Table Of ContentGraded Polynomial Identities for Matrices with
7
1
the Transpose Involution over an Infinite Field
0
2
b
Luís Felipe Gonçalves Fonseca ∗
e
F
7 Instituto de Ciências Exatas e Tecnlógicas
1 Universidade Federal de Viçosa
Florestal, MG, Brazil
]
A
R Thiago Castilho de Mello †‡
.
h
t Instituto de Ciência e Tecnologia
a
m Universidade Federal de São Paulo
[ São José dos Campos, SP, Brazil
2
v February 20, 2017
6
1
3
8
Abstract
0
1. Let F be an infinite field, and let Mn(F) be the algebra of n×n
0 matrices over F. Suppose that this algebra is equipped with an el-
7 ementary grading whose neutral component coincides with the main
1 diagonal. Inthispaper,wefindabasisforthegradedpolynomialiden-
:
v tities of Mn(F) with the transpose involution. Our results generalize
i for infinite fields of arbitrary characteristic previous results in the lit-
X
erature which were obtained for the field of complex numbers and for
r
a a particular class of elementary G-gradings.
1 Introduction
Let F be a field and A be an F-algebra. A polynomial identity of the
algebra A is a polynomial in noncommuting variables which vanishes under
∗[email protected]
†[email protected]
‡supported by Fapesp grant No. 2014/10352-4, Fapesp grant No.2014/09310-5, and
CNPq grant No. 461820/2014-5.
1
any substitution of these variables by elements of A.
One of the first important results about polynomial identities in algebras
is the Amitsur-Levitzki Theorem which proves thatthe standard polynomial
of degree 2n is a polynomial identity for M (F). Specht [16] raised the
n
question whether the T-ideal of all polynomial identities of a given algebra
is finitely generated as a T-ideal. This problem was answered by Kemer [13]
using the characteristic zero some decades latter. Although Kemer proved
that there always exists a finite basis for the identities of a given algebra in
characteristic zero, it is very difficult problem to find any such basis. The
explicit polynomial identities for concrete algebras are known in very few
cases. For instance, for Mn(C) such a basis is known only if n = 1 or 2.
In light of this, mathematicians started to work with ‘weaker’ polynomial
identities such as identities with trace, identities with involution and graded
identities. It is worth mentioning that the graded identities were also used
by Kemer in the solution of the Specht problem.
Subsequent to the pioneering work of Di-Vincenzo [6] about graded iden-
tities, many authors described the graded identities of M (F) and other
n
different important algebras in different contexts [1], [2], [3], [7], [17], [8], [4]
and [18]. Also, identities with involution for M (F) have been studied by
n
some authors [12].
Recently Haile and Natapov [11] exhibited a basis for the graded iden-
tities with involution of Mn(C), endowed with the transpose involution and
with a crossed-product grading. This grading is an elementary grading of
M (F)byagroupG ={g ,...,g }inducedbythen-tuple(g ,...,g ). The
n 1 n 1 n
authors used the graph theory as in [10].
Inthispaperwegeneralize theresultsofHaileandNatapovforabroader
class of gradings and for infinite fields of arbitrary characteristic. The main
tool here is the use of generic matrices, and we use ideas similar to those in
[1], [2], [4], [7], and [8].
2 Preliminaries
We denote by F aninfinite field of arbitrary characteristic. Allvector spaces
and algebras are over F. We denote the algebra of n×n matrices over F by
M (F) and a group with the unity e by G.
n
If A is an algebra and G is a group, a G-grading on A is a decomposition
of A as a direct sum of subspaces A = ⊕ A , indexed by elements of the
g∈G g
group G, which satisfy A A ⊆ A , for any g,h ∈ G. If a ∈ A − {0},
g h gh g
for some g ∈ G, we say that a is homogeneous of degree g and we denote
2
deg(a) = g. The support of the grading, is the subset of G, Supp(A) = {g ∈
G; A 6= {0}}.
g
If 1≤ i,j ≤ n, we denote by e the matrix with 1 on the position (i,j),
ij
and and 0 elsewhere. We call them elementary matrices, or matrix units.
