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The sovereign structure on categories of entwined modules
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Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa
The sovereign structure on categories of entwined modules ✩ Ling Jia Department of Information and Statistics, Ludong University, Yantai, Shandong 264025, China
a r t i c l e
i n f o
Article history: Received 12 April 2016 Received in revised form 3 July 2016 Available online xxxx Communicated by S. Kovács
a b s t r a c t In this paper we give sufficient and necessary conditions for which a category of entwined modules is a sovereign monoidal category. © 2016 Elsevier B.V. All rights reserved.
1. Introduction It is well-known that monoidal category theory is one of the key parts in theory of knots and quantum groups whose important role can be best summarized in the following words [8] “the category of framed tangles (or tangles on ribbons) is the free tortile (or ribbon) category generated by an object”. The notion of sovereign monoidal category was introduced by Freyd and Yetter [4,9] and what interests us is Deligne’s theorem which states that there is a twist on an autonomous braided monoidal category if and only if there is a sovereign structure on it. The more basic notions of tensor and monoidal category can be found in [10]. [6] investigated an algebraic structure on Hopf algebras corresponding to sovereign structures and introduced notion of cosovereign Hopf algebra. Motivated by the need for developing a general theory of non-commutative principal bundles, entwining structures were introduced [13] as generalized symmetries of such bundles. The categories associated to them unify many categories of modules well studied by Hopf-algebraists such as the categories of Hopf modules ([11]), relative Hopf modules, Doi–Hopf modules, Yetter–Drinfeld modules and modules graded by G-sets. Apart from being more general, entwining structures have a remarkable self-duality property, which essentially implies that for every statement involving the module structure of an entwined module there is a corresponding statement involving its comodule structure. The more preliminary results and the corresponding Smash product algebras defined for entwining structures can be found in [2,5,13,14]. [7] introduced notion of a monoidal entwining structure making the respective category of entwined modules into a braided monoidal category. The aim of this paper is to find sufficient and necessary conditions for it being a sovereign monoidal category. ✩
Project (No. ZR2012AL02) supported by the Natural Science Foundation of Shandong Province of China. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.jpaa.2016.08.008 0022-4049/© 2016 Elsevier B.V. All rights reserved.
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2. Convention Throughout the paper k denotes a field. We use the Sweedler notation for comultiplication and comodule. For a coalgebra C, Δ(c) = c1 ⊗c2 . For any right C-comodule M , ρ(m) = m(0) ⊗m(1) . For any k-space V , we denote idV the identity map from V to itself. In this paper all k-spaces are finite-dimensional. We assume the readers to be proficient in the theory of monoidal categories and Hopf algebras. We begin with autonomous monoidal categories and refer the reader to [1–3,13] for other basic notions. Definition 1. ([2]) A monoidal category is said to be left autonomous if and only if every object has a left dual. A left autonomous structure on a left autonomous monoidal category C is the choice for every object X of a left dual (X ∗ , εX , ηX ) such that I ∗ = I and εI = ηI = 1I . Definition 2. ([2]) A sovereign structure on a left autonomous category C consists in the choice of a left dual for each object of C (and hence a strong monoidal functor ?∗∗ : C → C) together with a monoidal morphism ς : 1C →?∗∗ . Note that, a sovereign structure ς : 1C →?∗∗ is automatically an isomorphism. A sovereign category is a left autonomous category endowed with an equivalence class of sovereign structures. Let C be a sovereign category, with chosen left duals (X ∗ , εX , ηX ) and sovereign isomorphisms ςX : X → −1 X ∗∗ . For each object X of C, set εX = εX ∗ (ςX ⊗ idX ∗ ) and ηX = (idX ∗ ⊗ ςX )ηX ∗ , then (X ∗ , εX , ηX ) is a right dual of X. Therefore C is autonomous. Definition 3. ([12]) A right–left entwining structure (A, C, ψ) over k consists of an algebra A, a coalgebra C ψ and a k-linear map ψ : C ⊗ A → A ⊗ C denoted by ψ(c ⊗ a) = a ⊗ cψ verifying the following conditions for any a, b ∈ A and c ∈ C:
ψ
(ab) ⊗ cψ =
aψ ⊗ cψ1 ⊗ cψ2
aψ bφ ⊗ cψφ . = aψφ ⊗ c1ψ ⊗ c2φ .
(1) (2)
1ψ ⊗ cψ = 1 ⊗ c.
(3)
aψ ε(cψ ) = ε(c)a.
(4)
If the map ψ occurs more than once in the same expression, then we use notations ψ, φ, δ, · · ·. In addition, if A and C are both bialgebras and the following equations are also satisfied, then (A, C, ψ) is called a monoidal entwining structure:
aψ 1 ⊗ aψ 2 ⊗ (cd)ψ =
1ψ ε(aψ ) = ε(a)1, a ∈ A.
a1 ψ ⊗ a2 φ ⊗ cψ dφ , a ∈ A, c, d ∈ C.
(5) (6)
Proposition 4. Let (A, C, ψ) be a right–left monoidal entwining structure. Then C ∗ A is a bialgebra with the i ψ multiplication given by (f a)(g b) = f u a bg(uiψ ), this leads to (f 1)(ε a) = f a. The comultiplication is defined by Δ(f a) = f1 a1 ⊗ f2 a2 , the counit ε(f a) = f (1)ε(a), f, g ∈ C ∗ , a, b ∈ A, ui is a basis of i C and u is the corresponding dual basis of C ∗ . In addition if A endowed with the antipode T and C endowed with the antipode R are both Hopf algebras, ψ then C ∗ A is a Hopf algebra with the antipode defined by S(f a) = ui T (a) f (R(uiψ )), f ∈ C ∗ , a ∈ A. Proof. First we prove that it is an associative algebra with the unit εC 1. In fact, for any f, g, h ∈ C ∗ , a, b, c ∈ A,
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(1)
[(f a)(g b)](h c) = (f a)[(g b)(h c)] =
3
f ui uj aψφ bδ cg(uiψ )h(ujφδ ).
(7)
f uj aφ bψ c(guj )(ujφ )h(uiψ ).
