang loop is Wilson loop if and only if α2∈ℵL for all α∈L. In [3], the concept of holomorph of a loop is given by Bruck. An extension work on holomorphism of a loop is done in [4].In the present paper, it is shown that we can obtain loop isotopes of Wilson loop under different autotopisms. In this way, we are able to give new forms of Wilson loops by opposite loops, and finally, we see conditions to prove that holomorph of a Wilson loop is a Wilson loop.We finish this section by giving an example of a noncommutative and nonassociative Wilson loop of order 12 in the form of Table1.


## 2. Results and Identities

A permutationP of a loop L is known as pseudoautotopism of L if and only if there exists an element c∈L such that P,PRc,PRc is an autotopism of L. The element c is called a companion of P. The pseudoautotopisms form a group denoted by PSL. The following proposition portrays that the pseudoautotopism is helpful to find a loop which is an isotope of the Wilson loop.Proposition 1.
LetL be a Moufang loop with the property α2∈ℵL for all α∈L and θ∈PSλL with companion c. Then, we have (θLc3,θ,θLc3) as an autotopism of the Wilson loop L.Proof.
Becauseθ∈PSλL with companion c, so(3)c⋅αθβθ=c⋅αβθfor allα,β∈L.
Considernc⋅αθβθ = n⋅cαθβθ = nc⋅αθβθ = ncαβθ = ncαβθ which shows that nc is also a companion of θ. By [1], L is the Wilson loop, so ℵλ = ℵχ = ℵρ. With the given property, α2 ∈ ℵL for all α ∈ L. So, c2⋅c is a companion of θ, and in Moufang loop L, α3 for all α∈L is well-defined. Therefore, c3⋅αθβθ = c3⋅αβθ⇒ (θLc3, θ, θLc3) is an autotopism of the Wilson loop L.We are able to find a class of loop isotopes of the Wilson loop with respect to different natural numbers. A loopL has the isotopy-isomorphy property provided each loop isotopic to L is isomorphic to L. Such a loop is called G-loop, and in [1], it is proved that the Wilson loop is an example of the G-loop.Lemma 1.
IfL is a Wilson loop and θ is the right pseudoautotopism on L with companion c, then c is in the nucleus of L as well as in the nucleus of the loop isotope of L. Moreover, θ has companions ci (not all distinct) for natural number i.Proof.
LetL be isomorphic to a loop isotope L1,c,⋄ under the isomorphism π:L,.⟶L1,c,⋄ defined by απ=αc for all α∈L where c is a companion of θ. By [1], we have c∈ℵρ. But, in Wilson loop, three nuclei are coincident, so c∈ℵL. Because c is an identity of L1,c,⋄ and ℵL1,c,⋄ is a subgroup of L1,c,⋄, ⇒c∈ℵL1,c,⋄. Since θ has companions c as well as nc for all n∈ℵL, therefore c⋅c,c2⋅c,… are the companions (not all distinct) of θ. Thus, θ has companions ci for natural number i.Note that the powers of companions play an important role as well as the left and the right inverse permutations are also important under an arbitrary autotopism.Lemma 2.
IfU,V,W is an autotopism of the Wilson loop L, then (TαV,LαJλWJρ, LαJλUJρ) and RαV,Tα−1JλWJρ,RαJλUJρ are also autotopisms on L for all α,β,γ∈L.Proof.
LetTα,Lα,Lα and Rα,Tα−1,Rα be autotopisms on L, so βTαγLα=βγLα, ⇒αβ/ααγ=αβγ, ⇒γLαLαβ/α=γLβLα, ⇒Lαβ/α=Lα−1LβLα.
Also,βRαγTα−1=βγRα, ⇒γαα/βα=γβα, ⇒γRαRα/βα=γRβRα, ⇒Rα/βα=Rα−1RβRα. In [5], V,JλWJρ,JλUJρ is also autotopism of L. Therefore, we obtain Tα,Lα,LαV,JλWJρ,JλUJρ=TαV,LαJλWJρ,LαJλUJρ. Hence, we have Rα,Tα−1,RαV,JλWJρ,JλUJρ=RαV,Tα−1JλWJρ,RαJλUJρ.We have autotopisms in the form of the left and right translations for the special subclass and Wilson loops of exponents 2 without depending upon the given arbitrary autotopism.Proposition 2.
