- Email: [email protected]
On the structure of graded commutative algebras
Linear Algebra and its Applications 447 (2014) 110–118
Contents lists available at SciVerse ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
On the structure of graded commutative algebras < Antonio J. Calderón Martín Department of Mathematics, University of Cádiz, 11510 Puerto Real, Cádiz, Spain
ARTICLE INFO
ABSTRACT
Article history: Received 1 December 2012 Accepted 25 February 2013 Available online 30 March 2013
Consider A a commutative algebra graded by means of an abelian Ag A−g group G. We show that, under the hypothesis A0 = g ∈ where = {g ∈ G : Ag = 0}, the algebra A is of the form A = Ij
Submitted by P. Sˇ emrl Dedicated to the memory of M. Neumann and U. Rothblum. AMS classification: 17Axx 17B70
j
with any Ij a well described graded ideal of A, satisfying Ij Ik = 0 if j = k. Under certain conditions, the simplicity of A is characterized and it is shown that A is the direct sum of the family of its minimal graded ideals, each one being a simple graded commutative algebra. © 2013 Elsevier Inc. All rights reserved.
Keywords: Graded algebra Commutative algebra Structure theory
1. Introduction and previous definitions The interest on gradings on different classes of algebras has been remarkable in the last years, motivated in part by their application in physics and geometry [2,3,6,12–16,18,20,21]. In the present paper we study graded commutative algebras, (where further identities on the product are not supposed), of arbitrary dimension and over an arbitrary base field K, by focussing on their structure. In this framework we have to mention the related recent Ref. [19] which has been motivated by the previous work [1]. By an algebra we just mean a vector space over K endowed with a bilinear product A×A
→A
(1)
(x, y) → xy. < Supported by the PCI of the UCA ‘Teoría de Lie y Teoría de Espacios de Banach’, by the PAI with project numbers FQM298, FQM7156 and by the project of the Spanish Ministerio de Educación y Ciencia MTM2010-15223. E-mail address: [email protected] 0024-3795/$ - see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2013.02.029
A.J. Calderón Martín / Linear Algebra and its Applications 447 (2014) 110–118
111
As usual, the algebra A is called commutative if xy = yx for any x, y ∈ A, but note that we do not suppose associativity or any other additional identity on the product (1). Definition 1.1. Let A be a commutative algebra and G a non-trivial abelian group. It is said that A is a graded commutative algebra, by means of G, if A
=
Ag
g ∈G
where any Ag is a linear subspace satisfying Ag Ah for any h ∈ G. We call the support of the grading to the set
⊂ Ag +h , (denoting by + the group operation in G),
:= {g ∈ G \ {0} : Ag = 0}. A graded ideal I of A is an ideal which splits as I
=
g ∈G
Ig with any Ig
= I ∩ Ag . A graded commutative
algebra A will be called simple if its product is nonzero and its only graded ideals are {0} and A. In Section 2 we develop connections techniques on the support of the grading so as to show that, Ag A−g , the algebra A is of the form A = Ij with any Ij a well under the hypothesis A0 = g ∈
j
described graded ideal of A, satisfying Ij Ik = 0 if j = k. In Section 3, and under certain conditions, the simplicity of A is characterized and it is shown that A is the direct sum of the family of its minimal graded ideals, each one being a simple graded commutative algebra. 2. Connections in the support: decompositions In the following, A denotes a graded commutative algebra with support and
A
=
g ∈G
⎛ Ag
= A0 ⊕ ⎝
⎞ Ag ⎠
g ∈
the corresponding grading. We begin by developing connections in the support techniques in this setting. Write − := {−g : g ∈ } and let us denote by
= {g ∈ : Ag A−g = 0} ∪ {g ∈ : (Ag + A−g )(Ah A−h ) = 0 for some h ∈ }. Note that the commutativity of the product gives us that g
∈ ⇒ −g ∈ .
For each g
∈ , a new variable θg is introduced and we denote by
= {θg : g ∈ } the set consisting of all these new symbols. Next, we consider the following operation, <
: (± ∪ ) × (± ∪ ) → G ∪ ,
(2)
112
A.J. Calderón Martín / Linear Algebra and its Applications 447 (2014) 110–118
as follows: ⎧ ⎨θ
• For g ∈ ± , g<(−g ) = ⎩
g,
if g
∈ ;
0, if g ∈ / . • For g , h ∈ ± with h = −g, we define g
θg
+ A−h ) = 0 or (Ag + A−g )(Ah A−h ) = 0;
• For θg , θh ∈ , θg <θh =
⎧ ⎨ h, if (A A )(A A ) g −g h −h ⎩ 0, otherwise.
= 0.
Note that Eq. (2) implies that if g<(−g )
= θg then −g
(3)
and that the commutativity of the product in A gives us that if θg <θh
= h then θh <θg = g and θ−g <θ−h = −h.