(8)
Applying x ∈ C on the first tensorand of the tensor product in (8), we obtain that
f (x1 )uj (x2 )aφ bψ cg(ujφ1 )h(uiψ )ui (ujφ2 ) = f (x1 )uj (x2 )aφδ bψ cg(uj1φ )h(uiψ )ui (uj2δ ) = f (x1 )aφδ bψ cg(x2φ )h(x3δψ ). (2)
Applying x ∈ C on the first tensorand of the tensor product in (7), we obtain that f (x1 )aφδ bψ cg(x2φ )h(x3δψ ). Hence [(f a)(g b)](h c) = (f a)[(g b)(h c)]. (4) (3) Also we have (f a)(ε 1) = f ui aψ ε(uiψ ) = f a and (ε 1)(f a) = ui 1ψ af (uiψ ) = f a. Next we prove that it is a bialgebra. It suffices to verify Δ and ε are algebra morphisms. In fact, for any f, g ∈ C ∗ , a, b ∈ A,
Δ((f a)(g b)) = Δ(f a)Δ(g b) =
f1 ui 1 aψ 1 b1 ⊗ f2 ui 2 aψ 2 b2 g(uiψ ). f1 ui a1 ψ b1 ⊗ f2 uj a2 φ b2 g1 (uiψ )g2 (ujφ ).
(9) (10)
Applying x ∈ C and y ∈ C on the first and third tensorand of the tensor product in (9) respectively, we obtain that
f1 (x1 )ui 1 (x2 )f2 (y1 )ui 2 (y2 )aψ 1 b1 ⊗ aψ 2 b2 g(uiψ ) (5) f (x1 y1 )a1 ψ b1 ⊗ a2 φ b2 g(x2ψ y2φ ) = f (x1 y1 )aψ 1 b1 ⊗ aψ 2 b2 g((x2 y2 )ψ ) = ψ φ i j = f1 (x1 )u (x2 )f2 (y1 )u (y2 )a1 b1 ⊗ a2 b2 g1 (uiψ )g2 (ujφ ). Applying x ∈ C and y ∈ C on the first and third tensorands of the tensor product in (10) respectively, we obtain that
f1 (x1 )ui (x2 )f2 (y1 )uj (y2 )a1 ψ b1 ⊗ a2 φ b2 g1 (uiψ )g2 (ujφ ).
So Δ((f a)(g b)) = Δ(f a)Δ(g b). (6)
And ε((f a)(g b)) = (f ui )(1)ε(aψ b)g(uiψ ) = ε(f a)ε(g b). Finally we have to show that it is a Hopf algebra. In fact, for any f ∈ C ∗ , a ∈ A, S((f a)1 )(f a)2 =
ψδ
f1 (R(uiψ ))ui uj T (a1 ) a2 f2 (ujδ ).
(11)
By naturally thinking of the two sides of (11) as elements of C ∗ A via the unit k → C ∗ A and applying x ∈ C on the first tensorand of the tensor product in (11) we obtain that
ψδ
ui (x1 )uj (x2 )T (a1 ) a2 f (R(uiψ )ujδ ) = (2) (4) ψ = T (a1 ) a2 f (R(xψ1 )xψ2 ) = ε(f a).
ψδ
T (a1 ) a2 f (R(x1ψ )x2δ )
Hence S((f a)1 )(f a)2 = ε(f a). Similarly one can verify that (f a)1 S((f a)2 ) = ε(f a). Thus we complete the proof. 2
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Lemma 5. Let (A, C, ψ) be a right–left monoidal entwining structure. A and C are both Hopf algebras with the antipode T and R respectively. Then for any a ∈ A, x ∈ C,
φ
T (aψ ) ⊗ R(xφ )ψ = T (a) ⊗ R(x).
(12)
Proof. Indeed, for any f ∈ C ∗ , a ∈ A, consider the equation S((ε a)(f 1)) = S(f 1)S(ε a). S((ε a)(f 1)) =
φ
uj T (aψ ) ui (R(ujφ ))f (uiψ ).
φ S(f 1)S(ε a) = (ui 1ψ f (R(uiψ )))(uj T (a) ε(ujφ )) (3),(4) i = (u f (R(ui )))(uj T (a)ε(uj )) = (f ◦ R 1)(ε T (a)) = f ◦ R T (a).
(13) (14)
Applying x ∈ C on the first tensorand of the tensor product in (13) we obtain that φ φ φ T (aψ ) f (R(xφ )ψ ). Therefore T (aψ ) ⊗R(xφ )ψ = T (a) ⊗R(x). 2 uj (x)T (aψ ) ui (R(ujφ ))f (uiψ ) =
Definition 6. Let (A, C, ψ) be a right–left entwining structure. A k-space M together with a right A-action (via · : M ⊗ A → A) and a left C-coaction (via ρ : M → C ⊗ M , m → m(−1) ⊗ m(0) ) is called an entwined module over (A, C, ψ) if for any m ∈ M , a ∈ A, ρ(m · a) = C
m(−1)ψ ⊗ m(0) · aψ .
(15)
M (ψ)A denotes the category of entwined modules.
Example 7. Let (A, C, ψ) be a right–left monoidal entwining structure. A and C are both Hopf algebras with the bijective antipode T and R respectively. Let M be a entwined module over (A, C, ψ). Then M ∗ = Homk (M, k) is an entwined module through the action given by (α · a)(m) = α(m · T −1 (a)) and the coaction given by α(0) (m)α(−1) = α(m(0) )R−1 (m(−1) ). And ∗ M = Homk (M, k) is an entwined module through the action given by (α · a)(m) = α(m · T (a)) and the coaction given by α<0> (m)α<−1> = α(m<0> )R(m<−1> ), m ∈ M , α ∈ Homk (M, k), a ∈ A. Obviously ∗ M is both a right A-module and a left C-comodule. We only need to prove that the compatible condition holds. In fact, for any m ∈ M , α ∈ M ∗ , a ∈ A, (α · a)<0> (m)(α · a)<−1> = α(m<0> · T (a))R(m<−1> ) (12) φ = α(m<0> · T (aψ ) )R(m<−1>φ )ψ (15) = α((m · T (aψ ))<0> )R((m · (T (aψ ))<−1> )ψ = (α<0> · aψ )(m)α(<−1>ψ .
Similarly one can check that M ∗ is also an entwined module over (A, C, ψ). Example 8. Let (A, C, ψ) be a right–left entwining structure with C being finite dimensional. Then C ⊗A ∈C M (ψ)A under the action given by (c⊗a)·b = cψ ⊗abψ and the coaction given by (c ⊗ a)(−1) ⊗ (c ⊗ a)(0) = c1 ⊗ c2 ⊗ a. And A ⊗ C ∈C M (ψ)A with the action defined by (a ⊗ c) ◦ b = ab ⊗ c and the coaction defined by (a ⊗ c)(−1) ⊗ (a ⊗ c)(0) = c1ψ ⊗ aψ ⊗ c2 , a, b ∈ A, c ∈ C. First C ⊗ A is clearly a left C-comodule. It suffice to check that it is a right A-module and satisfies the equation (15). Indeed, for any a, b, x ∈ A, c ∈ C,
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((c ⊗ a) · b) · x =
(1)
cψφ ⊗ abψ xφ =
5
ψ
cψ ⊗ a(bx) = (c ⊗ a) · (bx).
