An app oach o he iso heo y by means o ex ended pseudoiso opisms
R. M. Falcón and J. Núñez
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An App oach o he Iso heo y by Means o Ex ended
Pseudoiso opisms
R. M. Falcón∗and J. Núñez†
∗Dep o Applied Ma hema ics I, Uni e si y o Se ille (Spain)
†Dep o Geome y and Topology. Uni e si y o Se ille (Spain)
Abs ac . Based on he adi ional concep o iso opism, ex ended iso opisms we e in oduced by he au ho s in 2006 in o de
o p o ide a undamen al basis o he iso heo y o San illi. Since ha i s a emp , dis inc s udies on ex ended iso opisms ha e
ocused on he cons uc ion o pa ial La in squa es ha ing a San illi au o opism in hei au o opism g oup. In o de o deal
wi h new s uc u es, we in oduce in his pape he concep o ex ended pseudoiso opism. This is based on he use o on o
linea ans o ma ions ha a e no necessa ily injec i e. Some examples a e exposed h oughou he pape .
Keywo ds: Iso opism, iso heo y
PACS: 02.10.Ox, 11.10.Lm
INTRODUCTION
In 1942, Albe [1] in oduced he concep o iso opism o algeb as: Two algeb as (A1,·)and (A2,◦)a e iso opic i
he e exis h ee egula linea ans o ma ions
α
,
β
and
γ
om A1 o A2such ha
α
(u)◦
β
( ) =
γ
(u· ), o all u, ∈A1.(1)
The algeb a A2is hen said o be iso opic o A1and he iple Θ= (
α
,
β
,
γ
)is said o be an iso opism be ween bo h
algeb as A1and A2. I
α
=
β
=
γ
, hen his is an isomo phism.
Since he o iginal pape o Albe , a wide amoun o pape s ha e appea ed in he li e a u e ha deal wi h iso opisms
o dis inc ypes o algeb as as di ision [2, 3, 4, 5], Jo dan [6, 7, 8], al e na i e [9, 10], absolu e alued [11, 12],
s uc u al [13] and eal wo-dimensional commu a i e [14] algeb as. Iso opism o Lie algeb as we e al eady conside ed
by Albe himsel [1] and, sho ly a e , in 1944, by B uck [15]. Mo e ecen ly, in 1978, San illi [16] ook up he concep
in he ame o a dynamical sys em based on an uni a y Lie algeb a Lendowed wi h an inne p oduc ·, whose s a e
space is de e mined by a se So pa ame e s as coo dina es, eloci y, ime, empe a u e o densi y, among o he s.
He gene alized he associa i e p oduc u· be ween He mi ian gene a o s o he co esponding uni e sal en eloping
associa i e algeb a by conside ing he new p oduc
uˆ·s =u·ˆ
T(s)· , o all u, ∈Land s∈S,(2)
whe e ˆ
T:S→L {0}is called iso opic elemen 1. The commu a o p oduc [u, ] = uˆ·s − ˆ·sup ese es he Lie
axioms and is called he Lie-iso opic p oduc . The applica ion o Lie’s heo y (en eloping algeb as, Lie algeb as and
Lie g oups) ha eme ges om his new p oduc is he so-called Lie-San illi iso heo y [16, 17, 18, 19, 20, 21, 22].
Fo each s a e s∈S, he e a e also de ined he elemen s
ˆus=u·ˆ
I(s), o all u∈L,(3)
whe e ˆ
I:S→L {0}sa is ies ha
i. ˆ
T(s)·ˆ
I(s)and ˆ
I(s)·ˆ
T(s)coincide wi h he iden i y elemen in L, o all s∈S.
ii. The se ˆ
Ls=L·ˆ
I(s) = {ˆus:u∈L}coincides wi h L, o all s∈S.
1No e ha , in o de o cla i y he ela ion ha exis s be ween he iso heo y and he classical heo y o iso opisms, he no a ion ha is used h oughou
he cu en pape may di e om ha used in [16] and i s subsequen pape s.
In e na ional Con e ence o Nume ical Analysis and Applied Ma hema ics 2015 (ICNAAM 2015)
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iii. u·ˆ
I(s)= ·ˆ
I(s), o all s∈Sand u, ∈Lsuch ha u= .
