Identities and central polynomials of real graded division algebras

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by  Diogo Diniz

IDENTITIES AND CENTRAL POLYNOMIALS OF REAL
GRADED DIVISION ALGEBRAS
7
1 DIOGODINIZ,CLAUDEMIRFIDELIS,ANDSE´RGIO MOTA
0
2
Abstract. LetAbeafinitedimensionalrealalgebrawithadivisiongrading

n by afinite abelian groupG. In this paper weprovidefinite basis forthe TG-
a ideal of graded identities and for the TG-space of graded central polynomials
J forA.

5

]
A
1. Introduction
R
Let K be a field and A = ⊕ A a K-algebra graded by the group G, the
. g G g
h set of graded polynomial identitie∈s for A is an ideal of the free G-graded algebra
t
a KhX i invariant under graded endomorphisms, such ideals are called T -ideals.
G G
m A central problem in the theory of algebras satisfying polynomial identities (p.
[ i. algebras) is the Specht problem on the existence of a finite basis for the ideal

of identities for any algebra in a given class (Associative, Lie, Jordan, etc.). This
1
problemwassolvedbyA.Kemer[14]for(ungraded)associativealgebrasoverafield
v
6 of characteristic zero. The corresponding result for p.i. algebras graded by a finite
3 group G was proved by I. Sviridova [20] (for abelian groups) and by E. Aljadeff
4 and A. Kanel-Belov [1]. Explicit finite basis are known in a few cases only, for the
1
(ungraded)identities of the matrix algebraoforder2 a finite basis wasdetermined
0
by Yu. P. Razmyslov[16], a minimal basis was latter obtained by V. Drensky [11].
.
1 A polynomial in KhX i is a graded central polynomial for A if the result of every
G
0
admissiblesubstitutionisacentralelement. ThesetofcentralpolynomialsforAis
7
a subspace of KhX i invariant under graded endomorphisms, such subspaces are
1 G
: calledTG-subspaces. Centralpolynomialsplay animportant rolein p.i. theory,we
v
refer the reader to [12], [13] for more details and applications.
i
X InhisstudyofprojectiveplanesM.Hallprovedthatadivisionalgebraforwhich
r a certain polynomial (the Hall polynomial, see Theorem 4) is an identity is either
a
a field or a generalized quaternion algebra over its center. Latter I. Kaplansky
generalized this result and proved that the conclusion holds if the division algebra
satisfiesanypolynomialidentity. Divisiongradingsonsimplerealfinite-dimensional
algebraswereclassifiedbyYu. BahturinandM.Zaicev[6]andA.Rodrigo-Escudero
[18]. Moregenerallyfinite-dimensionalgradeddivisionalgebrasoverthefieldofreal
numbers were recently classified in [7]. In this paper we provide finite basis for the
T -ideal of graded identities and the T -space of graded central polynomials for
G G
the graded division algebras in [6], [18] and [7].
If an algebra has a division grading, with support G, such that the component
associated to the neutral element is one dimensional then every component is one
dimensional. In this case homogeneous elements commute up to a scalar and the
G-grading is regular (see Definition 1). The division algebras in [6], [18] and [7]
Thefirstauthoracknowleges supportbyCNPqgrant#303534/2013-3.
1
2 DIOGODINIZ,CLAUDEMIRFIDELIS,ANDSE´RGIOMOTA
are either regular, the tensor product of an algebra with a regular grading and a
basic algebra (presented in sections 4.1 and 5.1) or a complex division grading on
a matrix algebra over C viewed as a real algebra (Pauli grading). In [10] a basis
for graded identities of the tensor product of a graded algebra A and an algebra
with a regular grading is obtained from a basis for the graded identities for A. In
Section 3 we prove the analogous result for gradedcentral polynomials, we remark
that a particular case of this result was proved in [3]. Therefore a basis for the
graded identities and central polynomials of the tensor product division algebras
is obtained once a basis for the basic algebras is known. For most of the basic
algebras such basis are known, in sections 4.2.2 and 5.1 we obtain the description
fortheremainingbasicalgebras. InSection4.3weconsiderthePauligradingsthat
are not regular.