Nowlet(g ,...,g ) ∈ Gn beann-tupleofelementsofG. Foreachg ∈ G,
1 n
let R ⊆ M (F) be the subspace generated by the elementary matrices e
g n ij
for i and j satisfying g−1g = g. Then M (F) = ⊕ R is a G-grading on
i j n g∈G g
M (F) called elementary grading defined by (g ,...,g ).
n 1 n
We recall a known result from [5], which characterizes elementary grad-
ings on M (F).
n
Theorem 2.1. If G is any group, a G-grading of M (F) is elementary if
n
and only if all matrix units e are homogeneous.
ij
An involution on an algebra A is an antiautomorphism of the order two,
that is, a linear map ∗ : A −→ A satisfying (ab)∗ = b∗a∗ and (a∗)∗ = a, for
all a,b ∈A. A classic example of involution on M (F) is the transpose map.
n
A G-graded algebra A = A with involution ∗ is called a degree-
Lg∈G g
invertinginvolutionalgebra if(A )∗ = A forallg ∈ G. Inthiscase,wesay
g g−1
that ∗ is a degree-inverting involution on A. In this paper, if A is a degree-
inverting involution algebra, we say it is a (G,∗)-algebra. A typical example
of a (G,∗)-algebra is M (F) endowed with an elementary grading and with
n
the transpose involution. The degree-inverting involutions on M (F) have
n
been described by the authors in [9].
Remark 2.2. When dealing with identities with involution on algebras over
fields of characteristic different from 2, one usually consider the decomposi-
tion A = A+ ⊕A−, where A+ = {a ∈ A|a∗ = a} (symmetric component)
and A− = {a ∈ A|a∗ = −a} (skew-symmetric component) and set in the
free algebra, the set of symmetric and skew-symmetric variables. Note that
one cannot use this approach in the present case, since the symmetric and
skew-symmetric components are no longer homogeneous. In order to deal
with our case, we need to consider a free algebra where the grading and the
involution behave in the same way as in the algebra we want to study its
identities.
2.1 The free (G,∗)-algebra and (G,∗)-identities
To describe the identities of M (F) as a (G,∗)-algebra, we define what we
n
call the free (G,∗)-algebra.
For each g ∈ G, we define two countable sets Xg = {xk,g; k ∈ N}
and Xg∗ = {x∗k,g; k ∈ N}. Then, let X = ∪g∈GXg and X∗ = ∪g∈GXg∗.
3
Consider the free associative algebra FhX ∪X∗i, which is freely generated
by X∪X∗. Ofcourse, itisanalgebra withaninvolution defined inanatural
way. Now, we define a G-grading on this free algebra to make it a (G,∗)-
algebra. Let deg(1) = e, and for each k ∈ N and g ∈ G, let deg(xk,g) = g
and deg(x∗ ) = g−1. If m = xε1 ···xεl is a monomial in FhX ∪ X∗i,
where ε isk,∗g or nothing, we defini1e,gd1eg(mi)l,g=l deg(xε1 )···deg(xεl ).
i i1,g1 il,gl
If we define
(FhX ∪X∗i) = span {m = xε1 ···xεl | deg(m) = g},
g F i1,g1 il,gl
we obtain that FhX ∪ X∗i = (FhX ∪ X∗i) is a G-grading on the
Lg∈G g
algebra FhX ∪ X∗i, which makes it a (G,∗)-algebra. We denote such al-
gebra by FhX|(G,∗)i and call it the free (G,∗)-algebra. The elements of
FhX|(G,∗)i are called (G,∗)-polynomials.
If A and B are G-graded algebras with involution, we say that a ho-
momorphism φ : A −→ B is a homomorphism of graded algebras with
involutions, if φ(A ) ⊂ B , for all g ∈ G and φ(x∗)= φ(x)∗, for all x ∈ A.
g g
The algebra FhX|(G,∗)i satisfies a universal property: for any (G,∗)-
algebra A and for any map ϕ : X −→ A such that for all g ∈ G, ϕ(X ) ⊆
g
A , there exists a unique homomorphism of graded algebras with involution
g
φ :FhX|(G,∗)i −→ A, such that for all x ∈ X, φ(x) = ϕ(x).