(4)
Definition 2.1. Let g and h be two elements in . We say that g is connected to h if there exist g1 , . . . , gn ∈ ± ∪ such that 1. g1 = g 2. {g1
⊂ ± ∪
We also say that {g1 , . . . , gn } is a connection from g to h. Observe that {g } is a connection from g to itself, and in case −g ∈ also to −g. Our next goal is to show that the connection relation is of equivalence. For an easier notation, from now on we will denote by −θk := θ−k for k ∈ (see Eq. (2)). Lemma 2.1. For any g
∈ and h ∈ ± such that θg
1. h ∈ . 2. h<(−h) = θh . 3. θh
A.J. Calderón Martín / Linear Algebra and its Applications 447 (2014) 110–118
Lemma 2.2. Let g , k, l
113
∈ ± and h ∈ ± ∪ . The following assertions hold.
1. If g
∈ , then k = −g, h = l, l<(−h) = θl and ∈ , then k = −g; h = θl , l<(−l) = h and
Proof. 1. We have g = −h and h = k. Hence, if h ∈ ± then the mapping < in both expressions is just the usual sum in G and so k<(−h) = g. If h ∈ then g = k and the result follows from Lemma 2.1-4. 2. The facts k = −g and l = h are clear. So g
∈ ± and h ∈ ± ∪ . The following assertions hold.
1. If g
= θ−g and (−g<(−h))<(−k) = −l.
Proof. 1. Since g = −h, in case h ∈ ± we prove the assertion as in Lemma 2.2-1. In case h ∈ we have g = k and the result follows from Lemma 2.1-5. 2. Since g
∼ h if and only if g is connected to h, is an equivalence
Proof. {g } is a connection from g to itself and therefore g ∼ g. That is, the relation ∼ is reflexive. Now observe that given any connection {g1 , g2 , . . . , gn }, and taking into account condition 2 in Definition 2.1, a recursive argument with Lemma 2.3 and Eqs. (3) and (4), let us assert that if
(· · · ((g1
(· · · (((−g1 )<(−g2 ))<(−g3 ))< · · · )<(−gi ) = −((· · · ((g1
(5)
for any i = 2, . . . , n. Let us see the transitive character of ∼. Suppose g ∼ h and h ∼ l, and write {g1 , . . . , gn } for a connection from g to h and {h1 , . . . , hm } for a connection from h to l. If m = 1, then l ∈ ±h and so {g1 , . . . , gn } is a connection from g to l. If m 2, we have that {g1 , . . . , gn , h2 , . . . , hm } is a connection from g to l in case (· · · (g1
114
A.J. Calderón Martín / Linear Algebra and its Applications 447 (2014) 110–118
If g
∼ h, there exists a connection {g1 , g2 , g3 , . . . , gn−1 , gn } ⊂ ± ∪
from g to h satisfying conditions 1. 2. and 3. in Definition 2.1. If n = 1 then g1 = g ∈ ±h and so {h} is a connection from h to g. Suppose n 2. We can distinguish two possibilities. In the first one
((· · · ((g1
((· · · ((g1
(6)
Suppose we have the first one. Definition 2.1-2 gives us two options for the expression (· · · ((g1
∈ ± ,
Lemma 2.2-1 implies h<(−gn )
= (· · · ((g1
That is, the set {h, −gn } is a connection from h to (· · · ((g1
(7)
By the other hand, if (· · · ((g1
∈ ,
(· · · ((g1
= θgn ∈
and
(h<(−gn ))<(−gn−1 ) = (· · · ((g1
{h, −gn , −gn−1 } is a connection from h to (· · · ((g1
= θh ∈
∈ then
(8)
A.J. Calderón Martín / Linear Algebra and its Applications 447 (2014) 110–118
115
(h<(−h))<(θ−gn−1 ) = (· · · ((g1
{h, −h, θ−gn−1 } is a connection from h to (· · · ((g1
(9)
Now, we can argue in a similar way either from the partial connection
{g1 , g2 , g3 , . . . , gn−1 } from g to (· · · ((g1
∈ , we denote by
g = {h ∈ : g and h are connected}. Proposition 2.2. Let g
∈ . Then the following assertions hold.
1. g is a subsupport. 2. If l ∈ satisfies that l h ∈ g .
∈ / g , then Ah Al = 0, (Ah A−h )Al = 0 and (Ah A−h )(Al A−l ) = 0 for any
Proof. (1) Given h ∈ g , there exists a connection {g1 , . . . , gn } from g to h. It is clear that in case −h ∈ , the set {g1 , . . . , gn } also connects g to −h and therefore −h ∈ g . Given h, k ∈ g such that h
(A0 )0 := spanK {Ag A−g : g ∈ 0 } ⊂ A0
116
A.J. Calderón Martín / Linear Algebra and its Applications 447 (2014) 110–118
and V0
:=
A0
g ∈0
Ag . Finally we define the linear subspace
:= (A0 )0 ⊕ V0 .
