2. Notations and Preliminaries
Throughout this paper 𝕂 is a commutative ring, and all 𝕂-modules M are such that for all m∈ M, 2m = 0 implies m = 0.
be algebras over ground commutative ring 𝕂. Unadorned tensor product will denote the tensor product over 𝕂. For modules M
in A B
, symbols M
* , *M
* denote right dual, left dual and bidual of M
, and A B
) denotes the 𝕂-module of (A
)-bimodule maps M
. In what follows we shall concentrate on right dual of M
but similar observations can be made for the left dual as well.
For all ϕ
), let ϕ
* : N
* → M
* denote the right adjoint of ϕ i.e
) := g
We denote by (·)op : A → Aop the canonical anti-algebra isomorphism from the algebra A into its opposite Aop (which is the identity on the underlying 𝕂-modules), i.e., a = aop as module elements and (aa´)op = a´opaop for all a, a´∈ A.
The following facts are well known, but we recall them to set up the notation:
(i) If M
* ∈ Aop Bop
) = bf
Assume that M
is also finitely generated projective as a right B
, there exists a dual basis
, such that for any m
(ii) The mapping κM
→ M**, κM
) = f
is an isomorphism in A B
, with the inverse
. In fact κ
is a natural morphism between identity functor in A B
and the functor ()**
: A B
→ A B
(iii) If N
* ⊗ Bop N
* → (M ⊗ B N
)*, given by κM,N
⊗ Bop g
) = g
), is an isomorphism in Aop Cop
with the inverse
(iv) Let M
∈ A B
∈ B C
, P ∈C D
, where A
are algebras. Then the following diagram is commutative:
(v) Let M
∈ A B
be finitely generated projective as B
-module, with dual basis
, and let N
be finitely generated projective as a C
-module with dual basis
. Then M
⊗ B N
is finitely generated projective as a C
-module with a dual basis
The following terminology and theorems concerning corings and ring extensions are needed in this paper. For a review on coalgebras see: [5
]. For a review on corings see [3
Definition 2.1 C
is called a B-coring
if there exist morphisms ΔC
⊗ B C
In the sequel we shall use Sweedler’s notation ΔC
) = c(1)
⊗ B c(2)
. Given B
, a map ϕ
) is called a morphism of B-corings
) ◦ ΔC
. The category of B
-corings is denoted by CrgB
is called an extension of a ring B
if there exists an injective unital ring morphism
. Observe that
. Given ring extensions
, a ring morphism α :
is called a morphism of ring extensions
if α ◦
or, equivalently, if α ∈ B B
). The category of ring extensions of B
is denoted by RgeB
The full subcategory of CrgB (resp. RgeB) consisting of those B-corings (resp. ring extensions of B) that are finitely generated projective as right B-modules is denoted by r.f.g.pCrgB (resp. r.f.g.pRgeB).
Lemma 2.3 (i) If C
∈ CrgB then C
* ∈Bop Bopis a ring extension of Bop with multiplication
unit 1C* := εC and embedding map
(ii) If ϕ : C→D is any coring morphism then ϕ*: D* →C* is a ring extension morphism.
∈ r.f.g.pRgeB then * is a Bop-coring with comultiplication and counit
where is a (finite) dual basis of . (iv) If ϕ
→S is a morphism of right finitely generated projective ring extensions of B, then
: S* → * is a morphism of Bop-corings.
→ r.f.g.pRgeB is equivalent to the identity functor on r.f.g.pRgeB. For all
** is a ring extension isomorphism facilitating this equivalence.
(vi) Functor ()**: r.f.g.pCrgB → r.f.g.pCrgB is equivalent to the identity functor on r.f.g.pCrgB. For all C∈ r.f.g.pCrgB, κC : C→C** is a B-coring isomorphism facilitating this equivalence.
The statements (i) and (ii) are contained in Proposition 3.2 [8
], while (iii) and (v) are rephrasings of Theorem 3.7 [8
] (cf. [3
) Consider any ring extension morphism ϕ
be any finite dual basis of
, and let
be any finite dual basis of S
. For all s
Hence ϕ* is a coring map.