Let A be (G,∗)-algebra. A polynomial f ∈ FhX|(G,∗)i is called a
(G,∗)-polynomial identity of A if f ∈ Ker(φ) for any homomorphism of
graded algebra with involution φ : FhX|(G,∗)i → A. Equivalently, f van-
ishesunderanyadmissiblesubstitutionofvariablesbytheelementsofAwith
the condition that if x is substituted by a ∈ A , then x∗ is substituted
k,g g k,g
by a∗.
We observe that if A is a (G,∗)-algebra, then it is a graded algebra, and
if f is a graded polynomial identity of A, then it is also a (G,∗)-identity of
A. In particular Proposition 4.1 of [7] also holds for (G,∗)-algebras.
Proposition 2.3. Let G be a group and let g = (g ,...,g ) ∈ Gn be an
1 n
n-tuple of elements from G. Suppose M (F) is endowed with elementary
n
grading induced by g. The following assertions are equivalent
1. The neutral component of M (F) coincides with the main diagonal.
n
2. x x −x x is a graded identity of M (F).
1,e 2,e 2,e 1,e n
3. The elements of g are pairwise distinct.
A (two-sided) ideal I ⊂ FhX|(G,∗)i is called a T∗-ideal if I is closed
G
under all (G,∗)-endomorphism of FhX|(G,∗)i. We denote the set of all
4
(G,∗)-identities of A by T∗(A). Let S ⊂ FhX|(G,∗)i. We denote the
G
intersection of all TG∗-ideals containing S by hSiTG∗. Notice that TG∗(A) and
hSiTG∗ are TG∗-ideals of FhX|(G,∗)i. We say that S ⊂ FhX|(G,∗)i is a
basis for the (G,∗)-identities of A if TG∗(A) = hSiTG∗.
Proposition 2.4. Let G be a group and let M (F) be endowed with the
n
elementary grading induced by an n-tuple (g ,...,g ) of pairwise distinct el-
1 n
ements from G, andwiththe transpose involution. Thefollowingpolynomials
are (G,∗)-identities for M (F)
n
x x −x x (1)
1,e 2,e 2,e 1,e
x −x∗ (2)
1,e 1,e
x ,g 6∈ Supp(M (F)) (3)
1,g n
x x x −x x x ,g 6= e (4)
1,g 2,g−1 3,g 3,g 2,g−1 1,g
For more details about identities 1 and 4, see [3, Lemma 4.1]. Identity 2
follows from Proposition 2.3. Also, [11, Remark 2 of Theorem 8] shows that
identity 4 follows from identity 2.
When dealing with ordinary polynomials, it is well known that each T-
ideal is generated by its multi-homogeneous polynomials. In the case of
(G,∗)-polynomials, we need to slight modify this concept.
Definition 2.5. Let f = f(x ,...,x ,x∗ ,...,x∗ ) ∈ FhX|(G,∗)i.
1,g1 n,gn 1,g1 n,gn
Write f as
k
f = λ m
X l l
l=1
where λ ∈ F−{0}andm aremonomialsinFhX|(G,∗)i. Thepolynomialf
l l
is called strongly multi-homogeneous if for each t ∈ {1,...,m}, deg m +
xt,gt i
deg m = deg m + deg m for all i,j ∈ {1,...,k}. Here, the
x∗t,gt i xt,gt j x∗t,gt j
symbol deg m denotes the number of times the variable x appears in the
x i
monomial m .
i
Following the classic Vandermonde argument, we can prove that if I is a
(G,∗)-ideal, then I is generated by its strongly multi-homogeneous polyno-
mials.
3 The ∗-graded identities of M (F)
n
We start this section with the following theorem of [11], which we aim to
generalize for infinite fields and for a broader class of gradings by adding the
5
identities x = 0 for g 6∈ Supp(M (F)). We observe that in [11] the authors
g n
used the graph theory to prove this result.