(3) (c ⊗ a) · 1 = cψ ⊗ a1ψ = c ⊗ a. ((c ⊗ a) · b)(−1) ⊗ ((c ⊗ a) · b)(0) = cψ1 ⊗ cψ2 ⊗ abψ (2) = c1ψ ⊗ c2φ ⊗ abψφ = (c ⊗ a)(−1)ψ ⊗ (c ⊗ a)(0) · bψ . Next A ⊗ C is clearly a right A-module. We have to show that it is a left C-comodule and satisfies the equation (15). Indeed, for any a, b, x ∈ A, c ∈ C,
(a ⊗ c)(−1)1 ⊗ (a ⊗ c)(−1)2 ⊗ (a ⊗ c)(0) (2) = c ⊗ c1ψ2 ⊗ aψ ⊗ c2 = c1ψ ⊗ c2φ ⊗ aψφ ⊗ c3 1ψ1 = (a ⊗ c)(−1) ⊗ (a ⊗ c)(0)(−1) ⊗ (a ⊗ c)(0)(0) .
ε((a ⊗ c)(−1) )(a ⊗ c)(0) =
(4)
ε(c1ψ )aψ ⊗ c2 = a ⊗ c. ψ c1ψ ⊗ (ax) ⊗ c2 ((a ⊗ c) · x)(−1) ⊗ ((a ⊗ c) · x)(0) = (1) = c1ψφ ⊗ aψ xφ ⊗ c2 = (a ⊗ c)(−1)ψ ⊗ (a ⊗ c)(0) · xψ . Theorem 9. Let (A, C, ψ) be a right–left entwining structure with C being finite dimensional. Then there exists an isomorphism of C M (ψ)A ∼ = MC ∗ A . Proof. If M ∈C M (ψ)A , then M is a right C ∗ A-module via the action given by m ·(f a) = f (m(−1) )m(0) ·a, a ∈ A, m ∈ M , f ∈ C ∗ . Since for any a, b ∈ A, m ∈ M , f, g ∈ C ∗ , (15) (m · (f a)) · (g b) = m(0)(0) · aψ bf (m(−1) )g(m(0)(−1)ψ ) = m(0) · aψ bf (m(−1)1 )g(m(−1)2ψ ) = (f ui )(m(−1) )m(0) · aψ bg(uiψ ) = m · [(f a)(g b)]. m · (ε 1) = ε(m(−1) )m(0) = m.
Conversely if M ∈ MC ∗ A , then it naturally induces a right A-action and a left C-coaction. That is m · a = m · (ε a) and ui ⊗ m · (ui 1), a ∈ A, m ∈ M . It suffices to prove that the m(−1) ⊗ m(0) = equation (15) holds. Indeed, for any a ∈ A, m ∈ M , (m · a)(−1) ⊗ (m · a)(0) = ui ⊗ m · (uj aψ ui (ujψ )) (4) uiψ ⊗ m · (ui 1)(ε aψ ) = m(−1)ψ ⊗ m(0) · aψ . = uiψ ⊗ m · (ui aψ ) =
The others are trivial and thus we complete the proof. 2 ∗
Let (A, C, ψ) be a right–left monoidal entwining structure. Define two maps χ : N at(F ◦∗ (), F ◦ () ) → ∗ Homk (C, A) and κ : Homk (C, A) → N at(F ◦∗ (), F ◦ () ) as follows respectively: ∗
χ(α)(c) = (αA⊗C (ti ⊗ εC ))(1A ⊗ c)ti , α ∈ N at(F ◦∗ (), F ◦ () ), c ∈ C, where ti is a basis of A and ti is the corresponding dual basis of A∗ and (κ(f )(g))(m) = f ∈ Homk (C, A), g ∈∗ M , m ∈ M . Lemma 10. The two maps χ and κ are mutual inverses.
g(m(0) ·f (m(−1) )),
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Proof. It is trivial to prove that κ(f ) is a natural transformation. Now we claim that χ is the inverse of κ. Indeed, for any f ∈ Homk (C, A), c ∈ C, χ(κ(f ))(c) = (κ(f )A⊗C (ti ⊗ εC ))(1 ⊗ c)ti = (ti ⊗ εC )((1 ⊗ c)(0) · f ((1 ⊗ c)(−1) ))ti = (ti ⊗ εC )((1ψ ⊗ c2 ) · f (c1ψ ))ti (3) i = (t ⊗ εC )(f (c1 ) ⊗ c2 )ti = f (c). ∗
And for any α ∈ N at(F ◦∗ (), F ◦ () ), g ∈∗ M , m ∈ M by using naturality with respect to the map of the form A ⊗ C → M ⊗ C given by a ⊗ c → m · a ⊗ c, κ(χ(αM ))(g)(m) = g(m(0) · χ(α)(m(−1) )) = g(m(0) · ti )αA⊗C (ti ⊗ ε)(1 ⊗ m(−1) ) = αM ⊗C (g ⊗ ε)(ρ(m)) = αM (∗ ρ(g ⊗ ε))(m) = αM (g)(m). Therefore we complete the proof. 2 ∗
In what follows α ∈ N at(F ◦∗ (), F ◦ () ) and f ∈ Homk (C, A) are the corresponding elements as mentioned above. Lemma 11. α is monoidal if and only if the following equations hold for any c, d ∈ C, Δ(f (cd)) = f (c) ⊗ f (d).
(16)
ε(f (1)) = 1.