Theorem 2.1. Suppose A0 1. For any g
=
g ∈
Ag A−g . Then the following assertions hold.
∈ , the linear subspace
Ag
= (A0 )g ⊕ Vg
of A associated to the subsupport g is an ideal of A. 2. If A is simple, then there exists a connection from g to h for any g , h
∈ .
Proof. 1. We have by Proposition 2.2 that Ah Al = 0, (Ah A−h )Al = 0 and (Ah A−h )(Al A−l ) / g . As we also have g is a subsupport we get h ∈ g and l ∈ ⎛
⎛
Ag A = Ag ⎝A0
⊕⎝
⎛⎛
= 0 for any
⎞⎞ Ah ⎠⎠
h∈
⎞ ⎛ ⎞ ⎛ ⎞⎞
⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎟ = Ag ⎜ Ag A−g + Ag A−g ⎠⊕⎝ Ag ⎠ ⊕ ⎝ Ag ⎠⎠ ⊂Ag . ⎝⎝ g ∈g
2. The simplicity of A implies Ag Theorem 2.2. Suppose A0
A
=
[g ]∈ /∼
=
g ∈
g ∈ / g
g ∈g
g ∈ / g
= A and therefore g = .
Ag A−g . Then we have
I[g ] ,
where any I[g ] is one of the ideals Ag of A described in Theorem 2.1-1, satisfying I[g ] I[h]
= 0 if [g ] = [h].
Proof. By Proposition 2.1, we can consider the quotient set / ∼:= {[g ] : g ∈ }. Let us denote by I[g ] := Ag .We have I[g ] is well defined and by Theorem 2.1-1 an ideal of A. Therefore A
=
[g ]∈ /∼
I[g ] .
By applying Proposition 2.2-2 we also obtain I[g ] I[h] Let us denote by Z (A) Corollary 2.1. If A0 Theorem 2.2, A
=
[g ]∈ /∼
=
I[g ] .
= 0 if [g ] = [h].
= {x ∈ A : xA = 0} the center of A.
g ∈
Ag A−g and Z (A)
= 0, then A is the direct sum of the ideals given in
117
A.J. Calderón Martín / Linear Algebra and its Applications 447 (2014) 110–118
Proof. We just have to verify the direct character of the sum, but this follows from the facts I[g ] I[h] if [g ] = [h], and Z (A) = 0.
= 0,
3. The simple components In this section we study if any of the components in the decomposition given in Corollary 2.1 is simple. Under certain conditions we give an affirmative answer. We begin by introducing the concepts of maximal length and -multiplicativity in the framework of graded commutative algebras in a similar way than in the frameworks of graded Lie algebras, graded Leibniz algebras, graded associative algebras, graded Lie superalgebras, split color Lie algebras etc. (see [4,10,7,5,8,9,11,17]). Definition 3.1. We say that a graded commutative algebra A is of maximal length if A0 dim Ag = 1 for any g ∈ .
= 0 and
Definition 3.2. We say that a graded commutative algebra A is -multiplicative if given g , h ∈ ∪ {0} are such that g
∈ implies −g ∈ . From now
Theorem 3.1. Let A be a -multiplicative graded commutative algebra of maximal length and such that Ag A−g . Then A is simple if and only if it has all its elements in the support Z (A) = 0 and A0 = g ∈
connected. Proof. The first implication is Theorem 2.1-2. To prove the converse, consider I = Ig a nonzero g ∈G (I ∩ Ag )) where I := {g ∈ : I ∩ Ag = 0}. By the graded ideal of A and write I = (I ∩ A0 ) ⊕ ( g ∈I maximal length of A we can write I = I0 ⊕ ( Ag ). Observe that I = ∅ since in the opposite case g ∈I Ag A−g , there exist g , h ∈ such that Ag A−g ⊂ I and 0 = I ⊂ A0 and, since Z (A) = 0 and A0 = g ∈
(Ag A−g )(Ah A−h ) = 0. Then we would have by -multiplicativity that 0 = Ah ⊂ I , a contradiction. From here, we can take g0 ∈ I being so 0
= Ag 0 ⊂ I .
(10)
For any g ∈ , g = ±g0 , the fact that g0 and g are connected gives us a connection {g1 , g2 , . . . , gn } from g0 to g satisfying Definition 2.1. Consider g0 = g1 , g2 and g1
• In the first possibility, 0 = Ag0 Ag2 = Ag0
= A((···((g1
and either Ag
⊂ I or A−(g ) ⊂ I for any g ∈ . The fact A0 =
g ∈
Ag A−g gives us now
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A.J. Calderón Martín / Linear Algebra and its Applications 447 (2014) 110–118
A0
⊂ I.
