The antipode of a dual quasi-Hopf algebra with nonzero integrals is bijective
aa r X i v : . [ m a t h . QA ] M a y THE ANTIPODE OF A DUAL QUASI-HOPF ALGEBRA WITHNONZERO INTEGRALS IS BIJECTIVE
M. BEATTIE, M.C. IOVANOV, AND S¸. RAIANU
Dedicated to Fred Van Oystaeyen for his sixtieth birthday
Abstract.
For A a Hopf algebra of arbitrary dimension over a field K , it is well-knownthat if A has nonzero integrals, or, in other words, if the coalgebra A is co-Frobenius, thenthe space of integrals is one-dimensional and the antipode of A is bijective. Bulacu andCaenepeel recently showed that if H is a dual quasi-Hopf algebra with nonzero integrals,then the space of integrals is one-dimensional, and the antipode is injective. In this shortnote we show that the antipode is bijective. Introduction
The definition of quasi-Hopf algebras and the dual notion of dual quasi-Hopf algebrasis motivated by quantum physics and dates back to work of Drinfel’d [4]. The theory ofintegrals for quasi-Hopf algebras was studied in [9, 6, 2]. In [2], Bulacu and Caenepeelshowed that a dual quasi-Hopf algebra is co-Frobenius as a coalgebra if and only if it has anonzero integral. In this case, the space of integrals is one-dimensional and the antipode isinjective, so that for finite dimensional dual quasi-Hopf algebras the antipode is bijective.In this note, we use the ideas from a new short proof of the bijectivity of the antipode forHopf algebras by the second author [7] to show that the antipode of a dual quasi-Hopfalgebra with integrals is bijective, thus extending the classical result of Radford [10] forHopf algebras.In this paper we prove
Theorem 1.1.
Let H be a co-Frobenius dual quasi-Hopf algebra, equivalently, a dualquasi-Hopf algebra having nonzero integrals. Then the antipode of H is bijective. Preliminaries
In this section we briefly review the definition of a dual quasi-Hopf algebra over a field K . We refer the reader to [1, 3, 11] for the basic definitions and properties of coalgebrasand their comodules and of Hopf algebras. For the definition of dual quasi-Hopf algebrawe follow [8, Section 2.4]. Definition 2.1.
A dual quasi-bialgebra H over K is a coassociative coalgebra ( H, ∆ , ε ) together with a unit u : K → H , u (1) = 1 , and a not necessarily associative multiplication M : H ⊗ H → H . The maps u and M are coalgebra maps. We write ab for M ( a ⊗ b ) . The first author’s research was supported by an NSERC Discovery Grant.The second author was partially supported by the contract nr. 24/28.09.07 with UEFISCU “Groups,quantum groups, corings and representation theory” of CNCIS, PN II (ID 1002).
As well, there is an element ϕ ∈ ( H ⊗ H ⊗ H ) ∗ called the reassociator, which is invertiblewith respect to the convolution algebra structure of ( H ⊗ H ⊗ H ) ∗ . The following relationsmust hold for all h, g, f, e ∈ H : h ( g f ) ϕ ( h , g , f ) = ϕ ( h , g , f )( h g ) f (1) 1 h = h h (2) ϕ ( h , g , f e ) ϕ ( h g , f , e ) = ϕ ( g , f , e ) ϕ ( h , g f , e ) ϕ ( h , g , f )(3) ϕ ( h, , g ) = ε ( h ) ε ( g )(4)Here we use Sweedler’s sigma notation with the summation symbol omitted. Definition 2.2.