) It is enough to prove that κC
, is a coring map for any C
. Let C
be a B
-coring, and let
, be any finite dual basis of C
. Observe that
is a dual basis of C
*. Indeed, for any g
Hence, for all c∈C
Corollary 2.4 ()* is a duality functor between r.f.g.pRgeB and r.f.g.pCrgBop
3. An Extension for the Duality between Corings and Ring Extensions
Our aim in this section is to extend the duality between right finitely generated projective ring extensions and corings to the category of right finitely generated projective generalized Yang–Baxter structures.
We use the following terminology concerning the Yang–Baxter equation. Some references on this topic are: [9
Let B be a 𝕂-algebra. Given a (B, B)-bimodule V and a (B, B)-bilinear map R : V ⊗ B V → V ⊗ B V we write R12 = R ⊗ B id, R23 = id ⊗ B R : V ⊗ B V ⊗ B V → V ⊗ B V ⊗ B V where id : V → V is the identity map.
Definition 3.1 An invertible (B, B)-linear map R : V ⊗ B V → V ⊗ B V is called a generalized Yang–Baxter operator (or simply a generalised YB operator ) if it satisfies the equation
Definition 3.2 For an algebra B, we define the category YB strB whose objects are 4-tuples (V, φ, e, ε), where
(i) V is a (B, B)-bimodule;
(ii) φ : V ⊗ B V → V ⊗ B V is a generalized YB operator;
(iii) e∈ V such that for all b∈ B, eb = be, and for all x∈ V , φ(x ⊗ e) = e ⊗ B x, φ(e ⊗ B x) = x ⊗ B e;
(iv) ε : V → B is a (B, B)-bimodule map, such that (id ⊗ B ε) ◦φ = ε ⊗ B id, (ε ⊗ B id) ◦φ = id ⊗ B ε.
A morphism f :(V, φ, e, ε) → (V’, φ’,e’,ε’) in the category YB strB is a (B, B)-bilinear map f : V → V’ such that:
(v) (f ⊗ B f) ◦φ = φ´◦ (f ⊗ B f),
(vi) f(e) = e´,
(vii) ε´ ◦ f = ε.
Composition of morphisms is defined as the standard composition of B-linear maps. A full subcategory of YB strB consisting of all such (V, φ, e, ε) for which V is finitely generated projective as a right B-module is defined by r.f.g.pYBstrB.
Remark 3.3 Let R : V ⊗ B V → V ⊗ B V be a generalised YB operator . Then (V, R, 0, 0) is an object in the category YB strB.
Theorem 3.4 (i) There exists a functor:
Any ring extension map f is simply mapped into a (B, B) bimodule map.
(ii) F is a full and faithful embedding.
The proof that φR
is a generalised YB operator is left to the reader (cf. Proposition 2.1 from [12
, 0) is an object in the category YB strB
be a morphism of ring extensions. Then f
and 0 ◦ f = 0. Moreover
is a morphism in the category YB strB
) If F
= F S
, for some
, then obviously
, and the only thing which can differ is the multiplication. Denote by · the multiplication in
, and by ◦ the multiplication in S
. Then, as φR
, for all r
Multiplying tensor factors on both sides of this equation (whether using multiplication in
is irrelevant) yields 2(r
´) = 0, hence r
´, and so
as algebras. Therefore F
is an embedding.
Obviously, distinct ring extension maps are also distinct as (B, B)-bimodule morphisms, hence F is a faithful functor.
be a morphism in YB strB
. Then f
is unital, and
, hence, for all r
Multiplying factors in tensor products in both sides of the above equation yields 2(f(rr´) − f(r)f(r´)) = 0, hence f(rr´) = f(r)f(r´) and, as f is a (B, B)-bimodule map, it is a ring extension map. Therefore, F is a full functor.
Theorem 3.5 (i) There exists a functor
A coring morphism is mapped into a (B, B)-bimodule morphism.