(17)
Proof. If the conditions hold, then for any M, N ∈C M (ψ)A , m ∈ M , n ∈ N , g ∈∗ M , h ∈∗ N , αM ⊗N (g ⊗ h)(m ⊗ n) = (g ⊗ h)((m(0) ⊗ n(0) ) · f (m(−1) n(−1) )) = (g ⊗ h)(m(0) · f (m(−1) n(−1) )1 ⊗ n(0) · f (m(−1) n(−1) )2 ) = (g ⊗ h)(m(0) · f (m(−1) ) ⊗ n(0) · f (n(−1) )) = (αM ⊗ αN )(g ⊗ h)(m ⊗ n). By the equation (3) k is an object of C M (ψ)A via the trivial action and coaction. It is clearly αk = idk if and only if ε(f (1)) = 1. Therefore α is monoidal. Conversely we take M = N = C ⊗ A, then for any c, d ∈ C, θ, λ ∈∗ A, αC⊗A⊗C⊗A (ε ⊗ θ ⊗ ε ⊗ λ)(c ⊗ 1 ⊗ d ⊗ 1) = (ε ⊗ θ ⊗ ε ⊗ λ)((c2 ⊗ 1) · f (c1 d1 )1 ⊗ (d2 ⊗ 1) · f (c1 d1 )2 ) = θ(f (cd)1 )λ(f (cd)2 ) = (θ ⊗ λ)(Δ(f (cd))). (αC⊗A ⊗ αC⊗A )((ε ⊗ θ) ⊗ (ε ⊗ λ))(c ⊗ 1 ⊗ d ⊗ 1) = (ε ⊗ θ)((c2 ⊗ 1) · f (c1 )) ⊗ (ε ⊗ λ)((d2 ⊗ 1) · f (d1 )) (4)
= θ(f (c))λ(f (d)) = (θ ⊗ λ)(f (c) ⊗ f (d)).
Hence Δ(f (cd)) = f (c) ⊗ f (d).
2
Lemma 12. Let (A, C, ψ) be a right–left monoidal entwining structure. If A is also a Hopf algebra with the bijective antipode T , then α is an A-linear map if and only if for any c ∈ C, a ∈ A,
ψ
T −1 (a) f (cψ ) = f (c)T (a).
(18)
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Proof. If the equation (18) holds, for any M ∈C M (ψ)A , λ ∈∗ M , m ∈ M , a ∈ A, (18) ψ αM (λ · a)(m) = λ(m(0) · f (m(−1) )T (a)) = λ(m(0) · T −1 (a) f (m(−1)ψ )) = λ((m · T −1 (a))(0) f ((m · T −1 (a))(−1) )) = αM (λ)(m · T −1 (a)) = (αM (λ) · a)(m). Conversely we consider C ⊗ A ∈C M (ψ)A , then for any c ∈ C, a ∈ A, θ ∈∗ A, (αC⊗A (ε ⊗ θ) · a)(c ⊗ 1) = αC⊗A ((ε ⊗ θ) · a)(c ⊗ 1), ψ ψ (αC⊗A (ε ⊗ θ) · a)(c ⊗ 1) = (ε ⊗ θ)((cψ ⊗ T −1 (a) )(0) · f ((cψ ⊗ T −1 (a) )(−1) )) ψ ψ φ = (ε ⊗ θ)(cψ2φ ⊗ T −1 (a) f (cψ1 ) ) = θ(T −1 (a) f (cψ )). αC⊗A ((ε ⊗ θ) · a)(c ⊗ 1) = (ε ⊗ θ)((c2 ⊗ f (c1 )) · T (a)) ψ (4) = (ε ⊗ θ)(c2ψ ⊗ f (c1 )T −1 (a) ) = θ(f (c)T (a)). Hence
ψ
T −1 (a) f (cψ ) = f (c)T (a).
2
Dual to Lemma 12 by interchanging the roles of A and C we have the following lemma. Lemma 13. Let (A, C, ψ) be a right–left monoidal entwining structure. If C is also a Hopf algebra with the bijective antipode R, then α is C-colinear if and only if for any c ∈ C,
R(c2ψ ) ⊗ f (c1 )ψ =
R(c1 ) ⊗ f (c2 ).
(19)
Definition 14. Let (A, C, ψ) be a monoidal entwining structure such that A and C are both Hopf algebras. If there exists a k-linear map f : C → A satisfying the equations (16)–(19), then we call (A, C, ψ) a sovereign entwining structure and f is a generalized sovereign map in C M (ψ)A . Directly from Example 7, Example 8 and Lemmas 12–13 we have the following result. Theorem 15. Let (A, C, ψ) be a monoidal entwining structure with C being finite dimensional. A and C are both Hopf algebras with the bijective antipodes T and R respectively. Then C M (ψ)A is a sovereign category if and only if (A, C, ψ) is a sovereign entwining structure. Let H be a Hopf algebra with the antipode S. From [6] we know that a sovereign element of H is a group-like element π such that S 2 (h) = πhπ −1 for all h ∈ H. A sovereign Hopf algebra is a pair (H, π) where H is a Hopf algebra and π is a sovereign element of H. Moreover there is a fact that sovereign structures on the category of modules over H (being finite-dimensional) correspond to sovereign elements in H. Corollary 16. Let (A, C, ψ) be a monoidal entwining structure. Then C M (ψ)A is a sovereign category if and only if C ∗ A is a sovereign Hopf algebra. Proof. By Theorem 9 there exists an isomorphism between C M (ψ)A and MC ∗op A . If C M (ψ)A is a sovereign i category, then there is a generalized sovereign map f : C → A. It is very easy to prove that u ⊗ f (ui ) is the sovereign element in C ∗op A. Conversely if C ∗op A is a sovereign Hopf algebra with a sovereign element g ⊗ a ∈ C ∗op A, then it is trivial to verify that the map f : C → A, f (c) = g(c)a is the generalized sovereign map. 2
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Example 17. Let H be a sovereign Hopf algebra with the sovereign element . By [14] it is trivial to prove ψ that (H, H, ψ), ψ : H ⊗ H → H ⊗ H, h ⊗ gψ = h2 ⊗ S −1 (h3 )gh1 , h, g ∈ H is an entwining structure and H M (ψ)H is exactly H Y DH . Then it is very easy to verify that the map ζ : H → H, ζ(h) = ε(h) satisfies the conditions in Theorem 15 and so H Y DH is a sovereign monoidal category. References [1] [2] [3] [4]
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
A. Joyal, R. Street, Braided tensor categories, Adv. Math. 102 (1993) 20–73. A. Bruguières, A. Virelizier, Hopf monads, Adv. Math. 215 (2007) 679–733. C. Kassel, Quantum Groups, Grad. Texts Math., vol. 155, Springer, Berlin, 1995. D.N. Yetter, Framed tangles and a theorem of Deligne on braided deformations of Tannakian categories, in: Deformation Theory and Quantum Groups with Applications to Mathematical Physics, in: Contemp. Math., vol. 134, American Mathematical Society, Providence, RI, 1992, pp. 325–345. D. Hobst, B. Pareigis, Double quantum groups, J. Algebra 242 (2) (2001) 460–494. Julien Bichon, Cosovereign Hopf algebras, J. Pure Appl. Algebra 157 (2001) 121–133. L. Jia, Braided monoidal category of entwined modules, Acta Math. Sci. 29 (5) (2009) 1307–1311 (in Chinese). M.C. Shum, Tortile tensor categories, J. Pure Appl. Algebra 93 (1) (1993) 57–110. P. Freyd, D.N. Yetter, Coherence theorems via knot theory, J. Pure Appl. Algebra 78 (1992) 49–76. P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik, Tensor Categories, Math. Surv. Monogr., vol. 205, AMS, 2015, 344 pp. S. Montgonery, Hopf Algebras and Their Actions on Rings, CBMS Reg. Conf. Ser. Math., vol. 82, AMS, 1993. S. Caenepeel, E. De Groot, Modules over weak entwining structures, Contemp. Math. 267 (2000). T. Brzezinski, S. Majid, Coalgebra bundles, Commun. Math. Phys. 191 (1998) 467–492. T. Brzezinski, R. Wisbauer, Corings and Comodules, Lond. Math. Soc. Lect. Note Ser., vol. 309, Cambridge University Press, Mathematics, 2003, 476 pp.