From here, given any g ∈ , the -multiplicativity of A together with its maximal length allow us to assert Ag = A0 Ag ⊂ I for any g ∈ and so I = A. That is, A is simple. Theorem 3.2. Let A be a -multiplicative graded commutative algebra of maximal length and such that Ag A−g . Then A is the direct sum of the family of its minimal ideals. Each one Z (A) = 0 and A0 = g ∈
being a simple graded commutative algebra having all of the elements in its support connected. I[g ] is the direct sum of the ideals I[g ] = (A0 )g ⊕ Vg = Proof. By Corollary 2.1, A = [g ]∈ /∼ Ah A−h ) ⊕ ( Ah ) having any I[g ] its support, g , with all of its elements connected. We have ( h∈[g ]
h∈[g ]
that g has all of its elements g -connected, (connected through elements related to g ). We also have that any of the I[g ] is -multiplicative as consequence of the -multiplicativity of A. Clearly I[g ] is of maximal length, and finally ZI[g ] (I[g ] ) = 0, (where ZI[g ] (I[g ] ) denotes de center I[g ] in I[g ] ), as consequence of I[g ] I[h] = 0 if [g ] = [h], (Theorem 2.2), and Z (A) = 0. We can apply Theorem 3.1 to I[g ] satisfies the any I[g ] so as to conclude I[g ] is simple. It is clear that the decomposition A = assertions of the theorem.
[g ]∈ /∼
Remark 3.1. We note that it is possible to develop the results of the paper for graded anti-commutative algebras in a analogous way. Hence the results in this paper can be applied in particular to the frameworks of graded Jordan and graded Lie algebras. Acknowledgments We would like to thank the referee for the detailed reading of this work and for the suggestions which have improved the final version of the same. References [1] A. Alburquerque, S. Majid, Clifford algebras by twisting of groups algebras, J. Pure Appl. Algebra 171 (2002) 133–148. [2] E. Aljadeff, D. Haile, M. Natapov, Graded identities of matrix algebras and the universal graded algebra, Trans. Amer. Math. Soc. 362 (6) (2010) 3125–3147. [3] Y. Bahturin, M. Brešar, Lie gradings on associative algebras, J. Algebra 321 (1) (2009) 264–283. [4] A.J. Calderón, On the structure of graded Lie algebras, J. Math. Phys. 50 (10) (2009) 103513, 8 pp [5] A.J. Calderón, On graded associative algebras, Rep. Math. Phys. 69 (1) (2012) 75–86. [6] A.J. Calderón, C. Draper, C. Martín, Gradings on the real forms of the Albert algebra, of g 2 , and of f 4 , J. Math. Phys. 51 (5) (2010) 053516, 21 pp [7] A.J. Calderón, J.M. Sánchez, Split Leibniz algebras, Linear Algebra Appl. 436 (6) (2012) 1648–1660. [8] A.J. Calderón, J.M. Sánchez, On the structure of graded Lie superalgebras, Modern Phys. Lett. A 27 (25) (2012) 1250142, 18 pp [9] A.J. Calderón, J.M. Sánchez, On the structure of split Lie color algebras, Linear Algebra Appl. 436 (2) (2012) 307–315. [10] A.J. Calderón, J.M. Sánchez, On the structure of graded Leibniz algebras, Algebra Colloq., in press. [11] M. Chaves, D. Singleton, Phantom energy from graded algebras, Modern Phys. Lett. A 22 (1) (2007) 29–40. [12] C. Draper, C. Martín, Gradings on g 2 , Linear Algebra Appl. 418 (1) (2006) 85–111. [13] G. Dahl, The doubly graded matrix cone and Ferrers matrices, Linear Algebra Appl. 368 (2003) 171–190. [14] A. Elduque, Fine gradings on simple classical Lie algebras, J. Algebra 324 (12) (2010) 3532–3571. [15] H. Grosse, G. Reiter, Graded differential geometry of graded matrix algebras, J. Math. Phys. 40 (12) (1999) 6609–6625. [16] M. Havlíˇcek, J. Patera, E. Pelantová, On Lie gradings II, Linear Algebra Appl. 277 (1998) 97–125. [17] M. Kochetov, Gradings on finite-dimensional simple Lie algebras, Acta Appl. Math. 108 (1) (2009) 101–127. [18] W. Marcinek, Graded algebras and geometry based on Yang–Baxter operators, J. Math. Phys. 33 (5) (1992) 1631–1635. [19] S. Morier-Genoud, V. Ovsienko, Simple graded commutative algebras, J. Algebra 323 (6) (2010) 1649–1664. [20] C. Nastasescu, F. Van Oystaeyen, Methods of Graded Rings, Springer-Verlag, Berlin, 2004. [21] J. Patera, H. Zassenhaus, On Lie gradings I, Linear Algebra Appl. 112 (1989) 87–159.
