Cyclic homology of Hopf algebras

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1 Cyclic homology of Hopf algebras arxiv:math/9213v1 [math.qa] 25 Sep 2 Rachel Taillefer Abstract A cyclic cohomology theory adapted to Hopf algebras has been introduced recently by Connes and Moscovici. In this paper, we consider this object in the homological framework, in the spirit of Loday-Quillen ([LQ]) and Karoubi s work on the cyclic homology of associative algebras. In the case of group algebras, we interpret the decomposition of the classical cyclic homology of a group algebra ([B], [KV], [L]) in terms of this homology. We also compute both cyclic homologies for truncated quiver algebras. 2 Mathematics Subject Classification: 16E4, 16W3, 16W35, 17B37, 19D55, 57T5. Keywords: Cyclic homology, Hopf algebras, group algebras, quiver algebras. 1 Introduction In this paper, we study a cyclic homology theory for Hopf algebras, and we specialize our considerations mainly to group algebras and truncated quiver algebras. Connes and Moscovici introduced cyclic cohomology of Hopf algebras in [CM2]. They used it to compute an index pairing map associated to some convolution algebras A (see also [CM3]). To do this, they constructed some Hopf algebras H(n), endowed with extra data, and they defined a cyclic cohomology for these, subject to the following condition: there should be a characteristic map from the cyclic cohomology of the H(n) to the classical cyclic cohomology of A for each trace τ : A C satisfying some invariance condition (see [CM2], [CM3]). In [Cr], Crainic gives an alternative definition of this cyclic cohomology in terms of X complexes. He also computes some examples, and constructs a non-commutative Weil complex which is related to the cyclic cohomology of Hopf algebras. It was also used by Gorokhovsky in [G] to construct some cyclic cocycles associated to a specific vector bundle. In this paper, we consider the homological framework, inspired by Loday-Quillen and Karoubi s approach to the usual cyclic homology of algebras obtained from the cohomological theory considered by Connes. We proceed as follows: in the first section, we define a cyclic homology of Hopf algebras, and describe the SBI exact sequence in this situation. Next, we study the case of group algebras, and we interpret the decomposition of the classical cyclic homology of group algebras, established by Laboratoire G.T.A., Département de Mathématiques CC 51, Université Montpellier II, 3495 Montpellier Cedex 5. taillefr@math.univ-montp2.fr 1

2 Burghelea ([B]) and Karoubi and Villamayor ([KV]), in terms of Connes and Moscovici s homology. We also compute the cyclic homology of the cyclic group algebras (as Hopf algebras). Finally, we consider the family of truncated quiver algebras. The Hochschild homology of a truncated quiver algebra with coefficients in itself was computed by Skldberg in [S]; this enables us to compute its classical cyclic homology. We then specialize to the Taft algebras, which are noncommutative, non-cocommutative Hopf algebras (of finite representation type), and compute their cyclic homology as Hopf algebras. Unless specified, k is a commutative ring. 2 Connes and Moscovici cyclic (co)homology In [CM], A. Connes and H. Moscovici have defined a cyclic cohomology theory for Hopf algebras endowed with some specific data, including a grouplike element and a character satisfying some extra properties. This can be slightly extended to include a second grouplike element; we shall give here the dual version of this definition (cyclic homology of a Hopf algebra). Data: Let k be a commutative ring, and let H be a Hopf algebra over k. Suppose there are two characters α, β : H k on H, and a grouplike element π in H. Set S π = πs, and assume the following properties are satisfied: (i) α(π) = 1 = β(π), (cm) (ii) (α S π β) 2 = id. The second identity can be rewritten with Sweedler s notation: α(a (1) )α(s(a (4) ))πs 2 (a (3) )π 1 β(s(a (2) ))β(a (5) ), a H. Definition-Proposition 2.1 The following data define a cyclic module C (π,α,β) : C (π,α,β) n = H n for n, and d i : H n+1 H n α(a )a 1... a n if i =, a... a n a... a i 1 a i... a n if 1 i n, a... a n 1 β(a n ) if i = n + 1, s i : H n H n+1 a a i... a n 1 if i n 1, a 1... a n a 1... a n 1 1 if i = n, and t n : H n H n a 1... a n α(a (1) n )S π(a (2) 1...a (2) n ) a(3) 1... a (3) n 1β(a (3) n ), when n 1. When n =, the maps become: d = α, d 1 = β, s = η (the unit of H), and t = id k. 2

