n -ary comodules over n -ary (co)algebras
In the paper we study connections between (co)modules over n-ary and binary (co)algebras.
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irk-123456789-1533742019-06-15T01:26:21Z n -ary comodules over n -ary (co)algebras Zekovic, B. In the paper we study connections between (co)modules over n-ary and binary (co)algebras. 2008 Article n -ary comodules over n -ary (co)algebras / B. Zekovic// Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 4. — С. 80–89. — Бібліогр.: 7 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20N15, 20C05, 20C07, 16S34. http://dspace.nbuv.gov.ua/handle/123456789/153374 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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n -ary comodules over n -ary (co)algebras |
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n
-ary comodules over n
-ary (co)algebras / B. Zekovic// Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 4. — С. 80–89. — Бібліогр.: 7 назв. — англ. |
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Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 4. (2008). pp. 80 – 89
c© Journal “Algebra and Discrete Mathematics”
n-ary comodules over n-ary (co)algebras
B. Zeković
Communicated by V. A. Artamonov
Abstract. In the paper we study connections between
(co)modules over n-ary and binary (co)algebras.
Introduction
In this paper, the notations of right (left) (co)modules over (co)algeb-
ras are generalized from binary to n-ary case Definition 1.1 and 1.2 It
is proved that a module M is a right n-ary C-comodule if and only if
it is a left n-ary C∗-module Theorem 2.1. A dual statement is proved
in Theorem 3.1. Notations of right (left) n-ary (co)module algebras are
introduced and it is proved that an n-ary C-comodule algebra is an n-ary
C∗-module algebra, Theorem 2.2. Moreover, n-ary C-module algebra is
an n-ary C∗-comodule algebra Theorem 3.3.
All necessary notations and definitions can be founded in the papers
listed in References.
1. Basic notions
Let k be a ground commutative associative ring with a unit, C and M
modules over k. In what follows, ⊗ is a tensor product over k. All
homomorphisms are k-linear maps. In [Z1], the concept of n-ary algebras
(C, m) is defined, where m : C ⊗ · · · ⊗ C → C is n-ary multiplication,
2000 Mathematics Subject Classification: 20N15, 20C05, 20C07, 16S34.
Key words and phrases: right(left) n-ary (co)-module over (co)algebra,
right(left) n-ary (co)-module algebra.
B. Zeković 81
which is associative. It means that the following diagram is commutative:
C⊗(2n−1)
1⊗i
C
⊗m⊗1
⊗(n−i−1)
C
��
m⊗1
⊗(n−1)
C // C⊗n
m
��
C⊗n
m
// C
i.e. for any i = 1, . . . , n we have
m ·
(
m ⊗ 1
⊗(n−1)
C
)
= m ·
(
1⊗i
C ⊗ m ⊗ 1
⊗(n−i−1)
C
)
.
The concept of n-ary coalgebra (C,∆) is defined in [Z2], where ∆ : C →
C ⊗ · · · ⊗ C is n-ary comultiplication, which is coassociative, that is the
following diagram is commutative:
C
∆
��
∆ // C⊗n
1⊗i
C
⊗∆⊗1
⊗(n−i−1)
C
��
C⊗n
∆⊗1
⊗(n−1)
C
// C⊗(2n−1)
i.e. for any i = 1, . . . , n we have
(
∆ ⊗ 1
⊗(n−1)
C
)
· ∆ =
(
1⊗i
C ⊗ ∆ ⊗ 1
⊗(n−i−1)
C
)
· ∆.
Analogously, the concept of n-ary bialgebra (C, m,∆) , where m is as-
sociative n-ary multiplication and ∆ is coassociative n-ary comultiplica-
tion. It is shown that ∆ is a homomorphism of n-ary algebras. We do
not suppose an existence of the unit and counit. In [Z1] the notion of
homomorphism f : (C, mC) → (C ′, mC′) of n-ary algebras is defined as a
morphism f : C → C ′, such that the following diagram is commutative:
C⊗n
mC
��
f⊗n
// C ′⊗n
m
C′
��
C
f
// C ′
i.e. f · mC = mC′ · f⊗n.