This map ˆ
Iis called isouni , due o he ac ha i gene alizes he uni o he g ound ield o he o iginal algeb a. In
pa icula , since he p oduc ·is associa i e, he nex equali y holds
ˆusˆ·sˆ
I(s) = ˆus·ˆ
T(s)·ˆ
I(s) = ˆus=ˆ
I(s)·ˆ
T(s)·ˆus=ˆ
I(s)ˆ·sˆus, o all u∈Land s∈S.(4)
Fu he ,
ˆusˆ·sˆ s= (u·ˆ
I(s))·ˆ
T(s)·( ·ˆ
I(s)) = (u· )·ˆ
I(s), o all u, ∈Land s∈S.(5)
The se ˆ
Ls=Lcons i u es, he e o e, a Lie algeb a wi h he commu a o p oduc applied o he p oduc ˆ·s. I in ol es
he de ini ion o he isop oduc
ˆ
[ˆus,ˆ sˆ
]s=ˆusˆ·sˆ s−ˆ sˆ·sˆus= [u, ]ˆ·sˆ
I(s), o all u, ∈L.(6)
Fo each s a e s∈S, le
α
sbe he linea ans o ma ion om L o i sel , which is de ined so ha
α
s(u) = ˆus, o all
u∈L. The iden i y (6) is hen equi alen o
ˆ
[
α
s(u),
α
s( )ˆ
]s=
α
s([u, ]), o all u, ∈L.(7)
The amily o iples FS={(
α
s,
α
s,
α
s):s∈S}cons i u es, he e o e, a local s a e isomo phism o Lie algeb as
on he unde lying dynamical sys em. Ne e heless, o he bes knowledge o he au ho s, e en i dis inc pape s and
monog aphs ha e deal wi h he ounda ions o he Lie-San illi iso heo y and i s ex ension o o he algeb aic s uc u es
as g oups, ings o ec o spaces, among o he s [18, 19, 23, 24], he e does no exis a his ime any comp ehensi e
s udy ha ein e p e s he Lie-San illi iso heo y by means o he heo y o local s a e isomo phisms. A u he s udy in
his ega d is, he e o e, necessa y.
In 2006, in o de o gene alize he iso heo y o a non-isomo phic ame and b ing i close o he classical heo y
o iso opisms, he au ho s used he concep o ex ended iso opism in oduced in [25] as a way o ein e p e he
dependence o he isouni on he s a e space o he unde lying dynamical sys em as a amily o classical iso opisms.
A he ime, dis inc pape s on ex ended iso opisms ha e ocused on he cons uc ion o pa ial La in squa es ha ing a
San illi iso opism in hei au o opism g oup [26, 27, 28]. In o de o deal wi h new algeb aic s uc u es, we gene alize
in his pape he concep o ex ended iso opism by educing he condi ion o being injec i e. We in oduce in his way
he concep s o pseudoiso opism and ex ended pseudoiso opism, which cons i u e a new app oach o lay he ounda ion
o he iso heo y.
SANTILLI EXTENDED ISOTOPISMS
The main s eng h o he iso heo y consis s o he dependence o he isouni on he s a e space o he unde lying
dynamical sys em. In he con ex o he Lie-San illi iso heo y, le us e iew how he au ho s ein e p e ed his
dependence in [26] in o de o ela e i wi h he classical heo y o iso opisms.
Le (L,·)be a Lie algeb a, no necessa ily uni a y, associa ed o a dynamical sys em, whose s a e space is de e mined
by a se So pa ame e s. As a i s s ep, we include a new pa ame e in he se Swi h h ee possible s a es 1, 2 and 3.
The new se o pa ame e s is deno ed as S=S×{1,2,3}. Le us conside a map ˆ
I:S→L {0}such ha
i. The se ˆ
L(s, )=L·ˆ
I(s, ) = {u·ˆ
I(s, ):u∈L}coincides wi h L, o all (s, )∈S.
ii. u·ˆ
I(s, )= ·ˆ
I(s, ), o all (s, )∈Sand u, ∈Lsuch ha u= .
Fo each s∈S, le us conside he h ee linea ans o ma ions
α
s,
β
sand
γ
s om L o i sel so ha
α
s(u) = u·ˆ
I(s,1),
β
s(u) = u·ˆ
I(s,2)and
γ
s(u) = u·ˆ
I(s,3), o all u∈L. The condi ions (i) and (ii) imposed o ˆ
Iin ol e hese h ee
ans o ma ions o be on o and injec i e. As a consequence, he iple Θs= (
α
s,
β
s,
γ
s)cons i u es an iso opism be ween
he Lie algeb a (L,·)and he Lie algeb a (L,ˆ·s), whe e ˆ·sis he p oduc de ined so ha
uˆ·s =
γ
s(
α
−1
s(u)·
β
−1
s( )), o all u, ∈L.(8)
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I
α
s=
β
s, hen he iple Θscons i u es indeed an iso opism be ween he Lie algeb a Lendowed wi h he commu a o
p oduc [u, ] = u· − ·uand he Lie algeb a Lendowed wi h he commu a o p oduc
ˆ
[u, ˆ
]s=uˆ·s − ˆ·su=
γ
s(
α
−1
s(u)·
α
−1
s( ))−
γ
s(
α
−1
s( )·
α
−1
s(u)) =
γ
s([
α
−1
s(u),
α
−1
s( )]), o all u, ∈L.(9)
The amily o iples FS={(
α
s,
β
s,
γ
s):s∈S}is called an ex ended San illi iso opism o Lie-algeb as.
EXTENDED PSEUDOISOTOPISMS
Ex ended iso opisms make possible o gene alize he s a e isomo phism app oach o he iso heo y o a s a e iso opism
app oach. In he cu en sec ion we gene alize he la e by emo ing he injec i i y in he condi ions o he isouni ˆ
I. A
i s a emp in his ega d was al eady exposed by he au ho s in [26] o he cons uc ion o non-injec i e isoalgeb as
by means o San illi iso opisms. We o malize he ideas exposed in ha pape by in oducing he e he no ions o
pseudoiso opism and ex ended pseudoiso opism. The o iginal idea on which bo h concep s a e based was in oduced
in he Ph. D. Thesis o he i s au ho [29].