3 The cyclic homology of H is the homology of the usual bicomplexes associated to a cyclic module, or, when k contains Q, of the Hochschild complex factored by the cyclic action t. (see [L]). We shall denote this homology by HC π,α,β (H). Proof: Most of the relations which should be satisfied by a cyclic module are easy to check; the main difficulty arises for the relation t n+1 n = id. Let us first compute the square of t n : t 2 n(a 1... a n ) = α(a (1) n )α(s π (a (2) n ) (1) )α(s(a (2) n 1) (1) ))...α(s(a (2) 1 ) (1) ) α((a (3) 1 ) (1) )...α((a (3) n 1) (1) )S π ((a (3) n 1) (2) )S((a (3) n 2) (2) )...S((a (3) 1 ) (2) ) S(S π ((a (2) n )(2) )S((a (2) n 1) (2) )...S((a (2) 1 ) (2) )) S π ((a (2) n ) (3) )S((a (2) n 1) (3) )...S((a (2) 1 ) (3) ) (a (3) 1 ) (3)... (a (3) n 2) (3) α((a (3) n 1) (3) )α(a (3) n ) = α(a (1) n )α 1 (a (4) n )α 1 (a (4) n 1)...α 1 (a (4) 1 )α(a (5) 1...a (5) n 1) S π (a (6) n 1)S(a (6) n 2)...S(a (6) 1 )S 2 (a (3) 1 )...S 2 (a (3) n 1)S(S π (a (3) n )) S π (a (2) n )S(a(2) n 1)...S(a (2) 1 ) a (7) 1... a (7) n 2β(a (7) n 1)β(a (5) n ) = α(a (1) n )α 1 (a (4) n )S π (a (4) n 1)S(a (4) n 2)...S(a (4) 1 ) S 2 (a (3) 1 )...S 2 (a (3) n 1)S(S π (a (3) n )) S π(a (2) n )S(a(2) n 1)...S(a (2) 1 ) a (5) 1... a (5) n 2β(a (5) n 1a (5) n ) = α(a (1) n )α 1 (a (4) n )S π (a (4) n 1)S 2 (a (3) n 1)S(S π (a (3) n )) S π (a (2) n )S(a(2) n 1)...S(a (2) 1 ) a (3) 1... a (3) n 2β(a (5) n 1a (5) n ) = α(a (1) n )α 1 (a (4) n )πs(s π(a (3) n )) S π(a (2) n )S(a(2) n 1)...S(a (2) 1 ) a (3) 1... a (3) n 2β(a (3) n 1a (5) n ) = α(a (1) n )α 1 (a (4) n )S2 π (a(3) n ) S π(a (2) n )S(a(2) n 1)...S(a (2) 1 ) a (3) 1... a (3) n 2β(a (3) n 1a (5) n ) where α 1 (resp. β 1 ) denotes α S (resp. β S). By induction, we get: t j n(a 1... a n ) = α(a (1) n )α 1 (a (4) n )...α 1 (a (4) n j+2) When j = n + 1, this becomes: Sπ 2 (a(3) n j+2) Sπ 2 (a(3) n j+3)... Sπ 2 (a(3) n ) S π(a (2) n )S(a(2) n 1)...S(a (2) 1 ) a (3) 1... a (3) n jβ(a (3) n j+1a (5) n j+2...a (5) n ). t n+1 n (a 1 a n ) = (α S π β) 2 (a 1 )... (α S π β) 2 (a n ), which is equal to a 1... a n by condition (cm), thereby proving the result. Example: When H is the trivial Hopf algebra k, then HC π,α,β (k) is easy to compute, and it is equal to the classical cyclic homology HC (k) (the datum (π, α, β) is necessarily equal to (1, id k, id k )). 3