Contents lists available at SciVerse ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
On the structure of graded commutative algebras < Antonio J. Calderón Martín Department of Mathematics, University of Cádiz, 11510 Puerto Real, Cádiz, Spain
ARTICLE INFO
ABSTRACT
Article history: Received 1 December 2012 Accepted 25 February 2013 Available online 30 March 2013
Consider A a commutative algebra graded by means of an abelian Ag A−g group G. We show that, under the hypothesis A0 = g ∈ where = {g ∈ G : Ag = 0}, the algebra A is of the form A = Ij
Submitted by P. Sˇ emrl Dedicated to the memory of M. Neumann and U. Rothblum. AMS classification: 17Axx 17B70
j
with any Ij a well described graded ideal of A, satisfying Ij Ik = 0 if j = k. Under certain conditions, the simplicity of A is characterized and it is shown that A is the direct sum of the family of its minimal graded ideals, each one being a simple graded commutative algebra. © 2013 Elsevier Inc. All rights reserved.
Keywords: Graded algebra Commutative algebra Structure theory
1. Introduction and previous definitions The interest on gradings on different classes of algebras has been remarkable in the last years, motivated in part by their application in physics and geometry [2,3,6,12–16,18,20,21]. In the present paper we study graded commutative algebras, (where further identities on the product are not supposed), of arbitrary dimension and over an arbitrary base field K, by focussing on their structure. In this framework we have to mention the related recent Ref. [19] which has been motivated by the previous work [1]. By an algebra we just mean a vector space over K endowed with a bilinear product A×A
→A
(1)
(x, y) → xy. < Supported by the PCI of the UCA ‘Teoría de Lie y Teoría de Espacios de Banach’, by the PAI with project numbers FQM298, FQM7156 and by the project of the Spanish Ministerio de Educación y Ciencia MTM2010-15223. E-mail address: [email protected] 0024-3795/$ - see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2013.02.029
A.J. Calderón Martín / Linear Algebra and its Applications 447 (2014) 110–118
111
As usual, the algebra A is called commutative if xy = yx for any x, y ∈ A, but note that we do not suppose associativity or any other additional identity on the product (1). Definition 1.1. Let A be a commutative algebra and G a non-trivial abelian group. It is said that A is a graded commutative algebra, by means of G, if A
=
Ag
g ∈G
where any Ag is a linear subspace satisfying Ag Ah for any h ∈ G. We call the support of the grading to the set
⊂ Ag +h , (denoting by + the group operation in G),
:= {g ∈ G \ {0} : Ag = 0}. A graded ideal I of A is an ideal which splits as I
=
g ∈G
Ig with any Ig
= I ∩ Ag . A graded commutative
algebra A will be called simple if its product is nonzero and its only graded ideals are {0} and A. In Section 2 we develop connections techniques on the support of the grading so as to show that, Ag A−g , the algebra A is of the form A = Ij with any Ij a well under the hypothesis A0 = g ∈
j
described graded ideal of A, satisfying Ij Ik = 0 if j = k. In Section 3, and under certain conditions, the simplicity of A is characterized and it is shown that A is the direct sum of the family of its minimal graded ideals, each one being a simple graded commutative algebra. 2. Connections in the support: decompositions In the following, A denotes a graded commutative algebra with support and
A
=
g ∈G
⎛ Ag
= A0 ⊕ ⎝
⎞ Ag ⎠
g ∈
the corresponding grading. We begin by developing connections in the support techniques in this setting. Write − := {−g : g ∈ } and let us denote by
= {g ∈ : Ag A−g = 0} ∪ {g ∈ : (Ag + A−g )(Ah A−h ) = 0 for some h ∈ }. Note that the commutativity of the product gives us that g
∈ ⇒ −g ∈ .
For each g
∈ , a new variable θg is introduced and we denote by
= {θg : g ∈ } the set consisting of all these new symbols. Next, we consider the following operation, <
: (± ∪ ) × (± ∪ ) → G ∪ ,
(2)
112
A.J. Calderón Martín / Linear Algebra and its Applications 447 (2014) 110–118
as follows: ⎧ ⎨θ
• For g ∈ ± , g<(−g ) = ⎩
g,
if g
∈ ;
0, if g ∈ / . • For g , h ∈ ± with h = −g, we define g
θg
+ A−h ) = 0 or (Ag + A−g )(Ah A−h ) = 0;
• For θg , θh ∈ , θg <θh =
⎧ ⎨ h, if (A A )(A A ) g −g h −h ⎩ 0, otherwise.
= 0.
Note that Eq. (2) implies that if g<(−g )
= θg then −g
(3)
and that the commutativity of the product in A gives us that if θg <θh
= h then θh <θg = g and θ−g <θ−h = −h.