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The sovereign structure on categories of entwined modules ✩ Ling Jia Department of Information and Statistics, Ludong University, Yantai, Shandong 264025, China
a r t i c l e
i n f o
Article history: Received 12 April 2016 Received in revised form 3 July 2016 Available online xxxx Communicated by S. Kovács
a b s t r a c t In this paper we give sufficient and necessary conditions for which a category of entwined modules is a sovereign monoidal category. © 2016 Elsevier B.V. All rights reserved.
1. Introduction It is well-known that monoidal category theory is one of the key parts in theory of knots and quantum groups whose important role can be best summarized in the following words [8] “the category of framed tangles (or tangles on ribbons) is the free tortile (or ribbon) category generated by an object”. The notion of sovereign monoidal category was introduced by Freyd and Yetter [4,9] and what interests us is Deligne’s theorem which states that there is a twist on an autonomous braided monoidal category if and only if there is a sovereign structure on it. The more basic notions of tensor and monoidal category can be found in [10]. [6] investigated an algebraic structure on Hopf algebras corresponding to sovereign structures and introduced notion of cosovereign Hopf algebra. Motivated by the need for developing a general theory of non-commutative principal bundles, entwining structures were introduced [13] as generalized symmetries of such bundles. The categories associated to them unify many categories of modules well studied by Hopf-algebraists such as the categories of Hopf modules ([11]), relative Hopf modules, Doi–Hopf modules, Yetter–Drinfeld modules and modules graded by G-sets. Apart from being more general, entwining structures have a remarkable self-duality property, which essentially implies that for every statement involving the module structure of an entwined module there is a corresponding statement involving its comodule structure. The more preliminary results and the corresponding Smash product algebras defined for entwining structures can be found in [2,5,13,14]. [7] introduced notion of a monoidal entwining structure making the respective category of entwined modules into a braided monoidal category. The aim of this paper is to find sufficient and necessary conditions for it being a sovereign monoidal category. ✩
Project (No. ZR2012AL02) supported by the Natural Science Foundation of Shandong Province of China. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.jpaa.2016.08.008 0022-4049/© 2016 Elsevier B.V. All rights reserved.
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2. Convention Throughout the paper k denotes a field. We use the Sweedler notation for comultiplication and comodule. For a coalgebra C, Δ(c) = c1 ⊗c2 . For any right C-comodule M , ρ(m) = m(0) ⊗m(1) . For any k-space V , we denote idV the identity map from V to itself. In this paper all k-spaces are finite-dimensional. We assume the readers to be proficient in the theory of monoidal categories and Hopf algebras. We begin with autonomous monoidal categories and refer the reader to [1–3,13] for other basic notions. Definition 1. ([2]) A monoidal category is said to be left autonomous if and only if every object has a left dual. A left autonomous structure on a left autonomous monoidal category C is the choice for every object X of a left dual (X ∗ , εX , ηX ) such that I ∗ = I and εI = ηI = 1I . Definition 2. ([2]) A sovereign structure on a left autonomous category C consists in the choice of a left dual for each object of C (and hence a strong monoidal functor ?∗∗ : C → C) together with a monoidal morphism ς : 1C →?∗∗ . Note that, a sovereign structure ς : 1C →?∗∗ is automatically an isomorphism. A sovereign category is a left autonomous category endowed with an equivalence class of sovereign structures. Let C be a sovereign category, with chosen left duals (X ∗ , εX , ηX ) and sovereign isomorphisms ςX : X → −1 X ∗∗ . For each object X of C, set εX = εX ∗ (ςX ⊗ idX ∗ ) and ηX = (idX ∗ ⊗ ςX )ηX ∗ , then (X ∗ , εX , ηX ) is a right dual of X. Therefore C is autonomous. Definition 3. ([12]) A right–left entwining structure (A, C, ψ) over k consists of an algebra A, a coalgebra C ψ and a k-linear map ψ : C ⊗ A → A ⊗ C denoted by ψ(c ⊗ a) = a ⊗ cψ verifying the following conditions for any a, b ∈ A and c ∈ C:
ψ
(ab) ⊗ cψ =
aψ ⊗ cψ1 ⊗ cψ2
aψ bφ ⊗ cψφ . = aψφ ⊗ c1ψ ⊗ c2φ .
(1) (2)
1ψ ⊗ cψ = 1 ⊗ c.
(3)
aψ ε(cψ ) = ε(c)a.
(4)
If the map ψ occurs more than once in the same expression, then we use notations ψ, φ, δ, · · ·. In addition, if A and C are both bialgebras and the following equations are also satisfied, then (A, C, ψ) is called a monoidal entwining structure:
aψ 1 ⊗ aψ 2 ⊗ (cd)ψ =
1ψ ε(aψ ) = ε(a)1, a ∈ A.
a1 ψ ⊗ a2 φ ⊗ cψ dφ , a ∈ A, c, d ∈ C.
(5) (6)
Proposition 4. Let (A, C, ψ) be a right–left monoidal entwining structure. Then C ∗ A is a bialgebra with the i ψ multiplication given by (f a)(g b) = f u a bg(uiψ ), this leads to (f 1)(ε a) = f a. The comultiplication is defined by Δ(f a) = f1 a1 ⊗ f2 a2 , the counit ε(f a) = f (1)ε(a), f, g ∈ C ∗ , a, b ∈ A, ui is a basis of i C and u is the corresponding dual basis of C ∗ . In addition if A endowed with the antipode T and C endowed with the antipode R are both Hopf algebras, ψ then C ∗ A is a Hopf algebra with the antipode defined by S(f a) = ui T (a) f (R(uiψ )), f ∈ C ∗ , a ∈ A. Proof. First we prove that it is an associative algebra with the unit εC 1. In fact, for any f, g, h ∈ C ∗ , a, b, c ∈ A,
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(1)
[(f a)(g b)](h c) = (f a)[(g b)(h c)] =
3
f ui uj aψφ bδ cg(uiψ )h(ujφδ ).