4 Example: Suppose H is a group algebra kg. Then for any element π in the centre of G, and any characters α and β on H, provided they take the value 1 at π, we can consider the homology (kg). We shall study examples of this situation in more detail shortly. HC π,α,β As in the general theory of cyclic modules, there is a long periodic exact sequence, involving HC π,α,β (H) and the homology of the underlying simplicial module of C (π,α,β) ; view k as an H bimodule via: a.λ.b = β(a)λα(b), a, b H and λ k. Let β k α denote this module. Then this last homology is in fact the Hochschild homology of the underlying algebra of H with coefficients in β k α. Therefore, Proposition 2.2 There is a long exact sequence: H n (H, β k α ) HCn π,α,β (H) HC π,α,β n 2 (H) H n 1 (H, β k α ). We are now going to consider the case of a group algebra. 3 Cyclic homology of a group algebra 3.1 Case of trivial characters In this section, H is a group algebra kg, and the characters are both equal to the counit ε. D. Burghelea, M. Karoubi and O.E. Villamayor (see [B], [KV], [L]) have established a decomposition of the classical cyclic homology of a group algebra HC (kg). We are going to interpret this in terms of the cyclic homology of Connes and Moscovici. First, let us write the cyclic module maps (associated to (π, ε, ε)) for kg : d (g... g n ) = g 1... g n, d i (g... g n ) = g... g i 1 g i... g n for 1 i n, d n+1 (g... g n ) = g... g n 1, s i (g 1... g n ) = g g 1... g n for 1 i n + 1, and t n (g 1... g n ) = π(g 1...g n ) 1 g 1... g n 1, for any g,...g n in G. We can consider another cyclic module associated to G and π : Definition-Proposition 3.1 ([L] ) For any discrete group G and any element π in G, we define a cyclic module kγ. (G, π) as follows: as a set, Γ n (G, π) is the set of all (g,..., g n ) in G n+1 such that g... g n is conjugate to π, and the faces, degeneracies, and cyclic action are the usual ones: d i (g,..., g n ) = (g,..., g i g i+1,..., g n ) for i < n, d n (g,..., g n ) = (g n g, g 1,..., g n 1 ), s i (g,..., g n ) = (g,..., g i, 1, g i+1,..., g n ), and t n (g,..., g n ) = (g n, g,..., g n 1 ). 4

5 There is a canonical splitting of cyclic modules: C. (kg) = π G k(γ. (G, π)). We shall now make the link with Connes and Moscovici s cyclic homology: Proposition 3.2 The map (θ π ) n : C (π,ε,ε) n (kg π ) k(γ n (G, π)) g 1... g n π(g 1...g n ) 1 g 1... g n is a morphism of cyclic modules which induces an isomorphism on homology. Here G π is the centralizer of π in G. Proof: It is clear that θ π is a simplicial map; it also commutes with the cyclic operation, due to the fact that π is central in G π. At the simplicial module level, there is a factorization of θ π as C. π,ε,ε ψ (kg π ) = C. (G π, k) C. (G, k π ) φ 1 k(γ. (G, π)), where ψ(g 1,..., g n ) = (π; g 1,..., g n ), and φ(g,..., g n ) = (g...g n ; g,..., g n ). The module k π is induced by the inclusion map G π G from the trivial G π module k. Therefore Shapiro s lemma implies that ψ is a quasi-isomorphism. Since φ is also a quasi-isomorphism (see [L] ), so is θ π. Finally, the long periodic exact sequences for both cyclic modules, together with the five lemma, give an isomorphism of the cyclic homologies. Remark 3.3 This proof is very close to the proof of [L] These two propositions combined yield the following result: Theorem 3.4 For any discrete group G, there is a graded isomorphism: HC (kg) = π G HC π,ε,ε (kg π ). Remark 3.5 Using the results in [L] ( to ), there are various interpretations of HC π,ε,ε (kg π ) which we can give: (i) If G is a torsion free group, then HC π,ε,ε n (kg π ) = H n (G π /{π}, k) if π 1, H n (G) H n 2 (G)... if π = 1. Note that a similar result for Lie algebras was obtained by Connes and Moscovici in [CM] and [CM3], in the cohomological framework (for trivial character and grouplike). 5