Let C be an n-ary coalgebra and a finitely generated projective k-
module. Denote by C∗ the k-module Hom(C, k). Then C∗ is an n-ary
algebra with multiplication l1 ∗ · · · ∗ ln, where for c ∈ C:
(l1 ∗ · · · ∗ ln) (c) =
∑
c
l1
(
c(1)
)
· · · ln
(
c(n)
)
(1)
82 n-ary comodules over n-ary (co)algebras
if
∆(c) =
∑
c
c(1) ⊗ · · · ⊗ c(n) ∈ C⊗n.
Conversely, let C be an n-ary algebra and a finitely generated projective
k-module. Define n-ary comultiplication ∆∗ in C∗ = Hom(C, k), by the
rule:
(∆∗)(x1 ⊗ · · · ⊗ xn) = l(x1 · · ·xn) (2)
where xi ∈ C. Here we use the isomorphism of k-modules:
(
C⊗n
)∗
≃ (C∗)⊗n
because C is a finitely-generated projective k-module. Then, C∗ is a
n-ary coalgebra. If C is a n-ary (co)algebra, then (C∗)∗ ≃ C, [B].
Definition 1.1. Let C be a n-ary coalgebra. We say that k-module M is
a right n-ary C-comodule, if there exists a map ρ : M → M ⊗ C⊗(n−1),
such that the following diagram is commutative:
M
ρ
��
ρ // M ⊗ C⊗(n−1)
1M⊗1⊗i
C
⊗∆⊗1
⊗(n−i−2)
C
��
M ⊗ C⊗(n−1)
ρ⊗1
⊗(n−1)
C
// M ⊗ C⊗(2n−1)
i.e.
(
1M ⊗ 1⊗i
C ⊗ ∆ ⊗ 1
⊗(n−i−2)
C
)
· ρ =
(
ρ ⊗ 1
⊗(n−1)
C
)
· ρ
for any i.
Definition 1.2. Let C be a n-ary algebra. k-module M is called a left
n-ary C-module, if there exists a map γ : C⊗(n−1) ⊗ M → M , such that
the following diagram is commutative:
C⊗(n−1) ⊗ M
γ // M
C⊗(2n−1) ⊗ M
1
⊗(n−1)
C
⊗γ
OO
1⊗i
C
⊗m⊗1
⊗(n−i−2)
C
⊗1M
// C⊗(n−1) ⊗ M
γ
OO
i.e.
γ ·
(
1
⊗(n−1)
C ⊗ γ
)
= γ ·
(
1⊗i
C ⊗ m ⊗ 1
⊗(n−i−2)
C ⊗ 1M
)
for any i = 1, . . . , n.
B. Zeković 83
Definition 1.3. Let C be a n-ary bialgebra. n-ary algebra M is called a
left n-ary C-module algebra, if:
1) M is a left n-ary C-module;
2) for any c1, . . . , cn−1 ∈ C, m1, . . . , mn ∈ M
(c1 ⊗ · · · ⊗ cn−1) (m1 · · ·mn)
=
∑
c1,...,cn−1
(
c1(1) · · · cn−1(1)m1
)
· · ·
(
c1(n) · · · cn−1(n)mn
)
Definition 1.4. Let C be an n-ary bialgebra. n-ary algebra M is called
a right n-ary C-comodule algebra, if:
1) M is a right n-ary C-comodule with the structure morphism ρ : M →
M ⊗ C⊗(n−1);
2) ρ is a homomiorphism of n-ary algebras.
2. The relations between n-ary comodules and modules
Theorem 2.1. Let C be a n-ary coalgebra. Then M is a right n-ary
C-comodule if and only if M is a left n-ary C∗-module.