2. Preliminaries
Let K be a field of characteristic zero, G a group with identity element e. A
grading by the group G (or a G-grading) on the vector space V is a decomposition
Γ:V =⊕ V . LetΓ :W =⊕ W beanH-gradedvectorspace,amorphism
g G g ′ h H h
∈ ∈
of gradedvector spaces is a linear map f :V →W such that for every g ∈G there
existsh∈H suchthatf(V )⊆W . Iff hasaninversewhichisalsoamorphismof
g h
gradedvectorspaceswesaythatthegradingsΓandΓ areequivalent. AG-grading
′
on an algebra A is a grading on A as a vector space, Γ : A = ⊕ A , such that
g G g
∈
A A ⊆ A for every g,h in G. The subspaces A are called the homogeneous
g h gh g
components of A. An element a∈∪ A is called a homogeneous element, if a is
g G g
∈
non-zero there exists a unique g ∈ G such that a ∈ A , this element is called the
g
degree of a (in the G-grading) and denoted deg (a). The support of Γ is the set
G
supp Γ := {g ∈ G|A 6= 0}. Let Γ : A = ⊕ A be another grading by H on
g ′ h H h
∈
A, we say that Γ is a coarsening of Γ if for every g ∈ G there exists h ∈ H such
′
that A ⊆ A . Let Γ : A = ⊕ and Γ : B = ⊕ B be algebras graded by
g h g G ′ h H h
∈ ∈
the groups G and H respectively. The gradings Γ and Γ are equivalent if there
′
exists an isomorphism of algebras ϕ : A → B that is an equivalence of graded
vector spaces. The graded algebras A and B are weakly isomorphic if there exists
agroupisomorphismα:G→H andanalgebraisomorphismϕ:A→B suchthat
ϕ = B for every g in G. If G and H are the supports supp A and supp B,
Ag α(g)
respectively, and if the grading Γ is strong (i.e. A A = A for every g,h ∈ G)
g h gh
thenthe gradingsΓ andΓ areequivalentif andonly ifthey areweaklyisomorphic
′
(see [7, Section 2]). The tensor product A⊗B has a canonical G×H grading
where (A⊗B) =A ⊗B . The (graded) algebra Γ :A=⊕ A is a graded
(g,h) g h g G g
∈
division algebra (orΓisadivisiongradingonA)ifAhasaunitandeverynon-zero
homogeneous element is invertible.
We denote by KhX i the free G-graded algebra, freely generated by the set
G
X ={x |i∈N,g ∈G}. This algebra has a natural grading by G where, for any
G i,g
g in G, the homogeneous component (KhX i) is the subspace generated by the
G g
monomials x ···x such that g ···g = g. Let f(x ,...,x ) be a
i1,g1 im,gm 1 m 1,g1 m,gm
polynomial in KhX i, an m-tuple (a ,...,a ) such that a ∈A for i=1,...,m
G 1 m i gi
is called an f-admissible substitution (or simply an admissible substitution). If
f(a ,...,a ) = 0 for every admissible substitution (a ,...,a ) we say that the
1 m 1 m
polynomial f is a graded polynomial identity for A. We denote by Z(A) the centre
of A and we say that f is a graded central polynomial for A if f(a ,...,a ) ∈
1 m
IDENTITIES AND CENTRAL POLYNOMIALS OF REAL GRADED DIVISION ALGEBRAS 3
Z(A) for every admissible substitution (a ,...,a ). We denote by Id (A) and
1 m G
C (A) the set of all all graded identities and all graded central polynomials for A,
G
respectively. We say that the polynomial f is an proper graded central polynomial
for A if f ∈C (A)\Id (A), in other words f is G-graded central polynomial but
G G
not a G-graded identity for A.
It is clear that Id (A) and C (A) are subspaces of KhX i invariant under all
G G G
endomorphisms of KhX i, such subspaces are called T -spaces. Moreover, the
G G
set Id (A) of all graded identities for A is an ideal of KhX i invariant under
G G
all endomorphisms, such ideals are called T -ideals. The intersection of T -ideals
G G
(respect. T -spaces) of KhX i is also a T -ideal (respect. T -space). A subset
G G G G
S ⊂KhX i generates the T -ideal Id (A) (resp. T -space C (A)) if itequalsthe
G G G G G
intersectionofallT -ideals(resp. T -spaces)inKhX ithatcontainS. IfG={e}
G G G
werecoverthedefinitionofordinarypolynomialidentitiesandcentralpolynomials,
in this case we omit the sub index G and the sub indices that are elements of G
andusethe notationKhXiforthe freeassociativealgebraandx forthe variables.
i
We denote [x ,x ] = x x −x x the commutator of x and x . The standard
1 2 1 2 2 1 1 2
polynomial of degree n is the polynomial
S (x ,...,x )= (−1)σx ···x ,
n 1 n σ(1) σ(n)
σX∈Sn
where (−1)σ denotes the sign of the permutation σ.