6 (ii) If G is abelian, then HC π,ε,ε H (G/{π}, k) if π is of infinite order in G, (kg) = HC (k) H (G/{π}, k) if π is of finite order in G as a graded module. (iii) If k contains Q, then HC π,ε,ε H (G π /{π}, k) if π is of infinite order in G, (kg π ) = HC (k) H (G π /{π}, k) if π is of finite order in G as a graded module. In all these expressions, {π} is the cyclic subgroup of G generated by π. Remark 3.6 We shall see further on, on some examples (quiver algebras), that these decomposition formulae cannot be extended to general Hopf algebras (other than group algebras). Remark 3.7 These interpretations also enable us to compute explicitly HC π,ε,ε (kg) when G is a cyclic group, and therefore the classical HC (kg) also: Proposition 3.8 Let G be a cyclic group, and let π be an element in G. Let m π be the index of π in G. Let k be a ring, and view it as a trivial k (Z/m π Z) module. Then: HCn π,ε,ε k Ann(m π ) n/2 if n is even (kg) = (k/m π k) n+1/2 if n is odd, where Ann(m π ) = {λ k/ m π λ = }. Proof: Let σ π be a generator for Z/m π Z, and let N π = 1 + σ π + σπ σπ mπ 1 be its norm. The homology of Z/m π Z with coefficients in k is given as follows (see [W]): k/(σ π 1)k = k if n =, H n (Z/m π Z; k) = k G /N π k = k/m π k if n is odd, {λ k/n π λ = }/(σ π 1)k = Ann(m π ) if n is even >. Remark 3.5 (ii) yields the result. Corollary 3.9 With the same notations, we can express the classical cyclic homology of kg : k #G ( π G Ann(m π ) n/2) if n is even HC n (kg) = π G(k/m π k) n+1/2 if n is odd. Remark 3.1 This agrees with the result in [BACH2]. Remark 3.11 We can consider some special cases of these results. For instance, if m π is not a divisor of zero in k, then HCn π,ε,ε (kg) is equal to k when n is even, and to (k/m π k) n+1/2 when n is odd, so that HC n (kg) is equal to k #G when n is even, and to π G(k/m π k) n+1/2 when n is odd; if moreover the order of G is prime, this gives theorem 1 in [CGV]. Remark 3.12 Note that the results of proposition 3.8 and corollary 3.9 remain true for non-cyclic groups G, so long as G/{π} is cyclic. 6

7 3.2 Case of non-trivial characters when G is a cyclic group Set H = k (Z/mZ), and let α and β be characters on H. In the first place, we shall compute the Hochschild homology H (H, β k α ). For this, we shall use a simplified projective resolution defined in [CGV]: H 2 m 1 i= gi g m i H g gm 1 H 2 m 1 i= gi g m i H g gm 1 H 2 µ H where µ is the multiplication in H, g is a fixed generator in Z/mZ, and the maps are multiplication by the terms above the arrows. Tensoring this resolution by β k α over H e yields the following complex:... ζρ 1 1 βk α βk α ζρ βk α ζρ 1 1 βk α in which ζ = α(g) and ρ = β(g) are m th roots of unity. The homology of this complex is easy to compute: if α = β, then the homology is H n (H, α k α ) = k for all n N, and if α β, it is H (H, β k α ) =. The SBI exact sequence of proposition 2.2 yields the cyclic homology of H when α β. We shall now study the case α = β (we shall not use the Hochschild homology in this case). Let π = g s be an element in G with ζ s = 1 (ie. α(π) = 1). Consider the map χ π,ζ n : C n (π,α,β) (kg) C n (π,ε,ε) (kg) g i 1... g in ζ i 1+ +i n g i 1... g in. Then χ π,ζ. is an isomorphism of cyclic modules (the inverse is χ π,ζ 1. ). Therefore, HC π,α,β (kg) is isomorphic to HC π,ε,ε (kg). Using proposition 3.8, we finally have: Proposition 3.13 The cyclic homology of the Hopf algebra H = k (Z/mZ) is given as follows: HC π,α,β (H) = if α β, HCn π,α,α k Ann(m π ) n/2 (H) = (k/m π k) n+1/2 if n is even if n is odd. 4 Cyclic homologies of some truncated quiver algebras 4.1 Hochschild homologies of truncated quiver algebras Let be a finite quiver (that is a finite oriented graph), k a commutative ring, and let k be the algebra of paths in (k is the vector space with basis the set of paths in, and the multiplication is obtained via the concatenation of paths). For all p N, let p denote the set of paths of length p in, and consider an admissible ideal I in k. In [AG], Anick and Green introduced a new quiver Γ which gave them a minimal projective k /I resolution of the algebra k, graded by the lengths of paths. 7