Proof. Suppose that M is a right n-ary C-comodule and
ρ(m) =
∑
(m)
m(0) ⊗ m(1) ⊗ · · · ⊗ m(n−1), (3)
where m ∈ M and m(0) ∈ M, m(i) ∈ C for = 1, . . . , n−1. If l1, . . . , ln−1 ∈
C∗, m ∈ M , then we put
ρ∗ (l1 ⊗ · · · ⊗ ln−1 ⊗ m) =
∑
(m)
m(0)l1
(
m(1)
)
· · · sln−1
(
m(n−1)
)
∈ M. (4)
Further, if l1, . . . , l2n−2 ∈ C∗ and m ∈ M , then by Definition 1.2 we have:
(
ρ∗ ·
(
1
⊗(n−1)
C∗ ⊗ ρ∗
))
(l1 ⊗ · · · ⊗ l2n−2 ⊗ m)
= ρ∗
l1 ⊗ · · · ⊗ ln−1 ⊗
∑
(m)
m(0)ln
(
m(1)
)
· · · l2n−2
(
m(n−1)
)
=
∑
m(0)
m(0)(0)l1
(
m(0)(1)
)
· · · ln−1
(
m(0)(n−1)
)
ln
(
m(1)
)
· · · l2n−2
(
m(n−1)
)
.
(5)
84 n-ary comodules over n-ary (co)algebras
On the other hand,
(
ρ∗ ·
(
1
⊗(i−1)
C∗ ⊗ m∗ ⊗ 1
⊗(n−1−i)
C∗ ⊗ 1M
))
(l1 ⊗ · · · ⊗ l2n−2 ⊗ m)
= ρ∗ (l1 ⊗ · · · ⊗ li−1 ⊗ (li ∗ · · · ∗ li+n−1) ⊗ ln+i ⊗ · · · l2n−2 ⊗ m)
=
∑
m
m(0)l1
(
m(1)
)
· · · li−1
(
m(i−1)
)
(l1 ∗ · · · ∗ li+n−1)×
(
m(i)
)
li+n
(
m(i+1)
)
· · · l2n−2
(
m(n−1)
)
=
∑
m
m(0)l1
(
m(1)
)
· · · li−1
(
m(i−1)
)
×
∑
m(i)
l1
(
m(i)(1)
)
· · · li+n−1
(
m(i)(n)
)
li+n
(
m(i+1)
)
· · · l2n−2
(
m(n−1)
)
.
(6)
By Definition 1.2 we have:
∑
m
m(0)(0) ⊗ m(0)(1) ⊗ · · · ⊗ m(0)(n−1) ⊗ m(1) ⊗ · · · ⊗ m(n−1)
=
∑
m
m(0) ⊗ · · · ⊗ m(i−1) ⊗ m(i)(0) ⊗ · · ·
· · · ⊗ m(i)(n−1) ⊗ m(i+1) ⊗ · · · ⊗ m(n−1).
It proves the equallities (5) and (6), i.e. M is a left n-ary C∗-module.
Since, the module C is a finite-generated projective k-module, then
(C∗)∗ ≃ C and therefore the converse statement follows.
Theorem 2.2. Let C be an n-ary bialgebra. If M is a right n-ary C-
comodule algebra, then M is a left n-ary C∗-module algebra.
Proof. Suppose that M is a right n-ary C-comodule algebra and C is an
n-ary bialgebra. We shall show that M is left n-ary C∗-module algebra
with respect to the action (4). It is necessary to prove the following
equality:
ρ∗ [l1 ⊗ · · · ⊗ ln−1 ⊗ (m1 · · ·mn)]
=
∑
l1,...,ln−1
ρ∗
(
l(1)(1) ⊗ · · · ⊗ l(n−1)(1) ⊗ m1
)
×
· · · × ρ∗
(
l(1)(n) ⊗ l(n−1)(n) ⊗ mn
)
.
By (3) and (4), we have:
ρ∗ [l1 ⊗ · · · ⊗ ln−1 ⊗ (m1 · · ·mn)]
=
∑
(m1 · · ·mn)(0) l1
(
(m1 · · ·mn)(1)
)
· · · ln−1
(
(m1 · · ·mn)(n−1)
)
.