(4)
Definition 2.1. Let g and h be two elements in . We say that g is connected to h if there exist g1 , . . . , gn ∈ ± ∪ such that 1. g1 = g 2. {g1
⊂ ± ∪
We also say that {g1 , . . . , gn } is a connection from g to h. Observe that {g } is a connection from g to itself, and in case −g ∈ also to −g. Our next goal is to show that the connection relation is of equivalence. For an easier notation, from now on we will denote by −θk := θ−k for k ∈ (see Eq. (2)). Lemma 2.1. For any g
∈ and h ∈ ± such that θg
1. h ∈ . 2. h<(−h) = θh . 3. θh
A.J. Calderón Martín / Linear Algebra and its Applications 447 (2014) 110–118
Lemma 2.2. Let g , k, l
113
∈ ± and h ∈ ± ∪ . The following assertions hold.
1. If g
∈ , then k = −g, h = l, l<(−h) = θl and ∈ , then k = −g; h = θl , l<(−l) = h and
Proof. 1. We have g = −h and h = k. Hence, if h ∈ ± then the mapping < in both expressions is just the usual sum in G and so k<(−h) = g. If h ∈ then g = k and the result follows from Lemma 2.1-4. 2. The facts k = −g and l = h are clear. So g
∈ ± and h ∈ ± ∪ . The following assertions hold.
1. If g
= θ−g and (−g<(−h))<(−k) = −l.
Proof. 1. Since g = −h, in case h ∈ ± we prove the assertion as in Lemma 2.2-1. In case h ∈ we have g = k and the result follows from Lemma 2.1-5. 2. Since g
∼ h if and only if g is connected to h, is an equivalence
Proof. {g } is a connection from g to itself and therefore g ∼ g. That is, the relation ∼ is reflexive. Now observe that given any connection {g1 , g2 , . . . , gn }, and taking into account condition 2 in Definition 2.1, a recursive argument with Lemma 2.3 and Eqs. (3) and (4), let us assert that if
(· · · ((g1
(· · · (((−g1 )<(−g2 ))<(−g3 ))< · · · )<(−gi ) = −((· · · ((g1
(5)
for any i = 2, . . . , n. Let us see the transitive character of ∼. Suppose g ∼ h and h ∼ l, and write {g1 , . . . , gn } for a connection from g to h and {h1 , . . . , hm } for a connection from h to l. If m = 1, then l ∈ ±h and so {g1 , . . . , gn } is a connection from g to l. If m 2, we have that {g1 , . . . , gn , h2 , . . . , hm } is a connection from g to l in case (· · · (g1
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A.J. Calderón Martín / Linear Algebra and its Applications 447 (2014) 110–118
If g
∼ h, there exists a connection {g1 , g2 , g3 , . . . , gn−1 , gn } ⊂ ± ∪
from g to h satisfying conditions 1. 2. and 3. in Definition 2.1. If n = 1 then g1 = g ∈ ±h and so {h} is a connection from h to g. Suppose n 2. We can distinguish two possibilities. In the first one
((· · · ((g1
((· · · ((g1
(6)
Suppose we have the first one. Definition 2.1-2 gives us two options for the expression (· · · ((g1
∈ ± ,
Lemma 2.2-1 implies h<(−gn )
= (· · · ((g1
That is, the set {h, −gn } is a connection from h to (· · · ((g1
(7)
By the other hand, if (· · · ((g1
∈ ,
(· · · ((g1
= θgn ∈
and
(h<(−gn ))<(−gn−1 ) = (· · · ((g1
{h, −gn , −gn−1 } is a connection from h to (· · · ((g1
= θh ∈
∈ then
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A.J. Calderón Martín / Linear Algebra and its Applications 447 (2014) 110–118
115
(h<(−h))<(θ−gn−1 ) = (· · · ((g1
{h, −h, θ−gn−1 } is a connection from h to (· · · ((g1
(9)
Now, we can argue in a similar way either from the partial connection
{g1 , g2 , g3 , . . . , gn−1 } from g to (· · · ((g1
∈ , we denote by
g = {h ∈ : g and h are connected}. Proposition 2.2. Let g
∈ . Then the following assertions hold.
1. g is a subsupport. 2. If l ∈ satisfies that l h ∈ g .
∈ / g , then Ah Al = 0, (Ah A−h )Al = 0 and (Ah A−h )(Al A−l ) = 0 for any
Proof. (1) Given h ∈ g , there exists a connection {g1 , . . . , gn } from g to h. It is clear that in case −h ∈ , the set {g1 , . . . , gn } also connects g to −h and therefore −h ∈ g . Given h, k ∈ g such that h
(A0 )0 := spanK {Ag A−g : g ∈ 0 } ⊂ A0
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A.J. Calderón Martín / Linear Algebra and its Applications 447 (2014) 110–118
and V0
:=
A0
g ∈0
Ag . Finally we define the linear subspace
:= (A0 )0 ⊕ V0 .