(7)
f uj aφ bψ c(guj )(ujφ )h(uiψ ).
(8)
Applying x ∈ C on the first tensorand of the tensor product in (8), we obtain that
f (x1 )uj (x2 )aφ bψ cg(ujφ1 )h(uiψ )ui (ujφ2 ) = f (x1 )uj (x2 )aφδ bψ cg(uj1φ )h(uiψ )ui (uj2δ ) = f (x1 )aφδ bψ cg(x2φ )h(x3δψ ). (2)
Applying x ∈ C on the first tensorand of the tensor product in (7), we obtain that f (x1 )aφδ bψ cg(x2φ )h(x3δψ ). Hence [(f a)(g b)](h c) = (f a)[(g b)(h c)]. (4) (3) Also we have (f a)(ε 1) = f ui aψ ε(uiψ ) = f a and (ε 1)(f a) = ui 1ψ af (uiψ ) = f a. Next we prove that it is a bialgebra. It suffices to verify Δ and ε are algebra morphisms. In fact, for any f, g ∈ C ∗ , a, b ∈ A,
Δ((f a)(g b)) = Δ(f a)Δ(g b) =
f1 ui 1 aψ 1 b1 ⊗ f2 ui 2 aψ 2 b2 g(uiψ ). f1 ui a1 ψ b1 ⊗ f2 uj a2 φ b2 g1 (uiψ )g2 (ujφ ).
(9) (10)
Applying x ∈ C and y ∈ C on the first and third tensorand of the tensor product in (9) respectively, we obtain that
f1 (x1 )ui 1 (x2 )f2 (y1 )ui 2 (y2 )aψ 1 b1 ⊗ aψ 2 b2 g(uiψ ) (5) f (x1 y1 )a1 ψ b1 ⊗ a2 φ b2 g(x2ψ y2φ ) = f (x1 y1 )aψ 1 b1 ⊗ aψ 2 b2 g((x2 y2 )ψ ) = ψ φ i j = f1 (x1 )u (x2 )f2 (y1 )u (y2 )a1 b1 ⊗ a2 b2 g1 (uiψ )g2 (ujφ ). Applying x ∈ C and y ∈ C on the first and third tensorands of the tensor product in (10) respectively, we obtain that
f1 (x1 )ui (x2 )f2 (y1 )uj (y2 )a1 ψ b1 ⊗ a2 φ b2 g1 (uiψ )g2 (ujφ ).
So Δ((f a)(g b)) = Δ(f a)Δ(g b). (6)
And ε((f a)(g b)) = (f ui )(1)ε(aψ b)g(uiψ ) = ε(f a)ε(g b). Finally we have to show that it is a Hopf algebra. In fact, for any f ∈ C ∗ , a ∈ A, S((f a)1 )(f a)2 =
ψδ
f1 (R(uiψ ))ui uj T (a1 ) a2 f2 (ujδ ).
(11)
By naturally thinking of the two sides of (11) as elements of C ∗ A via the unit k → C ∗ A and applying x ∈ C on the first tensorand of the tensor product in (11) we obtain that
ψδ
ui (x1 )uj (x2 )T (a1 ) a2 f (R(uiψ )ujδ ) = (2) (4) ψ = T (a1 ) a2 f (R(xψ1 )xψ2 ) = ε(f a).
ψδ
T (a1 ) a2 f (R(x1ψ )x2δ )
Hence S((f a)1 )(f a)2 = ε(f a). Similarly one can verify that (f a)1 S((f a)2 ) = ε(f a). Thus we complete the proof. 2
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Lemma 5. Let (A, C, ψ) be a right–left monoidal entwining structure. A and C are both Hopf algebras with the antipode T and R respectively. Then for any a ∈ A, x ∈ C,
φ
T (aψ ) ⊗ R(xφ )ψ = T (a) ⊗ R(x).
(12)
Proof. Indeed, for any f ∈ C ∗ , a ∈ A, consider the equation S((ε a)(f 1)) = S(f 1)S(ε a). S((ε a)(f 1)) =
φ
uj T (aψ ) ui (R(ujφ ))f (uiψ ).
φ S(f 1)S(ε a) = (ui 1ψ f (R(uiψ )))(uj T (a) ε(ujφ )) (3),(4) i = (u f (R(ui )))(uj T (a)ε(uj )) = (f ◦ R 1)(ε T (a)) = f ◦ R T (a).
(13) (14)
Applying x ∈ C on the first tensorand of the tensor product in (13) we obtain that φ φ φ T (aψ ) f (R(xφ )ψ ). Therefore T (aψ ) ⊗R(xφ )ψ = T (a) ⊗R(x). 2 uj (x)T (aψ ) ui (R(ujφ ))f (uiψ ) =
Definition 6. Let (A, C, ψ) be a right–left entwining structure. A k-space M together with a right A-action (via · : M ⊗ A → A) and a left C-coaction (via ρ : M → C ⊗ M , m → m(−1) ⊗ m(0) ) is called an entwined module over (A, C, ψ) if for any m ∈ M , a ∈ A, ρ(m · a) = C
m(−1)ψ ⊗ m(0) · aψ .
(15)
M (ψ)A denotes the category of entwined modules.
Example 7. Let (A, C, ψ) be a right–left monoidal entwining structure. A and C are both Hopf algebras with the bijective antipode T and R respectively. Let M be a entwined module over (A, C, ψ). Then M ∗ = Homk (M, k) is an entwined module through the action given by (α · a)(m) = α(m · T −1 (a)) and the coaction given by α(0) (m)α(−1) = α(m(0) )R−1 (m(−1) ). And ∗ M = Homk (M, k) is an entwined module through the action given by (α · a)(m) = α(m · T (a)) and the coaction given by α<0> (m)α<−1> = α(m<0> )R(m<−1> ), m ∈ M , α ∈ Homk (M, k), a ∈ A. Obviously ∗ M is both a right A-module and a left C-comodule. We only need to prove that the compatible condition holds. In fact, for any m ∈ M , α ∈ M ∗ , a ∈ A, (α · a)<0> (m)(α · a)<−1> = α(m<0> · T (a))R(m<−1> ) (12) φ = α(m<0> · T (aψ ) )R(m<−1>φ )ψ (15) = α((m · T (aψ ))<0> )R((m · (T (aψ ))<−1> )ψ = (α<0> · aψ )(m)α(<−1>ψ .
