R-smash products of Hopf quasigroups

The theory of R-smash products for Hopf quasigroups is developed.


Introduction
Hopf quasigroups and Hopf coquasigroups were introduced in [4]. These are nonassociative (or non-coassociative) generalisations of Hopf algebras, in which the antipode provides one with a certain level of control over the non-associativity. In particular Hopf quasigroups are examples of unital coassociative H-bialgebras introduced in [5, Section 2]. They can be understood as linearisations of loops [1]. Also in [4] smash products of Hopf quasigroups were studied. It has been shown in [2] that a standard form of a smash product forces one to replace the conventional associativity of action (assumed in [4] from the onset) by a similar condition involving the antipode. In this note, which is a sequel to [2], we look at R-smash products [3] of Hopf quasigroups and, briefly, at W -smash coproducts of Hopf coquasigroups. This analysis reveals that some of the conventional requirements on the twisting map R need be replaced by similar conditions in which the antipode plays a prominent role; see Theorem 2.3 and Definition 2.1 for details.
All algebras and coalgebras are over a field k and they are assumed to be unital and counital respectively, but are not assumed to be associative or coassociative unless stated atherwise. Unadorned tensor product symbol represents the tensor product of kvector spaces. We use the standard Sweedler notation for coproducts ∆(h) = h (1) ⊗h (2) (summation understood) even if the coproduct ∆ is not assumed to be associative.

R-smash products of Hopf quasigroups
We begin by introducing terminology used in this note.
Definition 2.1. Let H be an algebra with product µ H and unit 1 H and a coalgebra with coproduct ∆ H and counit ε H that are algebra morphisms. Similarly, let A be an algebra with product µ A and unit 1 A and a coalgebra with coproduct ∆ A and counit ε A that are algebra morphisms. Consider linear maps S H : H → H and R : H⊗A → A⊗H. The map R is said to be: and it is said to be normal if it is both left and right normal; Dually, the map R is said to be: and it is said to be conormal if it is both left and right conormal; The action of R on elements is denoted by for all h ∈ H and a ∈ A. The reader is encouraged to write down all the above requirements on R in terms of this notation. For example, R is left multiplicative if for all g, h ∈ H. The identities (2.3)-(2.4) ensure that a Hopf quasigroup is an Hbialgebra with left division h \ g = S H (h)g and right division g/h = gS H (h); see [5,Definition 2]. It is proven in [4] that the antipode S H is antimultiplicative and anticomultiplicative and it immediately follows from the Hopf quasigroup identities that S H enjoys the standard antipode property.
Definition 2.2. Let H and A be Hopf quasigroups, R : H⊗A → A⊗H a k-linear map. An R-smash product of H and A is a Hopf quasigroup A >⊳ R H equal to A⊗H as a vector space, with tensor product coproduct, unit and counit, and the multiplication and antipode The aim of this note is to determine necessary and sufficient conditions for R to produce an R-smash product of Hopf quasigroups. These are listed in the following theorem, which is a Hopf quasigroup version of [3, Corollary 4.6] Theorem 2.3. Let H, A be Hopf quasigroups, R : H⊗A → A⊗H a k-linear map. If R is left multiplicative and left conormal, then the following statements are equivalent: (1) A >⊳ R H is an R-smash product Hopf quasigroup for H and A; (2) The map R is a coalgebra map that is normal, right S H -multiplicative and right S H -conormal.
Before the proof of Theorem 2.3 is given we state and prove three lemmata.
(1) Obviously, the right S H -multiplicativity of R implies (2.10). Conversely, the right S H -multiplicativity of R can be inferred from (2.10) by repetitive use of equations (2.8) in Lemma 2.4: The second statement is proven by a similar repetitive use of equations (2.8) in Lemma 2.4, and the proof is left to the reader. To prove (3), if R is right normal and right S H -multiplicative, then so the second of equations (2.11) is automatically satisfied. Now, we need to use the multiplicativity of the counit and (2.8) in Lemma 2.4 to compute Hence, the first of equations (2.11) is equivalent to right S H -conormality of R. ⊔ ⊓ Lemma 2.6. Let H and A be Hopf quasigroups, R : H⊗A → A⊗H a left normal and left multiplicative map. If R is also a coalgebra map and is left conormal, then (2.12) Furthermore, the first of equalities (2.11) implies that Proof. Take any a ∈ A and h ∈ H. Then, using the left multiplicativity and left conormality of R to make a start and to finish, we can compute This proves equality (2.12). The second assertion is proven by the following calculation, for all a ∈ A, h ∈ H, Thus the equality (2.13) holds as required. ⊔ ⊓ Proof of Theorem 2.3. (2) ⇒ (1) The normality of R immediately implies that 1 A ⊗1 H is the unit of A >⊳ R H. The left counitality of R together with the fact that a counit of a Hopf quasigroup is an algebra map imply that also the counit ε A ⊗ε H of A >⊳ R H is an algebra homomorphism. The coproduct ∆ of A >⊳ R H is obviously unital, and it is multiplicative since R is a coalgebra morphism. This part of the proof is not different from the standard Hopf algebra case; see [3].