Let A = ⊕ and B = ⊕ B be weakly isomorphic algebras graded by the
g G h H h
∈ ∈
groups G and H respectively and let α : G → H and ϕ : A → B be a group
isomorphism and an algebra isomorphism, respectively, such that ϕ = B for
Ag α(g)
every g in G. A polynomial f ∈ KhX i is a graded identity for A (resp. graded
G
central polynomial) if and only if Φ(f) is a graded identity (resp. graded central
polynomial) for B, where Φ : KhX i → KhX i is the isomorphism such that
G H
Φ(x ) = x . Moreover if S is a basis for the T -ideal Id (A) (resp. T -space
ig iϕ(g) G G G
C (A)) then Φ(S) is a basis for the T -ideal Id (B) (resp. T -space C (B)).
G G G G G
Let Γ:A=⊕ A be a G-gradeddivisionalgebra,the support H =supp Γ is
g G g
∈
a subgroupofG andA=⊕ A is an H-gradingonA. If S ⊂KhX i is a basis
h H h H
∈
for Id (A) (resp. C (A)) then the set S∪{x |g ∈G\H} is a basis for Id (A)
H H 1g G
(resp. C (A)).
G
Remark 1. Henceforth whenever Γ : A = ⊕ A is a G-graded division algebra
g G g
∈
we assume that G=supp Γ.
Next we recall the definition of a regular grading (see [2], [17]).
Definition 1. Let R be an algebra over the field K and
R= R ,
h
hMH
∈
a grading by the abelian group H. The H-grading above on R is said to be regular
if there exists a commutation function β :H ×H →K such that
×
(P1) For every n and every n-tuple (h ,...,h ) of elements of H, there exist
1 n
r ,...,r such that r ∈R , and r ···r 6=0.
1 n j hj 1 n
(P2) For every h ,h ∈H and for every a ∈A , a ∈A , we have
1 2 h1 h1 h2 h2
a a =β(h ,h )a a .
h1 h2 1 2 h2 h1
4 DIOGODINIZ,CLAUDEMIRFIDELIS,ANDSE´RGIOMOTA
The regular H-grading on R is said to be minimal if for any h 6= e there exists h
′
such that β(h,h)6=1.
′
In the definition above the commutation function β : H × H → K is a
×
skew-symmetric bicharacter, i.e., for every h ∈ H the maps h 7→ β(h ,h) and
0 0
h 7→ β(h,h ) are characters of the group H and β(h ,h ) = β(h ,h ) 1 for every
0 2 1 1 2 −
h ,h ∈ H. We say that the bicharacter β is non-degenerate if β(h ,h) = 1 for
1 2 0
every h∈H implies that h =e.
0
3. Graded Identities and Central Polynomials: From A To A⊗R
Let R be an algebra with a regular grading by the abelian group H and A
an algebra graded by the group G. Our main result in this section, Theorem 2,
providesa basisfor the centralpolynomialsofthe algebraA⊗R with its canonical
G×H-grading obtained from a basis for the G-graded central polynomials of A
that contains a basis for Id (A) as a T -space.
G G
Letg=(g ,··· ,g )be ann-tuple ofelementsofthe groupG. We denotebyP
1 n g
thesetofpolynomialsmultilinearinthe variablesx ,··· ,x . We fixsequence
1,g1 n,gn
h = (h ,...,h ) of elements of H. The description of the basis for A⊗R is in
1 n
terms of multilinear maps φ :P →P , where g×h=((g ,h ),··· ,(g ,h )),
h g g h 1 1 n n
introduced in [5]. A permutation τ ∈ S×, together with h, determines a non-zero
n
scalar λh such that
τ
(1) r ···r =λhr ···r ,
1 n τ τ(1) τ(n)
whenever r ∈R for i=1,...,n. We define
i hi
φ (x ···x )=λhx ···x ,
h gτ(1),τ(1) gτ(n),τ(n) τ (gτ(1),hτ(1)),τ(1) (gτ(n),hτ(n)),τ(n)
and extend to P by linearity.
g
Theorem 1. [10, Theorem 5.5] Let R be an algebra with a regular grading by the
abelian group H and let A be any G-graded algebra. If S ⊆ g GnPg is a basis, of
multilinear polynomials, for the T -ideal Id (A), then the sSetn∈∈N
G G
S ={φ (f) | f(x ,...,x )∈S, h∈Hn},
h 1g1 ngn
is a basis for the TeG H-ideal IdG H(A⊗R).