Le (G1,·)and (G2,◦)be wo g oupoids, ha is, a pai o se s G1and G2endowed wi h wo espec i e bina y
ope a ions ·:G1×G1→G1and ◦:G2×G2→G2. A iple Θ= (
α
,
β
,
γ
)o on o maps om G1 o G2is called a
pseudoiso opism om (G1,·) o (G2,◦)i he ollowing wo condi ions a e sa is ied
i.
γ
(u· ) =
γ
(u′· ′), o all u, ,u′
, ′∈G1such ha
α
(u) =
α
′(u)and
β
( ) =
β
′( ).
ii.
α
(u)◦
β
( ) =
γ
(u· ), o all u, ∈G1.
Obse e ha he second condi ion is consis en because o he i s condi ion. I he h ee maps
α
,
β
and
γ
a e injec i e,
hen he iple Θis a an iso opism. I
α
=
β
=
γ
, hen Θis called a pseudoisomo phism. I he e exis h ee elemen s
u
α
,u
β
and u
γ
in G1such ha
δ
(u) = u·u
δ
, o all u∈G1and
δ
∈ {
α
,
β
,
γ
}, hen he iple Θis called a San illi
pseudoiso opism. Finally, i Iis a se o indices, hen e e y amily F={(
α
i,
β
i,
γ
i)}i∈I o med by pseudoiso opisms
om (G1,·) o (G2,◦)is called an ex ended pseudoiso opism. I is a San illi ex ended pseudoiso opism i each iple o
he amily Fis a San illi pseudoiso opism. Le us inish he pape wi h some examples on his poin .
Example 1 In he ield o complex numbe s C, le G be a g oup gene a ed by i ∈C, endowed wi h he usual p oduc
·in C, ha is, G ={1,−1,i,−i}. Le us conside he subg oup H ={1,−1}o G and le
α
:G→H be such ha
α
(1) =
α
(−1) = 1and
α
(i) =
α
(−i) = −1. The iple (
α
,
α
,
α
)is a pseudoisomo phism om (G,·) o (H,·) ha
p ese es he s uc u e o g oup. ▹
Example 2 Le (R,+,×)be he ield o eal numbe s and le U be he se o di e en iable eal unc ions o one eal
a iable. This se cons i u es an R- ec o space wi h he na u al ope a o s
( +g)(x) = (x)+g(x), o all ,g∈U and x ∈R.(10)
(
λ
· )(x) =
λ
× (x), o all ∈U and x ∈R.(11)
Le us conside he map
α
:U→Rso ha
α
( ) =
∂
∂
x(2), o all ∈U. The la e is on o because, gi en a ∈R, he
unc ion (x) = a
4x2sa is ies ha
α
( ) = a. Howe e ,
α
is no injec i e. To see i , i is enough o conside he unc ions
1(x) = 15x2and 2(x) = 5x3, o which
α
( 1) =
α
( 2) = 60. The iple Θ1= (
α
,
α
,
α
)is a pseudoisomo phism om
(U,+) o (R,+) because
a+b=∪
,g∈U{
α
( +g):
∂
∂
x(2) = a,
∂
g
∂
x(2) = b}.
The iple Θ2= (Id,
α
,
α
)is a pseudoisomo phism om (U,·) o (R,×)whe e
a×b=∪
∈U{
α
(a· ):
∂
∂
x(2) = b}.
The pai (Θ1,Θ2)cons i u es he e o e a pseudoiso opism om (U,+,·) o (R,+,×).▹
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Example 3 Le us conside he polynomial ing F2[x]o e he ini e ield F2. Le us de ine he maps
α
0and
α
1 om F2[x] o i sel , so ha
α
(p) = p+ , o all polynomial p ∈F2[x]and ∈ {0,1}. The amily F=
{(
α
1,
α
2,
α
( 1+ 2) (mod 2))} 1, 2∈{0,1}is a San illi ex ended iso opism om (F2[x],+) o i sel . Speci ically,
p+q=
α
1(p− 1)+
α
2(q− 2) =
α
( 1+ 2) (mod 2)(p+q−(( 1+ 2) (mod 2))), o all 1, 2∈ {0,1}.
▹
CONCLUSIONS AND FURTHER STUDIES
Ex ended (pseudo)iso opisms a e in oduced he e o ex end he s a e isomo phism ame in which iso heo y is comp e-
hended o a mo e gene al s a e (pseudo)iso opism ame. A u he s udy on bo h concep s is necessa y. Speci ically, a
s udy ha ocuses on San illi (pseudo)iso opisms o pa ial La in ec angles can be an in e es ing s a ing poin in his
ega d.
ACKNOWLEDGMENTS
Au ho s a e e y g a e ul o P o . San illi o his aluable help in ou esea ch on he iso heo y and cong a ula e him
o his 80 bi hday.
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