(7)
B. Zeković 85
But, the map ρ : M → M⊗C⊗(n−1) is a homomorphism of n-ary algebras,
so:
ρ(m1 · · ·mn) =
∑
(m1 · · ·mn)(0) ⊗ · · · ⊗ (m1 · · ·mn)(n−1) ,
ρ(m1) · · · ρ(mn) =
(
∑
m1
m1(0) ⊗ · · · ⊗ m1(n−1)
)
· · ·
(
∑
mn
mn(0) ⊗ · · · ⊗ mn(n−1)
)
=
∑
m1,...,mn
(
m1(0) · · ·mn(0)
)
⊗ · · · ⊗
(
m1(n−1) ⊗ · · · ⊗ mn(n−1)
)
.
Consequently,
(m1 · · ·mn)(0) =
∑
m1(0) · · ·mn(0)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(m1 · · ·mn)(n−1) =
∑
m1(n−1) · · ·mn(n−1).
So in (7), we have:
ρ∗ (l1 ⊗ · · · ⊗ ln−1 ⊗ (m1 · · ·mn)
=
∑
(
m1(0) · · ·mn(0)
)
l1
(
m1(1) · · ·mn(1)
)
· · · ln−1
(
m1(n−1) · · ·mn(n−1)
)
=
∑
m,l
m1(0) · · ·mn(0)l1(1)
(
m1(1)
)
×
· · · l1(n)
(
mn(1)
)
· · · ln−1(1)
(
m1(n−1)
)
· · · ln−1(n)
(
mn(n−1)
)
.
But, it is equal to:
[
∑
m1
m1(0)l1(1)
(
m1(1)
)
· · · ln−1(1)
(
m1(n−1)
)
]
×
· · ·
[
∑
m1
mn(0)l1(n)
(
mn(1)
)
· · · ln−1(n)
(
mn(n−1)
)
]
.
3. Dual situation
Theorem 3.1. Let C be an n-ary algebra. Then M is a right n-ary
C-module M iff M is a left n-ary C∗-comodule.
Proof. Suppose that m ∈ M and the submodule mC⊗(n−1) ⊆ M is con-
tained in the span of linearly indepedent vectors a1, . . . , at . Then, for
c1, . . . , cn−1 ∈ C, we have:
m (c1 ⊗ cn−1) = f1 (c1 ⊗ · · · ⊗ cn−1) a1 + · · · + ft (c1 ⊗ · · · ⊗ cn−1) at (8)
86 n-ary comodules over n-ary (co)algebras
where f1, . . . , ft ∈
[
C⊗(n−1)
]∗
= (C∗)⊗(n−1)
, and so C is finitely gener-
ated. If the system of vectors {a1, . . . , at} can be enlarged by a vector
at+1 and then we can put ft+1 = 0.
Lemma 3.2. Let b1, . . . , bd ∈ 〈a1, . . . , at〉 be given and g1, . . . , gd ∈
(C∗)⊗(n−1)
, such that for all c1, . . . , cn−1 ∈ C:
m (c1 ⊗ · · · ⊗ cn−1) =
d
∑
i=1
gi (c1 ⊗ cn−1) bi.
Then
∑d
i=1 gi ⊗ bi =
∑t
j=1 fj ⊗ aj holds in (C∗)⊗(n−1) ⊗ M .
Proof. By the assumption bt =
∑t
j=1 αijaj where αij ∈ k. Then
m (c1 ⊗ · · · ⊗ cn−1) =
d
∑
i=1
gi (c1 ⊗ cn−1)
t
∑
j=1
αijaj
=
t
∑
j=1
[
d
∑
i=1
gi (c1 ⊗ · · · ⊗ cn−1) αij
]
aj .
Since a1, . . . , at are linearly indepedent, for all j by (8) we get:
fj (c1 ⊗ · · · cn−1) =
d
∑
i=1
gi (c1 ⊗ · · · ⊗ cn−1) αij .