× ×
Letβ :H×H →K denotethebicharacterassociatedtotheregularH-grading
×
on R. Denote
H ={h ∈H |β(h,h)=1∀h∈H},
′ ′ ′
itis clearthat Rh′ ⊆Z(R)for everyh′ ∈H′. Inthis sectionweassume thatR has
minimal center in the sense that
(2) Z(R)=⊕h′ H′Rh′.
∈
Remark 2. Let f(x ,··· ,x ) be a multilinear polynomial and h an n-tuple
g1,1 gn,n
in Hn. It follows from the definition of the map φ (see [5, Theorem 3.1]) that the
h
equality
(3) φ (f)(a ⊗r ,··· ,a ⊗r )=f(a ,...,a )⊗r ···r ,
h 1 1 n n 1 n 1 n
holds for every φ (f)-admissible substitution (a ⊗r ,··· ,a ⊗r ). This implies
h 1 1 n n
that φ (f) is a graded identity for A⊗R if and only if f is a graded identity for
h
A. If equality (2) holds then φ (f) is a proper central polynomial for A⊗R if and
h
only if f is a proper central polynomial for A and (h ···h )∈H .
1 n ′
IDENTITIES AND CENTRAL POLYNOMIALS OF REAL GRADED DIVISION ALGEBRAS 5
Our next theorem is the analogous result for graded central polynomials. In its
proof we need the following lemma.
Lemma 1. [10, Lemma 5.4]Let f = f(x ,...,x ) ∈ P , and w ,...,w
g11 gmm g 1 m
monomials in KhX i such that deg (w )=g for each i and f(w ,...,w )∈P ,
G G i i 1 m k
for some k ∈ Gn. If h ∈ Hn, there exist h ∈ Hm and homogeneous elements
′
b1,...,bm ∈KhX|G×Hi such that φh(f(w1,...,wm))=γφh′(f)(b1,...,bm).
Theorem2. LetAbeanalgebrawithagradingbyagroupGandletRbeanalgebra
with a regular grading by an abelian group H, with bicharacter β : H ×H →K ,
×
such that Z(R) = R , where H = {h ∈ H |β(h,h) = 1∀h ∈ H}. If
h H′ h ′ ′ ′
S = S1 ∪S2 ⊆ gLGn∈Pg is a basis, of multilinear polynomials, for the TG-space
Sn∈∈N
C (A) such that S generates Id (A) as a T -space then the set S = S ∪S ,
G 1 G G 1 2
where
e f f
S ={φ (f)|f(x ,...,x )∈S ,h∈Hn}
1 h g1,1 gn,n 1
and f
S ={φ (f)|f(x ,...,x )∈S ,h=(h ,...,h )∈Hn,h ···h ∈H }
2 h g1,1 gn,n 2 1 n 1 n ′
is afbasis for the T -space C (A⊗R).
G H G H
× ×
Proof. The inclusion S ⊆ C (A ⊗ R) follows from Remark 2, therefore
G H
×
hSi ⊆ C (A⊗R), where hSi denotes the T -space generated by
TG×H G×H e TG×H G×H
S. Let f(x ,...,x ) be a multilinear element of C (A⊗R). We
deenote g =(g(1g,h1,).,.1.,g ) a(gnnd,hkn),=n ((eg ,h ),...,(g ,h )). It is clGe×arHfrom the def-
1 n 1 1 n n
inition that the map φ is bijective. This implies that there exists a multilinear
h
polynomialf (x ,...,x )inP suchthatf =φ (f ). ItfollowsfromRemark
′ g1,1 gn,n g h ′
2 that f is a graded central polynomial for A. Therefore there exists polynomials
′
p ,...,p in S, p -admissible substitutions (wi,...,wi ) and scalars α such that
1 n i 1 ki i
n
(4) f′ = αipi(w1i,...,wkii).