Theorem 3.1 (Haile-Natapov, Theorem 8, [11]). Let G= {g ,...,g } be a
1 n
group of order n. The ideal of (G,∗)-identities of Mn(C) endowed with the
elementary grading induced by (g ,...,g ) and with the transpose involution
1 n
is generated as a T∗-ideal by the following elements
G
1. x x −x x
i,e j,e j,e i,e
2. x −x∗
i,e i,e
From now on, we consider M (F) endowed with elementary grading in-
n
duced by the n-tuple g = (g ,...,g ) ∈ Gn of pairwise distinct elements of
1 n
G, and we denote G = Supp(M (F)).
0 n
Let g ∈ G. We define
D(g) = {i ∈ {1,...,n}|g g ∈ {g ,...,g }}
i 1 n
and
Im(g) = {j ∈{1,...,n}|g g−1 ∈ {g ,...,g }}.
j 1 n
Notice that |D(g)| = |Im(g)|, D(g−1)= Im(g) and D(g) = ∅ if and only
if g ∈/ G . In that case, if i∈ D(g), there exists a unique j ∈ {1,...,n} such
0
that g g = g . If we define j = g(i), we obtain a bijective map
i j
b
g : D(g) −→ Im(g)
i 7−→ g(i)
b
b
Observe that for each i∈ {1,...,n}, we have eibg(i) ∈ (Mn(F))g and for each
g in support of M (F), g−1 = (g)−1.
n
d
b
Lemma 3.2. Let g,h ∈ G. If there exists i ∈ D(g)∩D(h) such that
g(i) = h(i),
b b
then g = h.
Proof. Let i ∈D(gˆ)∩D(hˆ). If j = gˆ(i) = hˆ(i), then g g = g = g h. We can
i j i
conclude that g = h.
6
Let Ω = {yik,bg(i)|g ∈ G,i ∈ D(g),k ∈ N} be a set of commuting variables
and F[Ω] be the algebra of commuting polynomials in Ω. We denote the set
of all matrices over F[Ω] by M (Ω). As in the case of matrices over F, if
n
g = (g ,...,g ) is an n-tuple of elements of G, then M (Ω) is endowed with
1 n n
an elementary G-grading induced by g.
Definition 3.3. For each g ∈ G0 and j ∈ N, the elements of Mn(Ω),
j
Aj,g = X yi,bg(i)eibg(i)
i∈D(g)
and
A∗j,g = X ybgj−1(i),ieigd−1(i)
i∈D(g−1)
are called generic (G,∗)-matrices. The subalgebra of M (Ω) generated by
n
{Aj,g,A∗j,g|g ∈ G0,j ∈ N} is called the algebra of generic (G,∗)-matrices
and we denote it by Gen.
Lemma 3.4. Let g,h ∈ G . If yj ∈ Ω is an entry of the matrices A and
0 i,k j,g
A , then g = h.
j,h
j
Proof. If y is an entry of A and of A then k = g(i) and k = h(i). Now
i,k j,g j,g
Lemma 3.2 implies that g = h. b b
Using classical arguments, we can prove the following proposition.
Proposition 3.5. The relatively free algebra FhX|(G,∗)i/T∗(M (F)) is
G n
isomorphic to Gen. Furthermore, T∗(M (F)) = T∗(Gen).
G n G
We now define the following maps, which by an abuse of notation will be
also denoted by ∗
∗ : G −→ G
g 7−→ g∗ = g−1
∗ : Ω −→ Ω
ykbg(k) 7−→ ykbg(k)∗ = ybg−1(k)k
Given hε1,hε2,...,hεr ∈ G , where h ∈ G and ε is ∗ or nothing, we
1 2 r 0 i i
considerthecomposition ν = hεr···hε1 ofthecorrespondingfunctions. This
r 1
may not be well defined, andcwe wilcl prove in Lemma 3.7 that in this case
the monomial xε1 ···xεr is a graded identity for M (F). Otherwise, its
1,h1 r,hr n
domain Dν = Dhdεrr···hdε11 is the set of i ∈ {1,...,n} for which the image
hεr(...(hε1(i))...) is well defined.