A dual quasi-bialgebra H is called a dual quasi-Hopf algebra if there existsan antimorphism S of the coalgebra H and elements α, β ∈ H ∗ such that for all h ∈ H : S ( h ) α ( h ) h = α ( h )1 , h β ( h ) S ( h ) = β ( h )1(5) ϕ ( h β ( h ) , S ( h ) , α ( h ) h ) = ϕ − ( S ( h ) , α ( h ) h , β ( h ) S ( h )) = ε ( h ) . (6)Let H be a dual quasi-Hopf algebra. As in the Hopf algebra case, a left integral on H is an element T ∈ H ∗ such that h ∗ T = h ∗ (1) T for all h ∗ ∈ H ∗ ; the space of left integralsis denoted by R l and by [2, Proposition 4.7] has dimension 0 or 1. Right integrals aredefined analogously with space of right integrals denoted by R r . Suppose 0 = T ∈ R l .It is easily seen that R l is a two sided ideal of the algebra H ∗ , and KT ⊆ Rat ( H ∗ )with right comultiplication given by T T ⊗
1. Since for co-Frobenius coalgebras
Rat ( H ∗ ) = Rat ( H ∗ H ∗ ) = Rat ( H ∗ H ∗ ), KT must have left comultiplication T a ⊗ T . Bycoassociativity, a is a grouplike element, called the distinguished grouplike of H . Then,for all h ∗ ∈ H ∗ , T h ∗ = h ∗ ( a ) T. (7)From [2, Proposition 4.2], the function θ ∗ : R l ⊗ H → Rat ( H ∗ H ∗ ) θ ∗ ( T ⊗ h ) = σ ( S ( h ) ⊗ α ( h ) h )( S ( h ) ⇀ T ) σ − ( S ( h ) ⊗ β ( S ( h )) S ( h ))(8)is an isomorphism of right H -comodules, where σ : H ⊗ H → H ∗ is defined by σ ( h ⊗ g )( f ) = ϕ ( f, h, g ), σ − is the convolution inverse of σ , and, as usual, ( h ⇀ T )( g ) = T ( gh ).3. Proof of the theorem
Let H be a quasi-Hopf algebra with 0 = T ∈ R l . As in [7], for each right H -comodule( M, ρ ), we denote by a M the left H -comodule structure on M defined by m a − ⊗ m a = aS ( m ) ⊗ m , where ρ ( m ) = m ⊗ m . Denote the induced right H ∗ -module structureon a M by m · a h ∗ = h ∗ ( m a − ) m a = h ∗ ( aS ( m )) m . By [2, Corollary 4.4] the antipode S of H is injective, and therefore has a left inverse S l . Then, for σ as above, we have thefollowing analogue of [7, Proposition 2.5]: HE ANTIPODE OF A DUAL QUASI-HOPF ALGEBRA WITH NONZERO INTEGRALS IS BIJECTIVE3
Proposition 3.1.
The map p : a H → Rat ( H ∗ ) defined by p ( h ) = σ ( S ( S l ( h )) ⊗ α ( S l ( h )) S l ( h )) ∗ ( h ⇀T ) ∗ σ − ( h β ( h ) ⊗ S ( h ))(9) is a surjective morphism of left H -comodules.Proof. Let Ψ := σ ( S ( S l ( h )) ⊗ α ( S l ( h )) S l ( h )). Then for c ∗ ∈ H ∗ and g ∈ H :( p ( h ) ∗ c ∗ )( g ) = p ( h )( g ) c ∗ ( g )(9) = Ψ( g ) T ( g h ) σ − ( h β ( h ) ⊗ S ( h ))( g ) c ∗ ( g )= Ψ( g ) T ( g h ) ϕ − ( g , h β ( h ) , S ( h )) c ∗ ( g )= Ψ( g ) T ( g h ) ϕ − ( g , h , S ( h )) c ∗ ( g β ( h ))(5) = Ψ( g ) T ( g h ) ϕ − ( g , h , S ( h )) c ∗ ( g ( h β ( h ) S ( h )))= Ψ( g ) T ( g h ) c ∗ ( ϕ − ( g , h , S ( h )) g ( h β ( h ) S ( h )))= Ψ( g ) T ( g h ) β ( h ) c ∗ ( ϕ − ( g , h , S ( h )) g ( h S ( h )))(1) = Ψ( g ) T ( g h ) β ( h ) c ∗ (( g h ) S ( h ) ϕ − ( g , h , S ( h )))= Ψ( g ) T ( g h )( S ( h ) ⇀ c ∗ )( g h ) β ( h ) ϕ − ( g , h , S ( h ))= Ψ( g )( T ∗ ( S ( h ) ⇀ c ∗ ))( g h ) β ( h ) ϕ − ( g , h , S ( h ))(7) = Ψ( g )( S ( h ) ⇀ c ∗ )( a ) T ( g h ) β ( h ) ϕ − ( g , h , S ( h ))= Ψ( g ) T ( g h ) σ − ( h β ( h ) ⊗ S ( h ))( g ) c ∗ ( aS ( h ))(9) = p ( h )( g ) c ∗ ( aS ( h ))= p ( c ∗ ( aS ( h )) h )( g )= p ( c ∗ ( h a − ) h a )( g )= p ( h · a c ∗ )( g ) . Thus p is left H -colinear. Finally, we note that p ◦ S = θ ∗ ( T ⊗ − ) where θ ∗ is theisomorphism from (8) so that p is surjective. (cid:3) Let c be a grouplike element of H . From [2, p.580], c is invertible with inverse S ( c ).We will show that left multiplication by c has an inverse too.Let θ c ∈ End( H ) be defined by θ c ( h ) = ch and define the coinner automorphisms q c and r c = q − c ∈ End( H ) by: q c ( h ) = ϕ − ( c, S ( c ) , h ) h ϕ ( c, S ( c ) , h ) and r c ( h ) = ϕ ( c, S ( c ) , h ) h ϕ − ( c, S ( c ) , h ) . Lemma 3.2.
