On the Zero Divisors of Hopf Algebras

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1 On the Zero Divisors of opf Algebras Ahed Roan Peter Linnell May 4, 2011 Abstract In an attept to study the zero divisors in infinite opf algebras, we study two non-trivial exaples of non-group ring infinite opf algebras and show that a variant of Kaplansky s classical zero divisor conjecture holds for these two opf algebras. 1 Introduction The zero divisor proble on group-rings over countably or uncountably infinite groups is an active area of research, and in fact was conjectured by Kaplansky in the 1940 s. The proble is siply to show that there are no non-trivial zero divisors in the group-ring where the group is torsion free. Dr. Peter Linnell then conjectured that a variant of Kaplansky s conjecture ay hold in opf algebras since their structure resebles uch of the structure of group-rings. Let be a opf algebra and let G be a torsion-free subgroup of its group-like eleents (definition 7). Then the conjecture is that if 0 α kg and 0 β, then αβ 0. We exaine two cases of non group-ring opf algebras, naely: = i 0 kgxi and = (CZ) 0 which is the set of C-linear aps that vanish on an ideal of finite co-diension. We show that this variant of the zero divisor conjecture holds for both and when G = Z. We plan on generalizing the arguents to arbitrary groups G with torsion free subgroups in future work.

2 2 Ahed Roan Peter Linnell 2 General Foralis We begin with the necessary definitions which are in [?] and [?]. Definition 1: A k-algebra (with a unit) is a k-vector space A together with two k-linear aps, unit u : k A, ultiplication : A A A, such that the following diagras are coutative: a) Associativity b)unit A id id u A k id u id A k A A The two lower aps in b) are scalar ultiplication. Also b) gives the identity eleent of A by allowing 1 A = u(1 k ). Definition 2 : A k-coalgebra (with a co-unit) is a k-vector space A together with two k-linear aps, co-ultiplication : A A A, co-unit ɛ : A k, such that the following diagras are coutative: a) co-associativity b)co-unit A id id A 1 A k A ɛ id 1 A k The upper aps of b) are given by a 1 a and a a 1 for all a A. It is noteworthy that is injective and is surjective by definition. id ɛ

3 On the Zero Divisors of opf Algebras 3 Definition 3: Let A be a k-algebra. The finite dual of A is A o = {f A f(i) = 0} for soe ideal I of A such that dia/i <. Definition 4: 1) Let A and B be co-algebras, with co-ultiplications A and B and co-units ɛ A and ɛ B, respectively. A ap θ : A B is a co-algebra orphis if B θ = (θ θ) A and if ɛ A = ɛ B θ 2) A subspace I A is a coideal if I I A + A I and if ɛ(i) = 0. Definition 5: A k-space A is a bialgebra if (A,, u) is an algebra,(a,, ɛ) is a co-algebra, and either of the following two (equivalent) conditions holds: 1) and ɛ are algebra orphiss. 2) and u are co-algebra orphiss. One can reason that a orphis is a bialgebra orphis if it is both an algebra and a co-algebra orphis. Definition 6: A subspace I A is a biideal if it is both an ideal and a co ideal. The quotient space A/I is a bialgebra precisely when I is a biideal of A. Definition 7: Let A be any co-algebra and let a A, then a is a group-like eleent if a = a a and if ɛ(a) = 1. Exaple: Let A be any algebra and define Alg(A, k) = {f A f is an algebra ap}. Then Alg(A, k) = G(A o ) is the set of group-like eleents in the co-algebra A o. Definition 8: For any k-spaces V and W, the twist ap τ : V W W V is given by τ (v w) = w v.

4 4 Ahed Roan Peter Linnell Definition 9: A opf algebra (,, u,, ɛ, S; k) over k is a bi-algebra over k equipped with a linear antipode ap S : obeying (S id) = (id S) = u ɛ. The additional axios that ake the co-algebra and algebra into a opf algebra are captured via the following coutative diagras: ɛ id τ id k u id S, S id ɛ ɛ k ɛ u u u k 3 Results Let G be an infinite cyclic group, and k be any field. Consider the opf algebra = k g, h, x gh = 1, gx = qxg = i 0 kgxi where 0 q k {ω n }, ω is a root of unity, n Z and g, h G are group-like eleents. We note that the co-ultiplication, co-unit ɛ and antipode S are defined as follows : (g) = g g, (h) = h h, (x) = g x + x 1 ɛ(g) = 1, ɛ(h) = 1, ɛ(x) = 0 S(g) = h, S(h) = g, S(x) = hx. Fro the definition of the above opf algebra antipode S we observe that S is