r 1
c c
7
Lemma 3.6. Let g,h ∈ G, then D(hg) ⊆ D(gh). Moreover, if i ∈ D(hg),
then hg(i) = gh(i). bb c bb
bb c
Proof. IfD(hg)= ∅,theresultisobvious. SupposeD(hg) 6= ∅. Ifi ∈D(hg),
let k = g(i)babnd j = h(k). Then, gk = gig and gj =bbgkh, and we obbtabin
gj = gi(gbh), that is, ghb(i) = j.
c
Lemma 3.7. Let hε11,hε22,...,hεrr ∈ G0. If Dhdεrr···hdε11 = ∅ then
Aε1 Aε2 ···Aεr = 0.
i1,h1 i2,h2 ir,hr
MAεio11r,he1oAveεi2r2,,h2if··t·hAeεirsr,ehtrDishdεrnro··n·hdzε1e1roisifnoannedmopntlyy tihfein∈thDehdεrir-t··h·hdε1l1in.eInofththisecmasaet,riixf
j = hεr ···hε1(i), the only nonzero entry in the i-th line is a monomial of Ω
r 1
in thce j-th ccolumn.
Proof. The proof is by induction on the length r of the product. The result
for r = 1 follows directly from Definition 3.3. Hence, we consider r > 1 and
assume the result for products of length r−1. Let us consider the first case
Dhdεrr···hdε11 6= ∅. In this case Dh\rεr−−11···hdε11 6= ∅ and we denote ν = h[εrr−−11···hcε11.
The induction hypothesis implies that there exists monomials m , where
i
i ∈ D , such that
h\εr−1···hdε1
r−1 1
Aεi11,h1Aεi22,h2···Aεirr,hr = X mieiν(i) X (yjhcr(j))εrejhdεrr(j)
i∈Dh\rεr−−11···hdε11 j∈Dhdεrr
(5)
Note that eiν(i)ejhdεrr(j) 6= 0 for some j if and only if i∈ Dhdεrr···hdε11. In this
case, the product equals eihdεrr(j). Hence, we obtain
Aεi11,h1Aεi22,h2···Aεirr,hr = X (mi(yjhcr(j))εr)eihdεrr(ν(i)),
i∈Dhdεrr···hdε11
and the result follows. Now, assume that Dhdεrr···hdε1r = ∅. If Dh\rεr−−11···hdε11 = ∅
then by the induction hypothesis Aε1 Aε2 ···Aεr−1 = 0 and the
i1,h1 i2,h2 ir−1,hr−1
8
result holds. Moreover, if D 6= ∅ then we may write the prod-
h\εr−1···hdε1
r−1 1
euicνt(i)Aejεih1d1,εrhr1(jA)εi2e2,qhu2a·l·s·Azeεirrro,hranadstihner(e5fo).re ASiεin11c,he1ADεi22hd,εrhr2····hd·ε1·rAεi=rr,hr∅,=e0v.ery product
Definition 3.8. Suppose h = (hǫ11,...,hǫmm) ∈ Gm such that Dhdεmm···hdε1r 6= ∅.
For each k ∈ Dhdεmm···hdε1r, we denote by sk(h) = (sk1,...,skm,skm+1) the
following sequence, inductively by setting:
(1) sk = k
1
(2) sk = h[εr−1(sk ), for r ∈ {2,...,m+1}.
r r−1 r−1
We denote by t (h) = (tk,...,tk ) the sequence defined by
k 1 m
tk = h (sk), r ∈ {1,...,m}
r r r
b
Lemma 3.9. Let hε11,...,hεmm ∈ G such that Dhdεmm···hdε11 6= ∅. Then
Aε1 ···Aεm = ωme ,
1,h1 m,hm k∈DhdPεmm···hdε11 k k,skm+1
where ωm = (y1 )ε1···(ym )εm. Furthermore, each matrix in the prod-
uct (Aε1k ,...,sAk1ε,tmk1 ) contsrkmib,tukmtes with exactly one factor of it in the prod-
1,h1 m,hm
uct (y1 )ε1···(ym )εm. For each p ∈ {1,...,m}, Aεp contributes with
(yp s)k1ε,pt.k1 skm,tkm p,hp
sk,tk
p p
Proof. The proof follows by induction on m. If m = 1, the result is obvious.