It remains to check the Hopf quasigroup identities (2.3) and (2.4), which is done by explicit calculations.
For all a, b ∈ A and g, h ∈ H, This proves the first of equations (2.3). Next (a⊗h) (1) (S((a⊗h) (2) )(b⊗g)) where also the normality was used to derive the penultimate euqality. This proves the second of relations (2.3). It is the proof of (2.4) where the right S H -multiplicativity of R, (2.2), is used. The first of identities (2.4) is proven by the following calculation ((b⊗g)(a⊗h) (1) )S((a⊗h) (2) ) The penultimate equality is a consequence of left comultiplicativity of R which is asserted by Lemma 2.4. In derivation of the last equality, the left conormality of R was also used. Finally, and again using Lemma 2.4 in the penultimate equality and left conormality of R in the last one, we compute ((b⊗g)S((a⊗h) (1) ))(a⊗h) (2) (2.6),(2.5) This completes the proof that A >⊳ R H is a Hopf quasigroup as required.
(1) ⇒ (2) The fact that 1 A ⊗1 H is the unit of the R-smash product Hopf quasigroup A >⊳ R H immediately implies the normality of R. The equalities, for all a ∈ A, h ∈ H, resulting from the multiplicativity of coproduct and counit imply that R is a coalgebra map. This is no different from the standard Hopf algebra case.
Developing the second of the Hopf quasigroup conditions (2.4) for A >⊳ R H as in the first part of the proof of the theorem, and using Lemma 2.6 and (2.8) repeatedly, we arrive at the following equality: Apply id A ⊗ε H to both sides of this equation, set b = R ,r a (1)Rr ε H (g (1)r S H (h) (1) )R), and shift Sweedler's indices as required to obtain: The fact that R is a coalgebra map, equation (2.9), implies Finally, the antipode property combined with the left conormality of R yield equation (2.10). Therefore, R is right S H -multiplicative by Lemma 2.5. Finally, making the same steps in the proof of the first Hopf quasigroup identity for A >⊳ R H as in the first part of the proof of the theorem, we conclude that the first of conditions (2.3) imply Applying id A ⊗ε H and evaluating this identity at a = 1 A and g = 1 H one immediately obtains the first of equations (2.11). In view of Lemma 2.5 this is tantamount to right S H -conormality of R. ⊔ ⊓ Remark 2.7. In [3,Corollary 4.6], which is a predecessor of Theorem 2.3, it is assumed that the R is compatible with the antipodes so that the equality (2.13) is satisfied. As explained in Lemma 2.6 this follows from other hypotheses made in Theorem 2.3, most notably from right S H -and left conormality of R, which are not assumed in [3].  (2) )·m) = ε H (h)m = S H (h (1) )·(h (2) ·m); (b) the H-action satisfies the following compatibility conditions:  [2]. That R satisfies all assumptions of Theorem 2.3 can be checked by straightforward calculations which are left to the reader. We only mention in passing that for the right S H -conormality of R both equations (2.14) are needed.
When written in terms of commutative diagrams, the definitions of a Hopf quasigroup and Hopf coquasigroup are formally dual to each other in the sense that one is obtained by reversing all the arrows in the definition of the other. Thus the theory of R-smash coproducts for Hopf coquasigroups can be obtained by dualising the theory of R-smash products for Hopf quasigroups. By this means one can first state Definition 2.9. Let H and A be Hopf coquasigroups and let W : H⊗A → A⊗H be a k-linear map. By a W -smash coproduct of H and A we mean a Hopf coquasigroup H W ◮ < A that is equal to H⊗A as a vector space with tensor product unit, multiplication and counit, and with comultiplication and antipode (1) H W ◮ < A is a W -smash coproduct Hopf coquasigroup of H and A; (2) The map W is an algebra map that is conormal, right S H -comultiplicative and right S H -normal.