Xi=1
Since f and the polynomials p are multilinear we may assume without loss of
′ i
generality that the wi are multilinear monomials. Lemma 1 implies that there
k
exists hi ∈ Hki, φhi(pi)-admissible substitutions (bi1,...,biki) and scalars γi ∈ K×
such that
φh(pi(w1i,...,wkii))=γiφhi(pi)(bi1,...,biki),
for i=1,...,n. It follows from (4) and the above equalities that
(5) f =φh(f′)= γi′φhi(pi)(bi1,...,biki),
Xi
where γ =α γ . If f is a gradedidentity for A⊗R then it follows from Remark 2
i′ i i
thatf isagradedidentityforA. Inthiscasewemayassumethatthepolynomials
′
p1,...,pn lie in S1, therefore the polynomials φhi(pi) lies in S1 (which is a subset
of S) and equality (5) implies that f lies in hSi . Now we assume that f is
TG×H f
not a graded identity for A⊗R. Since f is multilinear there exists an admissible
e e
substitution of the form (a ⊗r ,...,a ⊗r ) such that a ,...,a and r ,...,r
1 1 n n 1 n 1 n
arehomogeneouselementsinAandRrespectivelyandf(a ⊗r ,...,a ⊗r )6=0.
1 1 n n
6 DIOGODINIZ,CLAUDEMIRFIDELIS,ANDSE´RGIOMOTA
Thereforef(a ⊗r ,...,a ⊗r )isanon-zerocentralelement. Equality(3)implies
1 1 n n
that
f(a1⊗r1,...,an⊗rn)=f′(a1,...,an)⊗r1···rn,
thereforer ···r isahomogeneousnon-zerocentralelementinRofdegreeh ···h .
1 n 1 n
Since R satisfies the equality Z(R) = R we conclude that h ···h ∈ H .
h H′ h 1 n ′
Thepolynomialsp1,...,pnlieinSandLsinc∈eh1···hn ∈H′weconcludethatφhi(gi)
lies in S for i=1,...,n. Therefore equality (5) implies that f lies in hSi . (cid:3)
TG×H
Note that S = S ∪S , where S = {x [x ,x ]x } and S = {x }, is a basis
e 1 2 1 1 2 3 4 2 1 e
of multilinear polynomials for the central polynomials of the field K such that S
1
generates the identities of K as a T-space. Let R be an algebra with a regular
H-grading satisfying the hypothesis of the previous theorem. The algebra K ⊗R
with its {e}×H-grading is weakly isomorphic to R with its H-grading. Hence the
previous theorem yields the following:
Corollary 1. Let R denote an algebra with a regular grading by the abelian group
H,withcorrespondingbicharacterβ :H×H →K ,suchthatZ(R)= R ,
R × h H′ h
where H′ ={h′ ∈H |β(h′,h)=1∀h∈H}. The polynomials L ∈
(6) x1h,h∈H′
and
(7) x (x x −β (h ,h )x x )x , where h ∈H,i=1,...,4.
1h1 2h2 3h3 R 2 3 3h3 2h2 h4,4 i
form a basis for the T -space C (R).
H H
Remark 3. Analogously, as a corollary of Theorem 1, we conclude that the set
x x −β(h ,h )x x ,
1,h1 2,h2 1 2 2,h2 1,h1
where h ,h ∈H is basis for the H-graded identities of R.
1 2
Remark 4. Note that S ={x } is a basis for the T-space of central polynomials
2 1
of the field K, however no set of indeterminates x generates T -space C (R).
ih H H
Hence as noted in [3] the relation between a basis of C (A) and C (A⊗R) is
G G H
×
not direct.
4. Basis for the Graded Identities and Central Polynomials for
Division Gradings on Simple Real Algebras
Finitedimensionalsimplerealalgebraswithadivisiongradingbyafiniteabelian
group G were classified in [6] and [4]. In this section we present the main result
of [4] and describe basis for the T -ideal of graded identities and the T -space of
G G
graded central polynomials of these algebras.
4.1. Division Gradings on Simple Real Algebras. Next we present the ba-
sic division gradings that are the building blocks of this classification. One such
building block is the Z -grading on the field C of complex numbers:
2
Γ:C=C ⊕C ,
0 1
where C = h1i and C = hii. We will denote C(2) the above grading on C. The
0 1
other building blocks are division gradings on the algebras M (R), H and M (C).