IfL is the Wilson loop of exponent 2, then the following are autotopisms on L:(1)
Rα,Rα,Rα(2)
Lα,Lα,Lα(3)
RαLα,RαLα,RαLα(4)
LαRα,LαRα,LαRαProof.
Wilson identity in the form of left inverse isβαλα=γβαλγα and serves βγβρ=γρ for every α,β,γ∈L. Replacing γ by γβ⋅αλ and β by γα in the later identity γαγβ⋅αλγαρ = γβ⋅αλρ, ⇒γαβ⋅αλαρ= γβ⋅α. It implies that Rα,RαJλRαJρ,Rα is an autotopism on Wilson loop L. By using the same behavior, we can show that LαJρLαJλ,Lα,Lα is another autotopism for this L. Due to the exponent 2, Jλ and Jρ become identity mapping. Therefore, Rα,RαRα,Rα and LαLα,Lα,Lα are the autotopisms of L. So, Rα,Rα,Rα and Lα,Lα,Lα are the members of AUTL. Equations (3) and (4) are trivial by definition of AUTL.Theorem 1.
LetL be a Wilson loop and α, β, γ∈L. The following identities hold:(1)
LβJλRβLαLβLα−1 = JλLα/βα(2)
LβJλRβRαRβRα−1 = JλRαβ/α(3)
LβJλRβRβLαRαβ−1 = JλTα(4)
LβJλRβLβRαLβα−1 = JλTα−1(5)
RβJλLβLα−1Rγ−1RβLα = JλRαγ−1Rαβ(6)
RβJλLβRα−1Lγ−1LβRα = JλLγα−1LβαProof.
(1) In [2], it is shown that L is conjugacy closed loop (CCL) and weak inverse property loop (WIPL). We have αLβJλRβ = βαJλRβ = βαλRβ = βαλβ = αλ = αJλ and LβJλRβ = Jλ. Thus, in case of Wilson loop, we have LβJλRβLαLβLα−1 = JλLα/βα.
For (3), since by CCL, we haveLα−1LγLα = Lαγ/α, ⇒ βLγLα = βLαLαγ/α, ⇒ αγβ = αγ/ααβ, ⇒ γRβLα = γTαRαβ, ⇒ RβLαRαβ−1 = Tα. By WIPL, LβJλRβ = Jλ. Thus, LβJλRβRβLαRαβ−1 = JλTα.
For (5),Tα = RβLαRαβ−1, ⇒ Tα = RγLαRαγ−1, ⇒RβLαRαβ−1 = RγLαRαγ−1, ⇒ Lα−1Rγ−1RβLα = Rαγ−1Rαβ. Finally, RβJλLβLα−1Rγ−1RβLα = JλRαγ−1Rαβ.We have the following immediate corollary of the above theorem.Corollary 1.
LetL be a commutative Wilson loop with α2=e. Then, for all α, β, γ∈L, we have(1)
Rα,β = Rα−1(2)
Rα,β−1 = Rγ−13Rβ−1Rβγ(3)
Rα,β = Rαβ−1Rβ−1Rα
The Wilson loop is not ready to accept squaring property unless we select only those which are flexible. The next lemma indicates this notion.Lemma 3.
LetL be the flexible Wilson loop. Then, αβ2 = α2β2 for all α, β∈ L. Moreover, αℵβℵ2 = α2ℵβ2ℵ and show that Rαℵ−1Lαℵ,Lαℵ,Lαℵ is an autotopism on L/ℵ.Proof.
In [2], the Wilson loop L is an extra loop that satisfies the square property αβ2 = α2β2. So, αℵβℵ2 = αβℵ2 = αβ2ℵ = α2ℵβ2ℵ. L/ℵ entertains the autotopism Rαℵ−1Lαℵ,Lαℵ,Lαℵ.Lemma 4.
IfL/ℵ is a Wilson loop, then L need not to be a Wilson loop.Proof.
SinceL/ℵ is a Wilson loop, so for all αℵ, βℵ, γℵεL/ℵ, we have αℵαℵβℵρ = αℵγℵαℵβℵγℵρ⇒αℵαβρℵρ = αγℵαβγρℵρ⇒ααβρℵ = αγαβγρℵ. It implies that ααβρ≠αγαβγρ, i.e., L is not a Wilson loop.Theorem 2.
