REPRESENTATION THEORY OF LIFTINGS OF QUANTUM PLANES

arXiv:0712.1078v3 [math.QA] 12 Jan 2009

REPRESENTATION THEORY OF LIFTINGS OF QUANTUM PLANES WILLIAM CHIN AND LEONID KROP Abstract. We determine the regular representations, Gabriel quivers and representation type of all liftings of two-dimensional quantum linear spaces.

Introduction Liftings of quantum linear spaces were constructed and completely described in [3], and independently in [6]. These Hopf algebras belong to the class of pointed finite-dimensional Hopf algebras with an abelian group of group-likes and are, arguably, the simplest Hopf algebras constructed by the lifting method of N. Andruskiewitsch and H.-J. Schneider (see e.g. [4]). When the dimension of a quantum linear space is two, we colloquially refer to them as quantum planes. In this article we systematically study the representation theory of liftings of quantum planes. The group of grouplikes is arbitrary finite abelian in our construction, and the base field k is algebraically closed of characteristic zero. Liftings of quantum planes are generated as algebras by two skew-primitive elements along with the grouplikes G. The skew-primitives generalize the generators e, f in the restricted enveloping algebra of sl(2) and quantum analogs at roots of unity. If there is a nontrivial commutation relation between the skew-primitives we say that the lifting is linked. Let us summarize some existing related work. The simple representations of liftings of quantum planes were treated in [1, 2] where simple modules are described or reduced to known theory in most cases. The representation theory of various versions of quantized restricted enveloping algebras was studied in [7, 23, 21] using a variety of techniques. In [11] representations of quantum doubles of generalized Taft algebras were examined. The simple and projective modules were explicitly constructed within the regular representation, extending methods from [21]. These Hopf algebras are examples of liftings of quantum planes in the linked nilpotent case. In [13] rank one Hopf algebras were constructed and the structure of the regular representation of their doubles was obtained. These Hopf algebras are sometimes liftings of quantum planes; but this is not always the case as sometimes they are not even pointed. Date: 10/17/08. 1

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WILLIAM CHIN AND LEONID KROP

In this article we explicitly describe the simples, projectives, blocks, and Gabriel quiver in all cases. This enables us to determine the representation types of the blocks. We make use of certain central idempotents that are constructed from certain equivalence classes of characters of G to reduce to the corresponding two-sided summands, which we call class subalgebras. These idempotents were also used in [1, 2]. The class subalgebras we encounter are generated by three elements, one of which is a unit, and the other two elements are the images of skew-primitives. Our analysis breaks into the consideration of summands where both, just one, or neither of the generators is nilpotent. We shall refer to these cases as nilpotent, seminilpotent and unipotent, respectively. Another division is into the classes of linked and unlinked liftings, resulting in six distinct cases altogether. Let us give an outline of the results. In the unlinked nonnilpotent cases, the class subalgebras are the blocks, and are either Nakayama algebras, or skew group algebras over truncated polynomial rings. The Nakayama algebras are the simplest algebras of finite representation type, while the truncated polynomial rings are among simplest of the algebras of wild and tame representation type (depending on the degree of truncation). In the linked nonnilpotent cases the class subalgebras are direct sums matrix algebras over a Nakayama algebras and are thus of finite representation type. In the remaining linked nilpotent case the nonsimple blocks are special biserial algebras and therefore of tame representation type. The algebras in this case generalize the quantized restricted enveloping algebra of sl(2). The basic algebras and quivers that occur here also arose in the study of Hopf algebras in [11], and we can apply their results to this case. Here we use an alternative less explicit but much simpler determination of the simple and projective representations using analogs of “baby Verma modules” in the more general setting of seminilpotent and nilpotent liftings of quantum planes. We show that linked liftings are symmetric algebras and we provide an analog of the Casimir element. These tools allow a concise and efficient determination of the socle and Loewy series structure of the projective indecomposables, and the minimal central idempotents of nonsimple blocks. A more detailed description of material by sections is as follows. In section 1 we review the ideas leading up to the construction of liftings of quantum linear spaces. We define types of liftings and discuss some auxiliary facts about duality for finite abelian groups and group algebras used in the sequel. In section 2 we introduce the general theory of liftings of quantum planes. We define class idempotents, class subalgebras and compute bases for them. A quick argument proving Lemma 2.2 shows that the class idempotents are central. In §2.1 we give generators and relations for liftings and comment on the existence of unlinked or linked data. In section 3 we determine the structure of simples, projectives, Gabriel quiver and blocks for unlinked liftings. The nilpotent case is handled in Theorem 3.1, where the class subalgebras are the blocks and are essentially