5 On the Zero Divisors of opf Algebras 5 invertible. This is verified as follows: S 2 (x) = S( hx) = S(h)S(x) = gs(x) = g( hx) = ghx = x and oreover S 2 (g) = S(h) = g and siilarly for h. Also notice that is of infinite order since 0 q k is not a root of unity. 3.1 P roposition 1 In the case where G = Z, the resulting opf algebra = i 0 kgxi. If 0 α kg and 0 β, then αβ 0. proof: Suppose 0 α kg and 0 β. We wish to show that αβ 0. Fro the definition of, we observe that β = i α ix i where 0 α i for soe i Z since 0 β. Fro the definition of kg we note that α = j a jg j where 0 a j k for soe j Z since α 0. Now αβ = i αα ix i and it suffices to prove that αα i 0 for soe i Z. Since α i kg we see that α i = k i a ki g k i, where k i Z. It follows that for soe i Z, αα i = ( j a jg j )( k i a ki g k i ) is the product of two nonzero eleents of kg which is never zero. ence αα i 0 for soe i Z. Thus αβ = i αα ix i has at least one nonzero coefficient αα i and thus αβ 0 as desired. It is noteworthy that this result can be generalized to any infinite group. 3.2 P roposition 2 Let (CZ) 0 be the set of C-linear aps CZ C that vanish on an ideal of finite codiension. Let Z be an infinite cyclic subgroup of the group-like eleents. Suppose 0 α CZ and 0 β (CZ) 0, then αβ 0. proof: Let C denote the ultiplicative eleents of C. Let = (CZ) be the set of C linear aps Ψ : Z C and consider it as a opf algebra. Further consider a torsion free subgroup of the group-like eleents of, naely the set of hooorphiss θ : Z C such that θ(g) = π where π is a transcendental eleent of C, and g is a generator of Z. It follows that θ i (g) = π i. We can see that CZ (CZ) 0 (CZ). Let τ CZ, then τ = j Z θi (g)g j = j Z πj g j. Then α = τ + λ = λgj + j Z πj g j = (πj + λ)g j, i.e. α = (..., π 1 + λ, λ, π + λ,...). If π i = π j = λ, then π i j = 1 = i = j. It follows that at ost one co-ordinate

6 6 Ahed Roan Peter Linnell of α will vanish. Suppose that the vanishing co-ordinate is the i th co-ordinate, then α = (..., π i 1 + λ, 0, π i+1 + λ,..., ). Then the only eleent β such that αβ = 0 under co-ordinate-wise ultiplication is β = (..., 0, 0, 1, 0, 0,...) where 1 is in the i th co-ordinate and zeros are placed else where. Now (CZ) 0 vanishes on an ideal of finite co-diension. Inevitably the ideal is of the for I = (a 0 +a 1 g+...+a n g n ) and with out loss of generality a 0 0 and n Z +. Then β(i) = 0. Since I is an ideal, we see that (a 0 + a 1 g a n g n )g i I. Thus β vanishes on (a 0 + a 1 g a n g n )g i = (a 0 g i + a 1 g i a n g n+i ). Then β((a 0 g i + a 1 g i a n g n+i )) = 0. But β = (.., 0, 1, 0,...) where 1 is in the i th position and so (.., 0, 1, 0,...)(a 0 g i + a 1 g i a n g n+i ) = a 0 0 which presents a contradiction. Thus β (CZ) 0. That is if 0 α CZ and 0 β (CZ) 0, then αβ 0. W 5 Conclusion We have two non-trivial non-group ring opf algebras that satisfy Dr. Linnell s conjecture, for which our techniques should readily generalize fro G = Z to arbitrary infinite groups. Much research is required in order to solve the Zero Divisor proble. I will dedicate the next two years to solving the Zero Divisor proble on opf algebras. Acknowledgents The first author is grateful to Prof. Peter Linnell for explaining any technical details of opf algebras. References [1] Shahn Majid, Foundations of Quantu Group Theory (Cabridge University Press, 1995.) [2] Susan Montgoery, opf Algebras and Their Actions on Rings (Aerican Matheatical Society, Rhode Island, 1992), pages 1-5.