(ii) G is a full and faithful embbeding.
The proof that ψC
is a generalised YB operator (cf. Proposition 2.3 from [12
]) is left to the reader (ψC−1
). Furthermore, for all c
(c ⊗ B
0) = 0 = 0 ⊗ B c
(0 ⊗ B c
) = 0 = c ⊗ B
0. Moreover, for all c
Hence (C,ψC, 0,εC) is an object in YB strB. Let f : C→D be any morphism of B-corings. Then f is also a (B, B)-bimodule morphism, f(0) = 0, εD ◦ f = εC, and,
Therefore f :(C,ψC, 0,εC) → (D,ψD, 0,εD) is a morphism in YB strB.
(ii) Suppose that GC = GD for some B-corings C, D. This means that C = D as (B, B)-bimodules, εC = εD, and the only things which can differ are comultiplications. However, as ψC = ψD, we have
(ΔC − ΔD) ⊗ B εC = −εC ⊗ B (ΔC − ΔD)
Composing both sides of the above equation with ΔC yields 2(ΔC − ΔD) = 0 hence ΔC = ΔD and C = D as (B, B)-corings. Hence G is an embedding.
Obviously distinct B-coring morphisms are also distinct as (B, B)-bimodule morphisms, hence G is a faithful functor.
Let f :(C,ψC, 0,εC) → (D,ψD, 0,εD), where C, D are corings, be a morphism in YB strB. Then (B, B)-bimodule morphism f : C→D is counital, i.e., εD ◦ f = εC . Furthermore, (f ⊗ B f) ◦ ψC = ψD ◦ (f ⊗ B f), and hence (f ⊗ B f) ◦ ψC ◦ ΔC = ψD ◦ (f ⊗ B f) ◦ ΔC . Observe that ψC ◦ ΔC = ΔC . Therefore
i.e., 2(f ⊗ B f) ◦ ΔC = 2ΔD ◦ f, hence (f ⊗ B f) ◦ ΔC = ΔD ◦ f, and f is a B-coring map. Therefore G is full.
Proposition 3.6 Let (V, R, e, ε) ∈ r.f.g.pYB strB. Then
where e†(f) = f(e), and
is a natural isomorphism in r.f.g.pYB strB.
is invertible, hence
We shall prove that R†
satisfies the Yang–Baxter equation. Observe that
Indeed, let Γ ∈ (V
⊗ B V
*, and let
be a dual basis of V
Similarly we can prove the other equality. By virtue of (17,18), we can write
Hence R† is a generalised YB operator .
Proofs of bilinearity of e* and centrality of ε are the same as proofs of analogues properties of duals of units and counits in Lemma 2.4. Moreover, for all f∈ V * ,
Furthermore, for all x = f ⊗ Bop g∈ V * ⊗ Bop V*,
Hence (V*,R†, ε, e†) ∈ r.f.g.pYB strBop.
: () → ()** is natural in B B
, and as V is finitely generated projective, κV
is invertible. Therefore it suffices to prove that κV
is a morphism in r.f.g.pYB strB
. To this end, observe first that
κV (e) = f → f(e)op = e†
and, for all υ∈ V ,
is a dual basis of V
*. Therefore, for all Γ ∈ (V
* ⊗ Bop V
and so, for all υ, υ´ ∈ V ,
Therefore, κV is a morphism in r.f.g.pYB strB as required.
Proposition 3.7 Let
∈ r.f.g.pRgeB, C
∈ r.f.g.pCrgB. Then i.e.,
From Lemma 2.4 we know that 1R†
. Furthermore, for all c
Similarly, for all r, r´∈C* , rr´ = κC,C(r ⊗ Bop r´) ◦ ΔC, therefore for all r, r´∈C* ,
This completes the proof.
Remark 3.8 Put together the statements of Theorem 3.6, Theorem 3.5, Proposition 3.6 and Proposition 3.7, can be summarized in the following diagram:
This means that the duality between right finitely generated projective ring extensions of B and B corings extends to the category r.f.g.pYB strB.