2 2
IDENTITIES AND CENTRAL POLYNOMIALS OF REAL GRADED DIVISION ALGEBRAS 7
4.1.1. Division gradings on M (R). Let G = (a) ×(b) ∼= Z ×Z . A division
2 2 2 2 2
grading on S =M (R) may be obtained by means of Sylvester matrices below:
2
1 0 0 1 0 1
A= , B = , C = .
(cid:18) 0 −1 (cid:19) (cid:18) 1 0 (cid:19) (cid:18) −1 0 (cid:19)
We have A2 = B2 = I,AB = C = −BA, therefore a division grading by G is
obtained where the homogeneous components are
S =hIi, S =hAi, S =hBi, S =hCi,
e a b c
here c =ab and I is the identity matrix. This grading is regular and the center of
M (R) is the neutral component. We denote this gradeddivision algebra by M(4).
2 2
The decomposition
R =hI,Ci, R =hA,B,i
e a
whereaisthe generatorofthegroupH =(a) ∼=Z ,isacoarseningofM(4) which
2 2 2
is also a division grading on R=M (R). This grading will be denoted by M(2).
2 2
4.1.2. Division gradings on H. Let {1,i,j,k} be the canonical basis of the quater-
nion algebra H. The multiplication is determined by the relations
i2 =j2 =−1, ij =−ji=k.
SinceHisadivisionalgebra,anygradingmakesitagradeddivisionalgebra. There
is, up to weak isomorphism, two non trivial gradings on H that we describe next.
A grading by the group G=(a) ×(b) is the decomposition
2 2
R =h1i, R =hii, R =hji, R =hki
e a b c
where c = ab. This grading is regular and the neutral component is the center of
the quaternions. We denote this graded division algebra by H(4). A coarsening of
this grading is
S =h1,ii,S =hj,ki,
e a
this is a grading by the group H = (a) ∼= Z on S = H. This grading will be
2 2
denoted by H(2).
4.1.3. Division gradings on M (C). Let G = (a) ×(b) . A division G-grading on
2 4 2
R=M (C), which we denote M(8), is obtained with homogeneous components
2 2
R =hIi, R =hωAi, R =hiIi, R =hiωAi
e a a2 a3
R =hCi, R =hωBi, R =hiCi, R =hiωBi,
b ab a2b a3b
where w = 1 (1+i) is the 8-th root of 1 such that ω2 =i. A division grading on
√2
S =M (C) by the group H =(a) obtained as a coarsening of M(8) is
2 4 2
S =hI,Ci,S =hωA,ωBi,S =hiI,iCi,S =hiωA,iωBi,
e a a2 a3
this grading will be denoted by M (C,Z ).
2 4
8 DIOGODINIZ,CLAUDEMIRFIDELIS,ANDSE´RGIOMOTA
4.1.4. Pauli gradings. Now we describe the Pauli gradings onthe realalgebraR=
M (C). The algebra R is also an algebra over C. If the decomposition Γ : R =
n
⊕ R is a division grading by the group G for R as an algebra over C then it
g G g
is a∈lso a division grading for R over R. Such gradings are called Pauli gradings.
The division grading Γ is a regular grading for R as an algebra over C and can be
described in terms of the associated bicharacter β : G×G→ C . This grading is
×
regularforRasarealalgebraifandonlyiftheimageofβ liesinR . WerefertoΓ
×
as a non-regular Pauli grading if it is not a regulargrading for R as a realalgebra.
We remark that Γ is a regular Pauli grading if and only if G∼=Z2k for some K.
2
Our study of the gradedidentities of graded division algebrasrelies on the clas-
sification of division gradings on finite dimensional simple real algebras, this clas-
sification was done in [6] and [18]. The main result of [6] is the following:
Theorem 3. Any division grading on a real simple algebra M (D), D a real divi-
n
sion algebra, is weakly isomorphic to one of the following types:
D =R: (i) (M(4)) k ∼=Cgr(k,k);
2 ⊗
(ii) M(2)⊗(M(4)) (k 1), a coarsening of (i);
2 2 ⊗ −
(iii) M(4)⊗(M(4)) (k 2), a coarsening of (i);
4 2 ⊗ −
D =H: (iv) H(4)⊗(M(4)) k ∼=Cgr(k+1,k−1);
2 ⊗
(v) H(2)⊗(M(4)) k, a coarsening of (iv);
2 ⊗
(vi) H⊗(M(4)) k, a coarsening of (v);
2 ⊗
D =C:(vii) C(2)⊗(M2(4))⊗k ∼=Cgr(k+1,k);
(viii) C(2)⊗M(2)⊗(M(4)) (k 1), a coarsening of (vii);
2 2 ⊗ −
(ix) C(2)⊗H⊗(M(4)) k, a coarsening of (vii);
2 ⊗
(x) M(8)⊗(M(4)) (k 1);
2 2 ⊗ −
(xi) M (C,Z )⊗(M(4)) (k 1), a coarsening of (x);
2 4 2 ⊗ −
(xii) M(8)⊗H⊗(M(4)) (k 2), a coarsening of (x);
2 2 ⊗ −
(xiii) Pauli gradings.