The homomorphic image of a Wilson loop is a Wilson loop.Theorem 3.
The direct product of Wilson loops is again a Wilson loop.Wilson loop can be defined by another way of opposite loop in case of nonassociative finite invertible loop (NAFIL). Thus, we have the following consequence of the above discussion.Theorem 4.
LetL be any NAFIL. A loop L,⋅ is a Wilson loop if and only if L,⋄ is a Wilson loop, where L,⋄ is opposite loop of L,⋅. Thus, we have α⋅β = β⋄α for all α, β∈L.Proof.
LetL,⋅ be a Wilson loop, so ααβρ = αγα⋅βγρ, for all α, β, and γ∈L. Consider(4)γ⋄β⋄αλ⋄γ⋄α=ααβρ=ααβλ=β⋄αλ⋄α.
It shows thatL,⋄ is Wilson loop.
Conversely, now we suppose thatL,⋄ is a Wilson loop,(5)γβ⋅αλγα=α⋄γ⋄α⋄β⋄γλ=α⋄γ⋄α⋄β⋄γρ=α⋄α⋄βρ=α⋄α⋄βλ=α⋄βλα=βαλα.
It completes the proof.We finish this section with the following inclusions.Lemma 5.
LetL be a Wilson loop. Then, the following statements hold:(1)
{Wilson loop}⊂ {Osborn loop}(2)
{flexible Wilson loop}⊂ {S-loop}(3)
{Wilson loop}⊂ {automorphic inverse property loop (AIPL)}(4)
{extra loop}⊂ {Wilson loop}Proof.
First two results are obvious from [2, 6]. For (3), the Wilson loop ⇒ the Osborn loop ⇒ the Moufang loop ⇒ inverse property loop (IPL) ⇒ cross inverse property loop (CIPL) ⇒ AIPL.


## 3. Holomorphism of Wilson Loop

In this section, we describe the conditions under which the holomorph of Wilson loop is Wilson loop. Here is the first result.Lemma 6.
If holomorphHL of a loop L is Wilson loop, then L is a Wilson loop.Proof.
SinceI,L is a normal subloop of HL, moreover I,L is isomorphic to L, so L is a normal subloop of HL. Since HL is a Wilson loop, so does L.Theorem 5.
IfL is a Wilson loop, then for all Φ,α, ξ,γ∈HL, α, β, γ∈L, and Φ, ξ∈AUML, we have Φ,αξ,γΦ,α⋅I,βξ,γρ=Φ,αΦ,αI,βρ.Proof.
Necessary computations are given below(6)Φ,αξ,γΦ,α⋅I,βξ,γρ=Φ,αξ,γΦξ−1,αξ⋅βξ⋅γρΦξ−1=I,αξ⋅γξ−1Φ−1⋅αΦ−1⋅βΦ−1⋅γξ−1Φ−1ρ=I,αξ⋅γξ−1Φ−1⋅αΦ−1⋅βΦ−1⋅γξ−1Φ−1ρ=I,αΦ−1αΦ−1⋅βΦ−1ρ=ΦΦ−1,αΦ−1⋅αβΦ−1ρ=Φ,αΦ−1,αβρΦ−1=Φ,αΦ,αβρ=Φ,αΦI,αI⋅βρ=Φ,αΦ,αI,βρ.Thus, we have the following immediate corollary.Corollary 1.
The holomorphHL of a loop L is a Wilson loop if and only if L is a Wilson loop and AUML is trivial group.Lemma 7.
IfHL is a Wilson loop of a commutative loop L, then Jλ = Jρ = I.Proof.
With the aid of the last two results, loopL is commutative AIPL. Therefore, Jλ=Jρ=I.Lemma 8.
LetU = L,⋅ and V = L,⋄ be two loops and L is a nonempty set. If HU and HV are isomorphic Wilson loops, then binary operations are coincident.Proof.
Isomorphism ofHU and HV gives αδ⋄βξ = απ⋅βδ for all ξ, δ ∈ AUMV, α, β ∈ L, and π ∈ AUMU. Put π=ξ=δ=I, then we have αI⋄βI = (αI⋅β) I⇒α⋄β = α⋅β for all α, β∈L. Hence, we have two identical loops U and V.Similarly, we state the following obvious result.Lemma 9.