Theorem 2.1. Suppose A0 1. For any g
=
g ∈
Ag A−g . Then the following assertions hold.
∈ , the linear subspace
Ag
= (A0 )g ⊕ Vg
of A associated to the subsupport g is an ideal of A. 2. If A is simple, then there exists a connection from g to h for any g , h
∈ .
Proof. 1. We have by Proposition 2.2 that Ah Al = 0, (Ah A−h )Al = 0 and (Ah A−h )(Al A−l ) / g . As we also have g is a subsupport we get h ∈ g and l ∈ ⎛
⎛
Ag A = Ag ⎝A0
⊕⎝
⎛⎛
= 0 for any
⎞⎞ Ah ⎠⎠
h∈
⎞ ⎛ ⎞ ⎛ ⎞⎞
⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎟ = Ag ⎜ Ag A−g + Ag A−g ⎠⊕⎝ Ag ⎠ ⊕ ⎝ Ag ⎠⎠ ⊂Ag . ⎝⎝ g ∈g
2. The simplicity of A implies Ag Theorem 2.2. Suppose A0
A
=
[g ]∈ /∼
=
g ∈
g ∈ / g
g ∈g
g ∈ / g
= A and therefore g = .
Ag A−g . Then we have
I[g ] ,
where any I[g ] is one of the ideals Ag of A described in Theorem 2.1-1, satisfying I[g ] I[h]
= 0 if [g ] = [h].
Proof. By Proposition 2.1, we can consider the quotient set / ∼:= {[g ] : g ∈ }. Let us denote by I[g ] := Ag .We have I[g ] is well defined and by Theorem 2.1-1 an ideal of A. Therefore A
=
[g ]∈ /∼
I[g ] .
By applying Proposition 2.2-2 we also obtain I[g ] I[h] Let us denote by Z (A) Corollary 2.1. If A0 Theorem 2.2, A
=
[g ]∈ /∼
=
I[g ] .
= 0 if [g ] = [h].
= {x ∈ A : xA = 0} the center of A.
g ∈
Ag A−g and Z (A)
= 0, then A is the direct sum of the ideals given in
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Proof. We just have to verify the direct character of the sum, but this follows from the facts I[g ] I[h] if [g ] = [h], and Z (A) = 0.
= 0,
3. The simple components In this section we study if any of the components in the decomposition given in Corollary 2.1 is simple. Under certain conditions we give an affirmative answer. We begin by introducing the concepts of maximal length and -multiplicativity in the framework of graded commutative algebras in a similar way than in the frameworks of graded Lie algebras, graded Leibniz algebras, graded associative algebras, graded Lie superalgebras, split color Lie algebras etc. (see [4,10,7,5,8,9,11,17]). Definition 3.1. We say that a graded commutative algebra A is of maximal length if A0 dim Ag = 1 for any g ∈ .
= 0 and
Definition 3.2. We say that a graded commutative algebra A is -multiplicative if given g , h ∈ ∪ {0} are such that g
∈ implies −g ∈ . From now
Theorem 3.1. Let A be a -multiplicative graded commutative algebra of maximal length and such that Ag A−g . Then A is simple if and only if it has all its elements in the support Z (A) = 0 and A0 = g ∈
connected. Proof. The first implication is Theorem 2.1-2. To prove the converse, consider I = Ig a nonzero g ∈G (I ∩ Ag )) where I := {g ∈ : I ∩ Ag = 0}. By the graded ideal of A and write I = (I ∩ A0 ) ⊕ ( g ∈I maximal length of A we can write I = I0 ⊕ ( Ag ). Observe that I = ∅ since in the opposite case g ∈I Ag A−g , there exist g , h ∈ such that Ag A−g ⊂ I and 0 = I ⊂ A0 and, since Z (A) = 0 and A0 = g ∈
(Ag A−g )(Ah A−h ) = 0. Then we would have by -multiplicativity that 0 = Ah ⊂ I , a contradiction. From here, we can take g0 ∈ I being so 0
= Ag 0 ⊂ I .
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For any g ∈ , g = ±g0 , the fact that g0 and g are connected gives us a connection {g1 , g2 , . . . , gn } from g0 to g satisfying Definition 2.1. Consider g0 = g1 , g2 and g1
• In the first possibility, 0 = Ag0 Ag2 = Ag0
= A((···((g1
and either Ag
⊂ I or A−(g ) ⊂ I for any g ∈ . The fact A0 =
g ∈
Ag A−g gives us now
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A.J. Calderón Martín / Linear Algebra and its Applications 447 (2014) 110–118
A0
⊂ I.