For any grouplike element c and θ c , r c , q c as above, θ c ◦ θ c − = r c and thus θ c is bijective with inverse θ − c = θ c − ◦ q c = q c − ◦ θ c − .Proof. Using (1) and the fact that c − = S ( c ), we see that θ c ◦ θ c − ( h ) = c ( c − h ) = ϕ ( c, S ( c ) , h )( cS ( c )) h ϕ − ( c, S ( c ) , h ) = r c ( h ) . The same formula for c − = S ( c ) yields θ c − ◦ θ c = r c − and the statement then followsdirectly. (cid:3) M. BEATTIE, M.C. IOVANOV, AND S¸. RAIANU
We can now prove our main result.
Proof of Theorem 1.1.
We only need to prove the surjectivity. The proof goes along the lines of the proof of[7, Theorem 2.6], but with the difference that here the antipode is not necessarily ananti-morphism of algebras.Let π be the composition map a H p → Rat ( H ∗ H ∗ ) ∼ → H ⊗ R r ≃ H , where the last twoisomorphisms follow by left-right symmetry of the results of [2]. Since H is a co-Frobeniuscoalgebra, H H is projective by [5, Theorem 1.3] or [2, Theorem 4.5, (x)], and as π issurjective, there is a morphism of left H -comodules λ : H → a H such that πλ = Id H . Wethen have aS ( λ ( h ) ) ⊗ λ ( h ) = λ ( h ) a − ⊗ λ ( h ) a = h ⊗ λ ( h ) . Applying Id ⊗ επ , we get aS ( επ ( λ ( h ) ) λ ( h ) ) = h for any h ∈ H . Thus θ a ◦ S is surjectiveand since θ a is bijective by Lemma 3.2, S is surjective also. (cid:3) References [1] E. Abe,
Hopf Algebras , Cambridge Univ. Press, 1977.[2] D. Bulacu, S. Caenepeel, Integrals for (dual) quasi-Hopf algebras. Applications, J. Algebra (2003), no. 2, 552-583.[3] S. D˘asc˘alescu, C. N˘ast˘asescu, S¸. Raianu,
Hopf Algebras: an Introduction , Monographs and Text-books in Pure and Applied Mathematics , Marcel Dekker, Inc., New York, 2001.[4] V.G. Drinfel’d, Quasi-Hopf Algebras, Leningrad Math. J. (1990) 1419-1457.[5] J. Gomez-Torrecillas, C. N˘ast˘asescu, Quasi-co-Frobenius coalgebras, J. Algebra (1995), no.3, 909-923.[6] F.Hausser, F. Nill, Integral theory for quasi-Hopf algebras, arXiv:math/9904164v2.[7] M.C. Iovanov, Generalized Frobenius algebras and the theory of Hopf algebras, preprint,arXiv:0803.0775.[8] S. Majid, Foundations of Quantum Group Theory , Cambridge University Press, Cambridge, 1995.[9] F. Panaite, F. van Oystaeyen, Existence of integrals for finite dimensional Hopf algebras, Bull.Belg. Math. Soc. Simon Stevin (2000) 261-264.[10] D.E. Radford, Finiteness conditions for a Hopf algebra with a nonzero integral, J. Algebra (1977), no. 1, 189-195.[11] M.E. Sweedler, Hopf Algebras , Benjamin, New York, 1969.
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