None of the gradings of different types or of the same type but with different values
of k is weakly isomorphic to the other.
4.2. Graded Identities and Central Polynomials for Division Gradings on
Simple Real Algebras. Inthissectionwedescribebasisforthe gradedidentities
and central polynomials for the algebras in Theorem 3. The gradings are either
regular,atensorproductgradingofoneofthe building blockspresentedinSection
4.1 and an algebra with a regular grading or a non-regular Pauli grading.
4.2.1. Regular Division Gradings. The algebras in Theorem 3 with a regular divi-
sion grading are (i), (iv), (vii), (x) and the regular Pauli gradings in (xiii). The
tensorproductoftwoalgebraswithregulargradingshasaregulargrading,wehave
the following remark.
Remark 5. Let A,B be algebras with regular gradings by the abelian groups G and
H,respectively. Thetensor product gradingon A⊗B is regular andthecorrespond-
ing bicharacter β :(G×H)×(G×H)→K is given by
A B ×
⊗
β ((g ,g ),(h ,h ))=β (g ,g )β (h ,h ),
A B 1 2 1 2 A 1 2 B 1 2
⊗
IDENTITIES AND CENTRAL POLYNOMIALS OF REAL GRADED DIVISION ALGEBRAS 9
for g ,g ∈G and h h ∈H.
1 2 1 2
If β is the bicharacter corresponding to a regular grading on R by the finite
abeliangroupG, generatedby the finite setS ={s ,...,s },thenit isdetermined
1 k
by the set of values β(s ,s ), for 1 ≤ i < j ≤ k. Moreover each β(s ,s ) is a root
i j i j
of the unit and therefore equals ±1. Hence β is determined by the set of pairs
(s ,s ) such that β(s ,s )=−1. One determines this pairs for the building blocks
i j i j
in Section 4.1 by inspection, we then use Remark 5 to obtain the bicharacters for
the algebras (i), (iv), (vii), (x) in Theorem 3. The algebras in (i) and (iv) are
canonically graded by the group G =(a ) ×(b ) ×···×(a ) ×(b ) , and the
2k 1 2 1 2 k 2 k 2
algebrasin(vii)and(x)bythe groupsG =(a ) ×(a ) ×(b ) ×···×(a ) ×
2k+1 0 2 1 2 1 2 k 2
(b ) , andH =(a ) ×(b ) ×···×(a ) ×(b ) respectively. The bicharacters
k 2 k 1 1 4 1 2 k 2 k 2
−
are presented in the following table:
Algebra Grading group Bicharacter
(M(4)) k, H(4)×(M(4)) (k 1) G β (a ,b )=−1, i=1,...,k
2 ⊗ 2 ⊗ − 2k k i i
C(2)⊗(M(4)) k G γ (a ,b )=−1, i=1,...,k
2 ⊗ 2k+2 k i i
M(8)×(M(4)) (k 1) H δ (a ,b )=−1, i=1,...,k
2 2 ⊗ − 2k 1 k i i
−
A Pauligradingon M (C) by the groupG is regularif and only if G∼=Z2k and
n 2
n=2k forsomek. InthiscasethegradingisweaklyisomorphictoM (C) k where
2 ⊗
M (C) has its canonical Pauli grading. This algebra can be canonically endowed
2
with a grading by the group (a ) ×(b ) ×···×(a ) ×(b ) in such a way that
1 2 1 2 k 2 k 2
the corresponding bicharacter is β .
k
Note that the bicharacters β and δ are non-degenarate and
k k
H ={h ∈H |γ (h,h)=1∀h∈H}=(a ) .
′ ′ k ′ 0 2
Therefore a basis for the graded central polynomials of finite dimensional simple
real algebras with a regular division grading are obtained from Corollary 1.