Similarly one can check that M ∗ is also an entwined module over (A, C, ψ). Example 8. Let (A, C, ψ) be a right–left entwining structure with C being finite dimensional. Then C ⊗A ∈C M (ψ)A under the action given by (c⊗a)·b = cψ ⊗abψ and the coaction given by (c ⊗ a)(−1) ⊗ (c ⊗ a)(0) = c1 ⊗ c2 ⊗ a. And A ⊗ C ∈C M (ψ)A with the action defined by (a ⊗ c) ◦ b = ab ⊗ c and the coaction defined by (a ⊗ c)(−1) ⊗ (a ⊗ c)(0) = c1ψ ⊗ aψ ⊗ c2 , a, b ∈ A, c ∈ C. First C ⊗ A is clearly a left C-comodule. It suffice to check that it is a right A-module and satisfies the equation (15). Indeed, for any a, b, x ∈ A, c ∈ C,
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((c ⊗ a) · b) · x =
(1)
cψφ ⊗ abψ xφ =
5
ψ
cψ ⊗ a(bx) = (c ⊗ a) · (bx).
(3) (c ⊗ a) · 1 = cψ ⊗ a1ψ = c ⊗ a. ((c ⊗ a) · b)(−1) ⊗ ((c ⊗ a) · b)(0) = cψ1 ⊗ cψ2 ⊗ abψ (2) = c1ψ ⊗ c2φ ⊗ abψφ = (c ⊗ a)(−1)ψ ⊗ (c ⊗ a)(0) · bψ . Next A ⊗ C is clearly a right A-module. We have to show that it is a left C-comodule and satisfies the equation (15). Indeed, for any a, b, x ∈ A, c ∈ C,
(a ⊗ c)(−1)1 ⊗ (a ⊗ c)(−1)2 ⊗ (a ⊗ c)(0) (2) = c ⊗ c1ψ2 ⊗ aψ ⊗ c2 = c1ψ ⊗ c2φ ⊗ aψφ ⊗ c3 1ψ1 = (a ⊗ c)(−1) ⊗ (a ⊗ c)(0)(−1) ⊗ (a ⊗ c)(0)(0) .
ε((a ⊗ c)(−1) )(a ⊗ c)(0) =
(4)
ε(c1ψ )aψ ⊗ c2 = a ⊗ c. ψ c1ψ ⊗ (ax) ⊗ c2 ((a ⊗ c) · x)(−1) ⊗ ((a ⊗ c) · x)(0) = (1) = c1ψφ ⊗ aψ xφ ⊗ c2 = (a ⊗ c)(−1)ψ ⊗ (a ⊗ c)(0) · xψ . Theorem 9. Let (A, C, ψ) be a right–left entwining structure with C being finite dimensional. Then there exists an isomorphism of C M (ψ)A ∼ = MC ∗ A . Proof. If M ∈C M (ψ)A , then M is a right C ∗ A-module via the action given by m ·(f a) = f (m(−1) )m(0) ·a, a ∈ A, m ∈ M , f ∈ C ∗ . Since for any a, b ∈ A, m ∈ M , f, g ∈ C ∗ , (15) (m · (f a)) · (g b) = m(0)(0) · aψ bf (m(−1) )g(m(0)(−1)ψ ) = m(0) · aψ bf (m(−1)1 )g(m(−1)2ψ ) = (f ui )(m(−1) )m(0) · aψ bg(uiψ ) = m · [(f a)(g b)]. m · (ε 1) = ε(m(−1) )m(0) = m.
Conversely if M ∈ MC ∗ A , then it naturally induces a right A-action and a left C-coaction. That is m · a = m · (ε a) and ui ⊗ m · (ui 1), a ∈ A, m ∈ M . It suffices to prove that the m(−1) ⊗ m(0) = equation (15) holds. Indeed, for any a ∈ A, m ∈ M , (m · a)(−1) ⊗ (m · a)(0) = ui ⊗ m · (uj aψ ui (ujψ )) (4) uiψ ⊗ m · (ui 1)(ε aψ ) = m(−1)ψ ⊗ m(0) · aψ . = uiψ ⊗ m · (ui aψ ) =
The others are trivial and thus we complete the proof. 2 ∗
Let (A, C, ψ) be a right–left monoidal entwining structure. Define two maps χ : N at(F ◦∗ (), F ◦ () ) → ∗ Homk (C, A) and κ : Homk (C, A) → N at(F ◦∗ (), F ◦ () ) as follows respectively: ∗
χ(α)(c) = (αA⊗C (ti ⊗ εC ))(1A ⊗ c)ti , α ∈ N at(F ◦∗ (), F ◦ () ), c ∈ C, where ti is a basis of A and ti is the corresponding dual basis of A∗ and (κ(f )(g))(m) = f ∈ Homk (C, A), g ∈∗ M , m ∈ M . Lemma 10. The two maps χ and κ are mutual inverses.
g(m(0) ·f (m(−1) )),
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Proof. It is trivial to prove that κ(f ) is a natural transformation. Now we claim that χ is the inverse of κ. Indeed, for any f ∈ Homk (C, A), c ∈ C, χ(κ(f ))(c) = (κ(f )A⊗C (ti ⊗ εC ))(1 ⊗ c)ti = (ti ⊗ εC )((1 ⊗ c)(0) · f ((1 ⊗ c)(−1) ))ti = (ti ⊗ εC )((1ψ ⊗ c2 ) · f (c1ψ ))ti (3) i = (t ⊗ εC )(f (c1 ) ⊗ c2 )ti = f (c). ∗
And for any α ∈ N at(F ◦∗ (), F ◦ () ), g ∈∗ M , m ∈ M by using naturality with respect to the map of the form A ⊗ C → M ⊗ C given by a ⊗ c → m · a ⊗ c, κ(χ(αM ))(g)(m) = g(m(0) · χ(α)(m(−1) )) = g(m(0) · ti )αA⊗C (ti ⊗ ε)(1 ⊗ m(−1) ) = αM ⊗C (g ⊗ ε)(ρ(m)) = αM (∗ ρ(g ⊗ ε))(m) = αM (g)(m). Therefore we complete the proof. 2 ∗
In what follows α ∈ N at(F ◦∗ (), F ◦ () ) and f ∈ Homk (C, A) are the corresponding elements as mentioned above. Lemma 11. α is monoidal if and only if the following equations hold for any c, d ∈ C, Δ(f (cd)) = f (c) ⊗ f (d).