8 Let m denote the ideal of k generated by 1. When I is equal to m n, for an integer n, the set of vertices of this quiver Γ is 1... n 1, and the edges are given as follows: a e if a 1, e and the terminus of a is e, γ a if a 1, γ n 1 and the terminus of γ is the origin of a, a γ if a 1, γ n 1 and the terminus of a is the origin of γ. If Γ (i) denotes the set of paths of length i in Γ, then Γ (2c) can be identified with nc and Γ (2c+1) can be identified with nc First case: n 2 Using Anick and Green s resolution, Skldberg, in [S], constructed a resolution of the algebra A := k /m n for n 2 as follows: Theorem 4.1 ([S] Theorem 1) There is a projective A bimodule resolution of A, which is graded by the lengths of paths, as follows: where the differentials are defined by P A : di+1 d P i d i 2 d 1 d P1 P A, P i = A k kγ (i) k A, d 2c+1 (u a 1 a cn+1 v) = ua 1 a 2 a cn+1 v u a 1 a cn a cn+1 v, d 2c (u a 1 a cn v) = n 1 j= ua 1 a j a j+1 a (c 1)n+j+1 a (c 1)n+j+2 a cn v if c >, and the augmentation d : A k A = A k kγ () k o A A is defined by d (u v) = uv. Note that the differentials preserve the gradation, so that the Hochschild homology spaces also are graded. Let HH p,q (A) denote the q th graded part of the space HH p (A). By means of the resolution in theorem 4.1, Skldberg computes the Hochschild homology of A with coefficients in A; to state this result, we shall need some notation: let C denote the set of cycles in the quiver, and for any cycle γ in C, let L(γ) denote its length. There is a natural action of the cyclic group t γ of order L(γ) on γ; let γ denote the orbit of γ under this action, and let C denote the set of orbits of cycles. Theorem 4.2 ([S]) Set q = cn + e, for e n 1 (n 2). Then: k aq if 1 e n 1 and 2c p 2c + 1, r q(k (n r) 1 Ker(. n : k if e =, and < 2c = p, n r k))br HH p,q (A) = r q(k (n r) 1 Coker(. n : k if e =, and < 2c 1 = p, n r k))br k # if p = q =, and otherwise, 8

9 where a q is the number of cycles of length q in C, and b r is the number of cycles of length r in C which are not powers of smaller cycles, and n r is the greatest common divisor of n and r. Example: Suppose = (n) is the n crown, that is the quiver with n vertices e,..., e n 1, and n edges a,..., a n 1, each edge a i going from the vertex e i to the vertex e i+1 for i n 2, and the edge e n 1 going from e n 1 to e, as follows: n-1 n i i i Then the Hochschild homology of the Taft algebra Λ n := k /m n is given by: HH p,cn (Λ n ) = k n 1 if p = 2c or p = 2c 1 HH, (Λ n ) = k n HH p,q (Λ n ) = in all other cases. Remark 4.3 The cyclic cohomology of these truncated quiver algebras has been computed in [BLM] and [Li]. The resolution of theorem (4.1) also enables us to calculate other Hochschild homologies, useful for the computation of HC π,α,β (A). Let α and β be characters on A; we are going to compute the Hochschild homology H (A, β k α ), for n 2. This is equal to Tor Ae(A, βk α ), that is to the homology of the complex β k α A A P A. This complex is isomorphic to the complex β k α k e Γ ( ) with the following differentials: βk α k e Γ (2c) β k α k e Γ (2c 1) 1 b 1...b nc n 1 j= α(b (c 1)n+j+2...b cn )β(b 1...b j ) b j+1...b (c 1)n+j+1 βk α k e Γ (2c+1) β k α k e Γ (2c) 1 b 1...b nc+1 α(b 1 ) b 2...b nc+1 β(b nc+1 ) b 1...b nc. Note that the differentials do not preserve the gradation in this case. Now the list of characters on A is the following: for each vertex e i, there is one character α i which is worth 1 on e i, and on every other path in (the e i are orthogonal idempotents, and the edges either have different extremities, or have a power which is zero). In particular, the characters vanish on all the paths of length greater than 1, so the differentials vanish. Therefore, the homology is: H 2c (A, β k α ) = β k α k e k nc H 2c+1 (A, β k α ) = β k α k e k nc+1. 9