4.2.2. Tensor Product Division Gradings. A basis for the graded identities and
central polynomials for the algebras in (ii), (iii), (v), (vi), (viii), (ix), (xi) and
(xii), of Theorem 3, is obtained from Theorem 1 and Theorem 2 once a suitable
basis for the central polinomials of the algebras H, H(2),M(2) and M (C,Z ) is
2 2 4
known. Such a basis is known for the algebras H, H(2) and M(2).
2
Theorem 4. [11] Let K be a field of characteristic zero. The standard polyno-
mial S (x ,x ,x ,x ) and the Hall polynomial [[x ,x ]2,x ] form a basis for the
4 1 2 3 4 1 2 3
identities of M (K).
2
Theorem 5. [15] Let K be an infinite field of characteristic zero. The polynomials
[x ,x ][x ,x ]+[x ,x ][x ,x ] and x S (x ,x ,x ,x ) form a basis for the central
1 2 3 4 3 4 1 2 5 4 1 2 3 4
polynomials of M (K).
2
Remark 6. It is well known that the algebras H and M (R) satisfy the same
2
polynomial identities, hence the previous theorems provide basis for the polynomial
identities and central polynomials for H.
10 DIOGODINIZ,CLAUDEMIRFIDELIS,ANDSE´RGIOMOTA
It was proved in [4] that H(2) and M(2) satisfy the same graded polynomial
2
identities as R=M (R) with the (a) -grading
2 2
R 0 0 R
(8) R = R = .
e (cid:18) 0 R (cid:19) a (cid:18) R 0 (cid:19)
Basis for the graded identities and central polynomials for R were determined in
[9] and in [8].
Lemma 2. [9,Lemma2] The set consisting of the polynomials x x −x x and
1e 2e 2e 1e
x1,ax2,ax3,a−x3,ax2,ax1,a is a basis for the TG′-ideal of graded identities of R with
the grading in (8).
Theorem 6. [8, Theorem 4] The set of polynomials
x2 , x [x ,x ]x , x (x x x −x x x )x ,
1a 1g 2,e 3e 4g 1g 2a 3a 4a 4a 3a 2b 5g
where g ∈(a) generates the T -space C (R).
2 (a)2 (a)2
Remark 7. It is well known that the polynomial x2 in the theorem above may be
1a
replaced by the multilinear polynomial x x +x x to obtain a basis of multi-
1a 2a 2a 1a
linear polynomials.
Tocompleteourdescriptionwedetermineabasisforthegradedidentitiesofthe
algebra M (C,Z ). The basis is obtained using the following proposition.
2 4
Proposition 1. Let A=⊕ A be an algebra graded by the group G and assume
g G g
∈
that there exists an element g of the group G such that A contains an invertible
g
central element. If Q : KhX i → KhX i is the homomorphism such that
G G/ g
h i
Q(x ) = x , where q : G → G/hgi is the canonical homomorphism, and S is
i,g i,q(g) ′
a basis of multilinear polynomials for the T -ideal Id (A), with the grading
G/ g G/ g
induced by q, then the set h i h i
S ={f ∈KhXGi|Q(f)∈S′}
is a basis for the T -ideal Id (A).
G G
Proof.
Let f(x ,...,x ) be a multilinear polynomial. Denote by a the invertible
1,g1 n,gn
element in A . A homogeneous element of degree q(g) may be written as bak for
g
some b∈A and some integer k. Therefore any Q(f)-admissible substitution is of
g
the form (b1ak1,··· ,bnakn), where k1,...,kn are integers and (b1,...,bn) is an f-
admissiblesubstitution. Since a is centralitis clearthat Q(f)(b1ak1,··· ,bnakn)=
akf(b ,...,b ). HenceQ(f)is agradedidentityforAwiththe gradinginducedby
1 n
q if and only if f is a G-graded identity for A. Now assume that f ∈ Id (A), we
G
may write
n
Q(f)= g s (w1,...,wki)h ,
i ′i i i i
Xi=1
where s ∈ S , g ,h ,w1,...,wki ∈ KhX i. Now let R : KhX i → KhX i
′i ′ 1 i i i G/H G/H G
be a homomorphism such that R(x ) = x for i = 1,...,n. Then it is clear
i,q(gi) i,gi
that R(Q(f))=f and therefore we have
n
f = R(gi)R(s′i)(R(wi1),...,R(wiki))R(hi),
Xi=1