Thus, fj =
∑d
i=1 gjαij in (C∗)⊗(n−1). Then
t
∑
j=1
fj ⊗ aj =
t
∑
j=1
d
∑
i=1
giαij ⊗ aj =
d
∑
i=1
gi ⊗
t
∑
j=1
αijaj
=
d
∑
i=1
gi ⊗ bi.
Define the map ρ : M → (C∗)⊗(n−1) ⊗M by the following rule: if (8)
holds, then we put
ρ(m) = f1 ⊗ a1 + · · · + ft ⊗ at. (9)
By Lemma 3.2 this definition is correct. Let us show now that if M is a
right n-ary C-module, then M is a left n-ary C∗-comodule with respect
to (9). In fact,
(
1
⊗(n−1)
C∗ ⊗ ρ
)
ρ(m) = f1 ⊗ ρ(m1) + · · · + ft ⊗ ρ(mt).
B. Zeković 87
But ρ(mi) =
∑
j fij ⊗ mj where fij ∈ (C∗)⊗(n−1). So
(
1
⊗(n−1)
C∗ ⊗ ρ
)
ρ(m) =
∑
i,j
fi ⊗ fij ⊗ mj .
On the other hand,
(
1⊗i
C∗ ⊗ ∆ ⊗ 1
⊗(n−i−2)
C∗ ⊗ 1M
)
ρ(m)
=
∑
j
(
1⊗i
C∗ ⊗ ∆ ⊗ 1
⊗(n−i−2)
C∗
)
fj ⊗ mj .
Further, by associativity Definition 1.2, for all c1, . . . , c2n−2 ∈ C and all
m ∈ M , for any i we have
[m (c1 ⊗ · · · ⊗ cn−1)] (cn ⊗ · · · ⊗ c2n−2)
= m [c1 ⊗ · · · ⊗ ci ⊗ (ci+1 ⊗ · · · ⊗ ci+n) ⊗ ci+n+1 ⊗ · · · ⊗ c2n−2]
(10)
Suppose that m1, . . . , mt be as above. Then, as in (8)
mj (cn ⊗ · · · ⊗ c2n−2)
= gj1 (cn ⊗ · · · ⊗ c2n−2)m1 + · · · + gjt (cn ⊗ · · · ⊗ c2n−2) mt
where gj1, . . . , gjt ∈ (C∗)⊗(n−1). So, by (8) and (9) , we have
(
1
⊗(n−1)
C∗ ⊗ ρ
)
ρ(m) =
∑
i,j
fi ⊗ gij ⊗ mj .
By (10) for all j:
∑
i
fi (c1 ⊗ · · · ⊗ cn−1) gij (cn ⊗ · · · ⊗ c2n−2)
= fj [c1 ⊗ · · · ⊗ ci ⊗ (ci+1 ⊗ · · · ⊗ ci+n) ⊗ ci+n+1 ⊗ · · · ⊗ c2n−2] .
In other words we have that in (C∗)⊗(2n−2) that
∑
i
fi ⊗ gij =
(
1⊗i
C∗ ⊗ ∆C∗ ⊗ 1
⊗(n−i−2)
C∗
)
fj .
Tensor-multiplying by mj and summing on j, we obtain
∑
i,j
fi ⊗ gij ⊗ mj =
(
1⊗i
C∗ ⊗ ∆C∗ ⊗ 1
⊗(n−i−2)
C∗
)
∑
j
fj ⊗ mj .
But, the left side is equal to
(
1
⊗(n−1)
C∗ ⊗ ρ
)
ρ(m) and right side is equal
to
(
1⊗i
C∗ ⊗ ∆C∗ ⊗ 1
⊗(n−i−2)
C∗
)
ρ(m). Consequently, M is a left n-ary C∗-
comodule.
88 n-ary comodules over n-ary (co)algebras
Theorem 3.3. Let C be an n-ary bialgebra, now. If M is a right n-ary
C-module algebra, then M is a left n-ary C∗-comodule algebra.