LetL be any loop with holomorph HL. HL is Wilson CIPL if and only if L is Wilson CIPL and AUML = I.Theorem 6.
LetL be a loop with holomorph HL. HL is Wilson loop if and only if for all Ψ, ξ∈AUML, we have(7)αξ⋅αΨ⋅βξρ=αξ⋅γαΨξβξ⋅γρ.Proof.
For allΦ,α, Ψ,β∈HL,(8)Φ,αΦ,αΨ,βρ=Φ,αΦΨ−1,αΨ⋅βρΦΨ−1=Φ⋅Ψ−1Φ−1,αΨ−1Φ−1⋅αΨ⋅βρΨ−1Φ−1=Φ⋅Ψ−1Φ−1,ααΨ⋅βρΨ−1Φ−1,(9)Φ,αξ,γΦ,α⋅Ψ,βξ,γρ=Φξ,αξ⋅γΦ,α⋅Ψξ,βξ⋅γρ=Φξ,αξ⋅γΦ⋅Ψξ−1,αΨξ⋅βξ⋅γρΦ⋅Ψξ−1=Φξξ−1Ψ−1⋅Φ−1,αξ⋅γξ−1Ψ−1⋅Φ−1=αΨξ⋅βξ⋅γρξ−1Ψ−1⋅Φ−1=Φ⋅Ψ−1Φ−1,αξ⋅γαΨξ⋅βξ⋅γρξ−1Ψ−1⋅Φ−1.
Given hypothesis is(10)αξ⋅αΨ⋅βξρ=αξ⋅γαΨξβξ⋅γρααΨ⋅βρΦΨ−1=αξ⋅γαΨξ⋅βξ⋅γρξ−1ΦΨ−1ααΨ⋅βρΨ−1Φ−1=αξ⋅γαΨξ⋅βξ⋅γρξ−1Ψ−1⋅Φ−1.
Thus, equation (9) becomes(11)Φ,αξ,γΦ,α⋅Ψ,βξ,γρ=Φ⋅Ψ−1Φ−1,ααΨ⋅βρΨ−1⋅Φ−1.
We summarize the above discussion in the following manner.Theorem 7.
LetL be a loop with holomorph HL. Then, the following are equivalent:(1)
HL is Wilson loop(2)
L is Wilson loop with trivial group AUML(3)
αξ⋅αΨ⋅βξρ=αξ⋅γαΨξβξ⋅γρ for all Ψ,ξ∈AUMLLemma 10.
LetHL be the holomorph of the loop L. Then, AUML has right action on HL but not left.Proof.
We defineΦ,αδ = Φδ,αδ for all Φ,α∈HL,δ∈AUML.(12)Φ,αg=Φg,αg,Φ,αgh=Φgh,αgh=Φgh,αgh=Φ,αgh.
For allg, h∈AUML. Moreover,(13)Φ,αI=ΦI,αI=Φ,α.
So,AUML has right action on HL. Clearly, Φ,αgh≠Φ,αhg. AUML has not left action on HL.Remark 1.
AUML has both the left and right actions on Wilson loop HL.Proposition 3.
IfHL has the Wilson loop, then Jλ, Jρ∈S1L, where S1L is the group of weak inverse permutations of L.Proof.
SinceHL is Wilson loop, so AUML is trivial. By Wilson identity, ααβρ = αγα⋅βγρ for all α, β, and γ∈L. Put α = e, then βρ = γβγρ, L is WIPL. Jρ2 is an automorphism of this loop by [5], therefore Jρ2 is an identity mapping because AUML is trivial. To show Jρ∈S1L, we verify αρ = αJρρJρ.(14)αJρρJρ=αJρJρ2=αJρ=αρ.
It implies thatJρ is the weak right inverse permutation, so Jρ∈SρL. It can be shown easily(15)SρL=SλL=S1L.
Therefore,Jρ∈S1L. Similarly, we can prove Jλ∈S1L.Remark 2.
LetL be the NAFIL with trivial group AUML,⋄ = AUML,⋅. HL,⋅ is Wilson loop if and only if HL,⋄ is Wilson loop.



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