REPRESENTATION THEORY OF LIFTINGS

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skew group rings over truncated polynomial rings. The seminilpotent case is described in Theorem 3.2 where the class subalgebras either resemble the nilpotent case or are nonbasic Nakayama algebras. In this case the Gabriel quiver has vertices corresponding to certain cosets of a certain subgroup of the character group. In the unipotent case, we show in Theorem 3.3 that the class subalgebras are semisimple, and we explicitly construct the simple modules. We also give a criterion for the isomorphism of the simples in Theorem 3.4. We turn to the linked liftings in section 4, where we find an analog of the Casimir element from classical Lie theory, which is central by Lemma 4.1. In Proposition 4.2 we show that linked liftings are symmetric algebras by directly showing unimodularity and that the square of the antipode is inner. In §4.2 we study the image of the Casimir element C in each class subalgebra and compute the minimal polynomials. In the unipotent case we quickly find in Thereom 4.5 that each class subalgebra is isomorphic to a matrix algebra over the subalgebra generated by C. As a result, each class subalgebra is a direct sum of matrix rings over local rings. In the seminilpotent case, addressed in Theorem 4.7 where the blocks are matrix rings over the base field or over the truncated polynomial algebra k[v]/(v 2 ). The precise decompositions are expressed in a number of cases. For use in the seminilpotent and nilpotent cases, we introduce analogs of standard cyclic modules in §4.5, where they are shown to be simple in Proposition 4.8 in the seminilpotent case. Finally we study the lengthier nilpotent case in §4.6. Here the standard cyclic modules may no longer be simple, but have a unique maximal submodule. Paralleling Lie theory, we obtain precise results concerning the simple quotients. As a first step in Theorem 4.13, we dispose of the generic case where the class subalgebra is semisimple. We then look at the complementary case and we give detailed results about the Loewy factors of the standard cyclic modules. This enables the determination of the structure of the projective indecomposable modules in Theorem 4.18, along with a presentation of the basic algebras of blocks by quivers with relations in Theorem 4.20. We close by explicitly giving embeddings of the projective indecomposables into nonsimple blocks. 1. Preliminaries Notation: G a finite abelian group b the character group of G G k an algebraically closed field a1 , . . . , an ∈ G b χ1 , . . . , χn ∈ G qij = χj (ai ), qi = χi (ai ) For g ∈ G |g| denotes the order of g. In particular, for q ∈ k• |q| is the order of q. For κ ∈ k and N ∈ N, Rκ,N = {γ ∈ k|γ N = κ}

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1.1. Liftings of Quantum Linear Spaces. Recall [22, 19, 4] that the category G G YD consists of kG-modules and kG-comodules V such that the G-grading M V = Vg where Vg = {v ∈ V |ρ(v) = g ⊗ v} where ρ : V → kG ⊗ V is the comodule structure map, satisfies h.Vg ⊂ Vhgh−1 for every h, g ∈ G.

b Since G is abelian h.Vg = Vg and V has a basis, say, {vi |i ∈ I} of G and Geigenvectors indexed by a set I. In other words, (1.1)

g.vi = χi (g)vi

(1.2)

ρ(vi ) = ai ⊗ vi

b and for all g ∈ G, i ∈ I. for some ai ∈ G and χi ∈ G, We say that an element v ∈ V is bihomogeneous of degree (a, χ) if the equations (1.1) and (1.2) hold for v, g and χ and we write a = gv , χ = χv . Definition 1.1. ([3]) An n- dimensional Yetter- Drinfel’d module V is called a quantum linear space if χi (aj )χj (ai ) = 1 for all i 6= j

in some bihomogeneous basis for V .

We review a construction of the Nichols algebra B(V ) associated to the quantum linear space V (cf. [3, Lemma 3.4]). As an algebra B(V ) is defined via the relations vi vj = χj (ai )vj vi v ni = 0 for all i 6= j and ni = |qi |. One can see immediately that the set {v1i1 · · · vnin } is a basis of B(V ). The action and coaction of G on V both extend uniquely to B(V ) by requiring B(V ) to be an algebra in G G YD. Explicitly g.v1i1 · · · vnin = χi11 · · · χinn (g)v1i1 · · · vnin

ρ(v1i1 · · · vnin ) = ai11 · · · ainn ⊗ v1i1 · · · vnin . Note that every monomial v1i1 · · · vnin is bihomogeneous. The coalgebra structure maps δ, ǫ are defined by first setting ǫ(v1i1 · · · vnin ) = δ0,i1 · · · δ0,in

and for δ : B(V ) → B(V ) ⊗ B(V ) is given by

δ(vi ) = vi ⊗ 1 + 1 ⊗ vi

on generators and extends to B(V ) as follows. Following Lusztig [14] or [4] we give B(V ) ⊗ B(V ) a new multiplication by setting (r ⊗ s)(t ⊗ u) = χt (gs )rt ⊗ su