10 Example: For the Taft algebras, the results are: H 2c (Λ n, β k α ) = k if α = β, H 2c+1 (Λ n, β k α ) = H 2c (Λ n, β k α ) = if α = α i+1 and β = α i, H 2c+1 (Λ n, β k α ) = k and in all other cases the homology is in all degrees Second case: n = or n = 1 We shall now look at the cases n = and n = 1. The case n = 1 is quite simple: k /m is equal to k = s ks, so that HH p (k /m) = s HH p (ks) = ks if p =, s if p >. Finally, for the case n =, we can state: Proposition 4.4 The Hochschild homology of k is given by: HH (k ) = kc, HH 1 (k ) = { L(γ) 1 i= t i γ(γ)/ γ C, L(γ) 1} HH p (k ) = if p 2. Proof: We shall use the following resolution (see for instance [C2]): Lemma 4.5 ([C2] theorem 2.5) There is a k bimodule projective resolution of k given by... k k k 1 k k k k k k. Tensoring by k over k e yields the following complex:... k k e k 1 δ kc. ν a νa aν The space HH (k ) is generated by the cycles in C, subjected to the relations given by the image of δ. Since δ(ν a) = νa t νa (νa), the relations identify two cycles in the same orbit, and HH (k ) = kc. The complex is C graded; therefore, to find HH 1 (k ) = ker δ, it is sufficient to consider elements of type x = L(γ) 1 i= λ i t i γ (γ), in which ν a is identified with νa, and the λ i belong to k. We have δ(x) = iff λ = λ 1 =... = λ L(γ) 1, and the result follows. 1

11 Remark 4.6 We may there again compute the homology H (A, β k α ), when n is equal to or 1. For n =, we shall once more use the resolution of lemma 4.5. Tensoring it by β k α over k e, we obtain the following complex:... β k α k e k 1 β k α k e k. All the maps in this complex are zero (the characters vanish on the edges). Given any character χ, let e χ denote the unique vertex such that χ(e χ ) = 1. Then H (k, β k α ) = β k α k e k = eα k 1 e β, and H 1 (k ) = β k α k e k is equal to if α β, and to ke α if α = β. For n = 1, we have H (ks, β k α ) = H (k, k) if α = β or if both characters differ from α s ; in the other cases, H (ks, β k α ) =. Hence, H (k /m, β k α ) = k # if α = β, k # 2 if α β, and H p (k /m, β k α ) = for all p >. 4.2 Cyclic homology of graded algebras In this paragraph, k is a commutative ring which contains Q. When A is a graded k algebra, Connes SBI exact sequence splits in the following way: Theorem 4.7 ([L] Theorem ) Let A be a unital graded algebra over k containing Q. Define HH p (A) = HH p (A)/HH p (A ) and HC p (A) = HC p (A)/HC p (A ). Connes exact sequence for HC reduces to the short exact sequences: HC n 1 HH n HC n. This will enable us to compute the classical cyclic homology of truncated quiver algebras. Let us first consider the cases n = and n = 1. Combining the results for Hochschild homology and theorem 4.7 yields the following: Proposition 4.8 The cyclic homology of k and of k /m are given by: for all c N. HC 2c (k /m) = s ks HC 2c+1 (k /m) = The case n 2 involves the same methods: Proposition 4.9 Suppose n 2. Then: and HC (k ) = kc HC 2c (k ) = k # HC 2c+1 (k ) =, dim k HC 2c (k /m n ) = # + n 1 e=1 dim k HC 2c+1 (k /m n ) = r n(r 1)b r. a cn+e r (c + 1)n n / rn (r n 1)b r 11

12 Proof: In the first place, A is equal to k, so that we know the homologies of A (see proposition 4.8). Next, we have HC (A) = HH (A) = k # n 1 + e=1 ae. Then, using theorem 4.7, we get the following formula: dim k HC 2c (A) + dim k HC 2c+1 (A) = # + In particular, dim k HC 1 (A) = r n(r 1)b r. An induction on c yields the result. r (c+1)n Corollary 4.1 When is the n crown, then the results are: HC 2c (Λ n ) = k n, HC 2c+1 (Λ n ) = k n 1 for c N. (r n 1)b r + n 1 e=1 a cn+e. Example: Let A be the quotient algebra k[x]/(x n ). It has a presentation by quiver and relations (the quiver has one vertex and one loop), and its cyclic homology is k n in even degree, and vanishes in odd degree. Remark 4.11 This agrees with the general results given in [BACH] and [BACH2] (in which k may be a field of positive characteristic, or in fact any commutative ring). 4.3 Connes and Moscovici homology of some truncated algebras Here, k is a commutative ring which contains a primitive n th root of unity q. The Taft algebra Λ n is then a Hopf algebra (see [C1]), with the following structure maps: ε(e i ) = δ i,, ε(a i ) =, (e i ) = e j e k, (a i ) = (e j a k + q k a j e k ), j+k=i j+k=i S(e i ) = e i, S(a i ) = q i+1 a i 1, where δ is the Kronecker symbol. We can therefore consider the homology HC π,α,β (Λ n ). First of all, we need to find out which characters and grouplikes (π, α, β) satisfy the necessary conditions (cm). There are n grouplike elements in Λ n, given as follows: π i = n 1 l= A triple (π i, α u, α v ) satisfies (cm) iff ui (mod n), vi (mod n), and v u i (mod n). For instance, when u = v, this means that i is necessarily equal to -1, and u and v are equal to (that is α u = ε = α v ); when v = u 1, it means that i is equal to, that is that π i is equal to 1. We shall not need to know the details of the other possibilities. Using the long periodic exact sequence for HC π,α,β (proposition 2.2), we obtain the following results: q il e l. 12