Suppose m > 1. By the induction hypothesis, we obtain
Aε1 ···Aεm = (Aε1 ···Aεm−1 )Aεm
1,h1 m,hm 1,h1 m−1,hm−1 m,hm
= X ωkm−1ek,skm X (yim,hcm(i))εmeihcm(i)
k∈Dhεm\m−−11···hdε11 i∈D(hdεmm)
= ωm−1(ym )εme
X k skm,hcm(skm) k,skm+1
k∈Dhεm\m−−11···hdε11
Now, the proof follows once one observes that tk = h (sk ) and ωm =
m m m k
ωm−1(ym )εm. c
k sk ,tk
m m
9
ε ε
Definition 3.10. Let σ ∈ S . For m = x σ(1) ...x σ(n) , and any
m iσ(1),hσ(1) iσ(n),hσ(n)
two integers 1 ≤ k ≤ l ≤ n, we denote m[k,l] the subword obtained from m
by deleting the first k−1 and the last m−l variables.
m[k,l] = xεσ(k) ...xεσ(l) .
i i
σ(k),hσ(k) σ(l),hσ(l)
Lemma 3.11. Let xε1 ···xεr and xη1 ···xηl be two monomials, with
ε and η being ∗ oir1,hn1othingir,,hsruch thaj1t,ht′1he mjalt,hr′lices Aε1 ···Aεr and
k k i1,h1 ir,hr
Aη1 ···Aηs have in the same position, the same nonzero entry. Then,
j1,h′1 js,h′s
r = l and there exists a permutation σ ∈ S such that j = i and h′ =
r q σ(q) q
h for all q ∈ {1,...,r}. In particular, xε1 ···xεr −xη1 ···xηl is a
σ(q) i1,h1 ir,hr j1,h′1 jl,h′l
stronglymulti-homogeneous polynomial. Ifthisentryis(yi1 )ε1···(yir )εr,
sk,tk sk,tk
1 1 r r
then (yiσ(q) )εσ(q) = (yjq )ηq for q = 1,...,r.
sk ,tk s′k,t′k
σ(q) σ(q) q q
Proof. Let Aε1 ···Aεr = ωre , and Aη1 ···Aηl =
i1,h1 ir,hr k∈DhdPεmr···hdε11 k k,skr+1 j1,h′1 jl,h′l
ω˜le as in Lemma 3.9. Let k be the row in which these
k k,sk
k∈DhdP′εl···hd′ε1 m+1
l 1
two matrices have the same nonzero entry. Let h = (h ,...,h ) and h′ =
1 r
(h′,...,h′) and consider the sequences s (h) = (sk,...,sk ), s (h′) =
1 l k 1 r+1 k
(s′k,...,s′k ), t (h) = (tk,...,tk ), t (h′) = (t′k,...,t′k ) as in Def-
1 r+1 k 1 r+1 k 1 r+1
inition 3.8. Then ωr = ω˜l, where ωr = (yi1 )ε1···(yir )εr and ω˜l =
k k k sk,tk sk,tk k
1 1 r r
(yj1 )η1···(yjl )ηl. Ofcourse,wehaver = l,andforeachq ∈ {1...,r},
s′k,t′k s′k,t′k
1 1 l l
there exists p ∈ {1...,r} such that (yjq )ηq = (yip )εp.
s′kq,t′kq skp,tkp
Let us now consider two cases:
Case 1: ε = η . Then i = j , sk = s′k and tk = t′k. Since t′k = h′(s′k)
p q p q p q p q q q q
and tk = h (sk), we obtain h (sk) = h′(sk) and Lemma 3.4 impliebs that
p p p p p q p
h = h′. c c b
p q
Case 2: ε 6= η . We suppose ε = ∗. Then, we have
p q p
(yjq ) = (yip )∗ = yip .
s′kq,hc′q(s′kq) skp,hcp(skp) hd−p1(skp),skp
Bycomparingtheindexes,weobtaini = j ,sk = h′(s′k)ands′k = h−1(sk).
p q p q q q p p
Hences′k = h−1(h′(s′k))andbyLemmas3.2anbd3.6,weobtainhd′h−1 =
q p q q q p
e and h′ = h . d b
q p
10