Proof. Assume that M is a right n-ary C-module algebra. It means that
for all c1, . . . , cn−1 ∈ C and all m1, . . . , mn ∈ M we have:
(m1 · · ·mn) (c1 ⊗ · · · ⊗ cn−1)
=
∑
c1,...,cn−1
m1
(
c1(1) ⊗ · · · ⊗ cn−1(1)
)
· · ·mn
(
c1(n) ⊗ · · · ⊗ cn−1(n)
) (11)
Let us show now that M is a left n-ary C∗-comodule algebra, i.e. the map
(9) is a homomorphism of n-ary algebras. Suppose that m1, . . . , mr ∈
M and a1, . . . , at is a base in
∑r
j=1 mjC
⊗(n−1). Assume that ρ(mi) =
∑t
j=1 fij ⊗ aj where fij ∈ (C∗)⊗(n−1) But,
∑
c1,...,cn−1
m1
(
c1(1) ⊗ · · · ⊗ cn−1(1)
)
· · ·mn
(
c1(n) ⊗ · · · ⊗ cn−1(n)
)
=
∑
c1,...,cn−1
∑
j1
f1,j1
(
c1(1) ⊗ · · · ⊗ cn−1(1)
)
aj1
times
· · · ×
∑
jn
f1,jn
(
c1(n) ⊗ · · · ⊗ cn−1(n)
)
ajn
∑
j1,...,jn
aj1 · · · ajn
[
∑
f1,j1
(
c1(1) ⊗ · · · ⊗ cn−1(1)
)
×
· · · × fn,jn
(
c1(n) ⊗ · · · ⊗ cn−1(n)
)]
For all j1, . . . , jn we have:
∑
c1,...,cn−1
f1,j1
(
c1(1) ⊗ · · · ⊗ cn−1(1)
)
· · · fn,jn
(
c1(n) ⊗ · · · ⊗ cn−1(n)
)
= (f1,j1 ∗ · · · ∗ fn,jn
) (c1 ⊗ · · · ⊗ cn−1) .
In that way, the right side of (11) is equal to:
∑
j1,...,jn
(f1,j1 ∗ · · · ∗ fn,jn
) (c1 ⊗ · · · ⊗ cn−1) aj1 · · · ajn
.
By Lemma 3.2 b∗ = aj1 · · · ajn
for all j1, . . . , jn and by (11), we obtain
that
ρ (m1 · · ·mn) =
∑
j1,...,jn
(f1,j1 ∗ · · · ∗ fn,jn
) ⊗ aj1 · · · ajn
=
n
∏
s=1
∑
js
(fs,js
⊗ ajs
)
= ρ(m1) · · · ρ(mn).
B. Zeković 89
References
[A] Artamonov V. A., Structure of Hopf algebra. Itogi Nauki I techniki, Algebra.
Topology, Geometry, v29, Moscow, Viniti, 1999, pp.3–63.
[Ab] Abe Eiichi, Hopf algebras, Cambridge University Press, Cambridge, 1980.
[M] Montgomery Susan, Hopf algebras and Teir Actions on Ring, Amer. Math. Soc.,
Providence, Rhode Island, Nat. Sci. Found, 1993.
[Z1] Zekovich B., On n-ary bialgebras (I), Tchebyshev sbornik 4 N3, 2003, pp.65–73.
[Z2] Zekovich B., On n-ary bialgebras (II), Tchebyshev sbornik 4 N3, 2003, pp.73–
80.
[B] Bourbaki N, Alge‘bre commutative, Hermann, Paris, 1961–1965.
[Z3] Zekovich B., Ternary Hopf algebras, Algebra and Discrete mathematics, N3,
2005, pp.96–106.
Contact information
B. Zeković Faculty of Science, University of Montene-
gro, P. O. Box 211, 81000 Podgorica, Mon-
tenegro
E-Mail: biljanaz@cg.yu
Received by the editors: 03.09.2008
and in final form 02.10.2008.
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