13 Proposition 4.12 HC π n 1,ε,ε p (Λ n ) = HC 1,αu,α u 1 p (Λ n ) = k p/2+1 if p is even, if p is odd, if p is even, k p+1/2 if p is odd, for all u {,..., n 1}, HC π,α,β (Λ n ) = in every other case. Remark 4.13 Note that none of these homologies are direct factors in the classical HC (Λ n ), which seems to preclude any possibility of decomposing HC (Λ n ) as a sum of Connes and Moscovici homologies. References [AG] Anick, D. and Green, E. L., On the homology of quotients of path algebras, Comm. Alg. 15 (1987), pp [BACH] Buenos Aires Cyclic Homology Group, Cyclic homology of algebras with one generator, K-Theory 5 (1991), no. 1, pp Jorge A. Guccione, Juan Jos Guccione, María Julia Redondo, Andrea Solotar and Orlando E. Villamayor participated in this research. [BACH2] Buenos Aires Cyclic Homology Group, Cyclic homology of monogenic algebras, Comm. Algebra 22 (1994), no. 12, pp Jorge A. Guccione, Juan Jos Guccione, María Julia Redondo, Andrea Solotar and Orlando E. Villamayor participated in this research. [BLM] Bardzell, M.J., Locateli, A.C., Marcos, E.N., On the Hochschild cohomology of truncated cycle algebras, Comm. Algebra 28 (2), no. 3, pp [B] Burghelea, D., The cyclic homology of the group rings, Comment. Math. Helv. 6 (1985), no. 3, pp [C1] Cibils, C., A quiver quantum group, Comm. Math. Phys. 157 (1993), no. 3, pp [C2] Cibils, C., Rigid monomial algebras, Math. Ann. 289 (1991), no. 1, pp [CGV] Cortinas, G., Guccione, J. and Villamayor, O.E., Cyclic homology of K[Z/pZ], Proceedings of Research Symposium on K-Theory and its Applications (Ibadan, 1987), K- Theory 2, 5 (1989), pp [CM] Connes, A. and Moscovici, H., Cyclic cohomology and Hopf algebras, Lett. Math. Phys. 48 (1999), no. 1, pp [CM2] Connes, A. and Moscovici, H., Hopf algebras, cyclic cohomology and the transverse index theorem, Comm. Math. Phys. 198 (1998), no. 1, pp [CM3] Connes, A. and Moscovici, H., Cyclic Cohomology and Hopf Symmetry, preprint no. math.oa/2125, Jussieu. 13

14 [Cr] Cyclic cohomology of Hopf algebras, and a non-commutative Chern-Weil theory, preprint no. math.qa/ , Jussieu. [G] Gorokhovsky, A., Secondary Characteristic Classes and Cyclic Cohomology of Hopf Algebras, preprint no. math.oa/2126, Jussieu. [KV] Karoubi, M., and Villamayor, O.E., Homologie cyclique d algbres de groupes, C. R. Acad. Sci. Paris Sr. I Math. 311 (199), no. 1, pp 1-3. [Li] Locateli, A. C., Hochschild cohomology of truncated quiver algebras, Comm. Algebra 27 (1999), no. 2, pp [L] Loday, J-L., Cyclic homology, Appendix E by María O. Ronco, Springer-Verlag, Berlin (1992). [LQ] Loday, J-L. and Quillen, D., Cyclic homology and the Lie algebra homology of matrices, Comment. Math. Helv. 59 (1984), no. 4, pp [M] Montgomery, S., Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (1993). [S] Skldberg, E., The Hochschild homology of truncated and quadratic monomial algebras, J. London Math. Soc. (2) 59 (1999), no. 1, pp [W] Weibel, C., An introduction to homological algebra, Cambridge University Press, Cambridge (1994). 14