From here, given any g ∈ , the -multiplicativity of A together with its maximal length allow us to assert Ag = A0 Ag ⊂ I for any g ∈ and so I = A. That is, A is simple. Theorem 3.2. Let A be a -multiplicative graded commutative algebra of maximal length and such that Ag A−g . Then A is the direct sum of the family of its minimal ideals. Each one Z (A) = 0 and A0 = g ∈
being a simple graded commutative algebra having all of the elements in its support connected. I[g ] is the direct sum of the ideals I[g ] = (A0 )g ⊕ Vg = Proof. By Corollary 2.1, A = [g ]∈ /∼ Ah A−h ) ⊕ ( Ah ) having any I[g ] its support, g , with all of its elements connected. We have ( h∈[g ]
h∈[g ]
that g has all of its elements g -connected, (connected through elements related to g ). We also have that any of the I[g ] is -multiplicative as consequence of the -multiplicativity of A. Clearly I[g ] is of maximal length, and finally ZI[g ] (I[g ] ) = 0, (where ZI[g ] (I[g ] ) denotes de center I[g ] in I[g ] ), as consequence of I[g ] I[h] = 0 if [g ] = [h], (Theorem 2.2), and Z (A) = 0. We can apply Theorem 3.1 to I[g ] satisfies the any I[g ] so as to conclude I[g ] is simple. It is clear that the decomposition A = assertions of the theorem.
[g ]∈ /∼
Remark 3.1. We note that it is possible to develop the results of the paper for graded anti-commutative algebras in a analogous way. Hence the results in this paper can be applied in particular to the frameworks of graded Jordan and graded Lie algebras. Acknowledgments We would like to thank the referee for the detailed reading of this work and for the suggestions which have improved the final version of the same. References [1] A. Alburquerque, S. Majid, Clifford algebras by twisting of groups algebras, J. Pure Appl. Algebra 171 (2002) 133–148. [2] E. Aljadeff, D. Haile, M. Natapov, Graded identities of matrix algebras and the universal graded algebra, Trans. Amer. Math. Soc. 362 (6) (2010) 3125–3147. [3] Y. Bahturin, M. Brešar, Lie gradings on associative algebras, J. Algebra 321 (1) (2009) 264–283. [4] A.J. Calderón, On the structure of graded Lie algebras, J. Math. Phys. 50 (10) (2009) 103513, 8 pp [5] A.J. Calderón, On graded associative algebras, Rep. Math. Phys. 69 (1) (2012) 75–86. [6] A.J. Calderón, C. Draper, C. Martín, Gradings on the real forms of the Albert algebra, of g 2 , and of f 4 , J. Math. Phys. 51 (5) (2010) 053516, 21 pp [7] A.J. Calderón, J.M. Sánchez, Split Leibniz algebras, Linear Algebra Appl. 436 (6) (2012) 1648–1660. [8] A.J. Calderón, J.M. Sánchez, On the structure of graded Lie superalgebras, Modern Phys. Lett. A 27 (25) (2012) 1250142, 18 pp [9] A.J. Calderón, J.M. Sánchez, On the structure of split Lie color algebras, Linear Algebra Appl. 436 (2) (2012) 307–315. [10] A.J. Calderón, J.M. Sánchez, On the structure of graded Leibniz algebras, Algebra Colloq., in press. [11] M. Chaves, D. Singleton, Phantom energy from graded algebras, Modern Phys. Lett. A 22 (1) (2007) 29–40. [12] C. Draper, C. Martín, Gradings on g 2 , Linear Algebra Appl. 418 (1) (2006) 85–111. [13] G. Dahl, The doubly graded matrix cone and Ferrers matrices, Linear Algebra Appl. 368 (2003) 171–190. [14] A. Elduque, Fine gradings on simple classical Lie algebras, J. Algebra 324 (12) (2010) 3532–3571. [15] H. Grosse, G. Reiter, Graded differential geometry of graded matrix algebras, J. Math. Phys. 40 (12) (1999) 6609–6625. [16] M. Havlíˇcek, J. Patera, E. Pelantová, On Lie gradings II, Linear Algebra Appl. 277 (1998) 97–125. [17] M. Kochetov, Gradings on finite-dimensional simple Lie algebras, Acta Appl. Math. 108 (1) (2009) 101–127. [18] W. Marcinek, Graded algebras and geometry based on Yang–Baxter operators, J. Math. Phys. 33 (5) (1992) 1631–1635. [19] S. Morier-Genoud, V. Ovsienko, Simple graded commutative algebras, J. Algebra 323 (6) (2010) 1649–1664. [20] C. Nastasescu, F. Van Oystaeyen, Methods of Graded Rings, Springer-Verlag, Berlin, 2004. [21] J. Patera, H. Zassenhaus, On Lie gradings I, Linear Algebra Appl. 112 (1989) 87–159.