(16)
ε(f (1)) = 1.
(17)
Proof. If the conditions hold, then for any M, N ∈C M (ψ)A , m ∈ M , n ∈ N , g ∈∗ M , h ∈∗ N , αM ⊗N (g ⊗ h)(m ⊗ n) = (g ⊗ h)((m(0) ⊗ n(0) ) · f (m(−1) n(−1) )) = (g ⊗ h)(m(0) · f (m(−1) n(−1) )1 ⊗ n(0) · f (m(−1) n(−1) )2 ) = (g ⊗ h)(m(0) · f (m(−1) ) ⊗ n(0) · f (n(−1) )) = (αM ⊗ αN )(g ⊗ h)(m ⊗ n). By the equation (3) k is an object of C M (ψ)A via the trivial action and coaction. It is clearly αk = idk if and only if ε(f (1)) = 1. Therefore α is monoidal. Conversely we take M = N = C ⊗ A, then for any c, d ∈ C, θ, λ ∈∗ A, αC⊗A⊗C⊗A (ε ⊗ θ ⊗ ε ⊗ λ)(c ⊗ 1 ⊗ d ⊗ 1) = (ε ⊗ θ ⊗ ε ⊗ λ)((c2 ⊗ 1) · f (c1 d1 )1 ⊗ (d2 ⊗ 1) · f (c1 d1 )2 ) = θ(f (cd)1 )λ(f (cd)2 ) = (θ ⊗ λ)(Δ(f (cd))). (αC⊗A ⊗ αC⊗A )((ε ⊗ θ) ⊗ (ε ⊗ λ))(c ⊗ 1 ⊗ d ⊗ 1) = (ε ⊗ θ)((c2 ⊗ 1) · f (c1 )) ⊗ (ε ⊗ λ)((d2 ⊗ 1) · f (d1 )) (4)
= θ(f (c))λ(f (d)) = (θ ⊗ λ)(f (c) ⊗ f (d)).
Hence Δ(f (cd)) = f (c) ⊗ f (d).
2
Lemma 12. Let (A, C, ψ) be a right–left monoidal entwining structure. If A is also a Hopf algebra with the bijective antipode T , then α is an A-linear map if and only if for any c ∈ C, a ∈ A,
ψ
T −1 (a) f (cψ ) = f (c)T (a).
(18)
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Proof. If the equation (18) holds, for any M ∈C M (ψ)A , λ ∈∗ M , m ∈ M , a ∈ A, (18) ψ αM (λ · a)(m) = λ(m(0) · f (m(−1) )T (a)) = λ(m(0) · T −1 (a) f (m(−1)ψ )) = λ((m · T −1 (a))(0) f ((m · T −1 (a))(−1) )) = αM (λ)(m · T −1 (a)) = (αM (λ) · a)(m). Conversely we consider C ⊗ A ∈C M (ψ)A , then for any c ∈ C, a ∈ A, θ ∈∗ A, (αC⊗A (ε ⊗ θ) · a)(c ⊗ 1) = αC⊗A ((ε ⊗ θ) · a)(c ⊗ 1), ψ ψ (αC⊗A (ε ⊗ θ) · a)(c ⊗ 1) = (ε ⊗ θ)((cψ ⊗ T −1 (a) )(0) · f ((cψ ⊗ T −1 (a) )(−1) )) ψ ψ φ = (ε ⊗ θ)(cψ2φ ⊗ T −1 (a) f (cψ1 ) ) = θ(T −1 (a) f (cψ )). αC⊗A ((ε ⊗ θ) · a)(c ⊗ 1) = (ε ⊗ θ)((c2 ⊗ f (c1 )) · T (a)) ψ (4) = (ε ⊗ θ)(c2ψ ⊗ f (c1 )T −1 (a) ) = θ(f (c)T (a)). Hence
ψ
T −1 (a) f (cψ ) = f (c)T (a).
2
Dual to Lemma 12 by interchanging the roles of A and C we have the following lemma. Lemma 13. Let (A, C, ψ) be a right–left monoidal entwining structure. If C is also a Hopf algebra with the bijective antipode R, then α is C-colinear if and only if for any c ∈ C,
R(c2ψ ) ⊗ f (c1 )ψ =
R(c1 ) ⊗ f (c2 ).
(19)
Definition 14. Let (A, C, ψ) be a monoidal entwining structure such that A and C are both Hopf algebras. If there exists a k-linear map f : C → A satisfying the equations (16)–(19), then we call (A, C, ψ) a sovereign entwining structure and f is a generalized sovereign map in C M (ψ)A . Directly from Example 7, Example 8 and Lemmas 12–13 we have the following result. Theorem 15. Let (A, C, ψ) be a monoidal entwining structure with C being finite dimensional. A and C are both Hopf algebras with the bijective antipodes T and R respectively. Then C M (ψ)A is a sovereign category if and only if (A, C, ψ) is a sovereign entwining structure. Let H be a Hopf algebra with the antipode S. From [6] we know that a sovereign element of H is a group-like element π such that S 2 (h) = πhπ −1 for all h ∈ H. A sovereign Hopf algebra is a pair (H, π) where H is a Hopf algebra and π is a sovereign element of H. Moreover there is a fact that sovereign structures on the category of modules over H (being finite-dimensional) correspond to sovereign elements in H. Corollary 16. Let (A, C, ψ) be a monoidal entwining structure. Then C M (ψ)A is a sovereign category if and only if C ∗ A is a sovereign Hopf algebra. Proof. By Theorem 9 there exists an isomorphism between C M (ψ)A and MC ∗op A . If C M (ψ)A is a sovereign i category, then there is a generalized sovereign map f : C → A. It is very easy to prove that u ⊗ f (ui ) is the sovereign element in C ∗op A. Conversely if C ∗op A is a sovereign Hopf algebra with a sovereign element g ⊗ a ∈ C ∗op A, then it is trivial to verify that the map f : C → A, f (c) = g(c)a is the generalized sovereign map. 2
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Example 17. Let H be a sovereign Hopf algebra with the sovereign element . By [14] it is trivial to prove ψ that (H, H, ψ), ψ : H ⊗ H → H ⊗ H, h ⊗ gψ = h2 ⊗ S −1 (h3 )gh1 , h, g ∈ H is an entwining structure and H M (ψ)H is exactly H Y DH . Then it is very easy to verify that the map ζ : H → H, ζ(h) = ε(h) satisfies the conditions in Theorem 15 and so H Y DH is a sovereign monoidal category. References [1] [2] [3] [4]
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
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