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for monomials r, s, t, u, and denote this algebra by B(V )⊗B(V ). We also endow B(V ) ⊗ B(V ) with the usual tensor product module and comodule structures over kG. A tedious, but straightforward calculation shows that B(V )⊗B(V ) is an algebra in G G YD, δ is well-defined and δ, ǫ are G- linear and G- colinear algebra maps. The above mentioned properties amount to saying that B(V ) is a bialgebra in G G YD. We turn B(V ) into a Hopf algebra G in G YD by defining an antipode S as the linear map P

S(v1i1 · · · vnin ) = (−1)

ij

n Y (ij ) qj 2 v1i1 · · · vnin .

j=1

We can use now a result of Radford [20], recast by Majid [15] in categorical terms, to the effect that the biproduct or bosonization B(V )#kG is a Hopf algebra. The algebra B(V )#kG is is known as the trivial lifting of V . Definition 1.2. A Hopf algebra H is a lifting of V if the graded Hopf algebra associated to the coradical filtration of H is isomorphic to B(V )#kG. Next we derive a general property of liftings. Proposition 1.3. Every lifting of V has a structure of left-left YetterDrinfel’d module. Proof: Let H be a lifting of V . By [3] H0 = kG hence H is naturally a G-module under the action of G by conjugation, h 7→ g.h := ghg−1 for every h ∈ H, g ∈ G. Moreover, H is generated by G and skew-primitives xi satisfying g.xi = χi (g)xi . By [3, 5.2] the set {gxi11 · · · xinn |g ∈ G, 0 ≤ P ij < nj } is a basis for H. Let I be the span of all gxi11 · · · xinn with ij > 0. By the quantum binomial formula [12] one can see readily that I is a coideal and it complements H0 . Therefore we have a coalgebra projection π : H → kG. Then ρ := (π ⊗ id)∆ : H → kG ⊗ H equips H with a kGcomodule structure. Further, every u = gxi11 · · · xinn is bihomogeneous of degree (gai11 · · · ainn , χi11 · · · χinn ). Therefore for every h ∈ G ρ(h.u) = χi11 · · · χinn (h)ρ(u) = gai11 · · · ainn ⊗ h.u

and the proof is complete. We apply the previous proposition and consider the braided commutator [ , ]c : H ⊗ H → H corresponding to the braiding c : H ⊗ H → H ⊗ H arising from the Yetter- Drinfel’d module structure on H [4]. This commutator is given by [a, b]c = µ(id − c)(a ⊗ b) for a, b ∈ H where µ is the multiplication in H. When a, b are bihomogeneous we have [a, b]c = ab − χb (ga )ba.

We shall later have use of the following braided commutator rules. Let x, y be bihomogeneous elements of degrees (a, χ) and (b, χ−1 ), respectively, satisfying (1.3)

[x, y]c = ab − 1

and χ(a) = χ(b).

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Proposition 1.4. Let H be a lifting of V . Suppose u, v ∈ H are bihomogeneous and x, y are as above. Then (a) [x, uv]c = [x, u]c v + χu (a)u[x, v]c (b) Let q = χ−1 (a). For every s ≥ 1 (i) [x, y s ]c = (s)q y s−1 (q s−1 ab − 1) (ii) [y, xs ]c = q −s (s)q x−s+1 (q −(s−1) ab − 1). Proof: (a) is seen by a direct inspection. (b) (i). The formula holds for s = 1 by 1.3. We induct on s assuming it holds for the given s. By part (a) we calculate [x, y s+1 ]c = [x, y s · y]c = (s)q y s−1 (q s−1 ab − 1)y + q s y s (ab − 1) because y s is a G- eigenvector of weight χ−s . Since aby = q 2 yab the first term on the right is (s)q y s (q s+1 ab − 1). Therefore the right hand side equals γy s ((q s + (s)q q s+1 )ab − (q s + (s)q )) = γy s (s + 1)q (q s ab − 1) the last equality holds by the identities 1 + (s)q q = (s + 1)q and q s + (s)q = (s + 1)q . Part (b)(ii) is proven similarly using the relation yx − q −1 xy = −q −1 (ab − 1) and (s)q−1 = q −s+1 (s)q . 1.2. Duality for abelian groups and group algebras. We collect some basic facts that will be used freely henceforth. b as before. Then Let G be a finite abelian group and its dual group G there is an inclusion-reversing correspondence between the subgroups of G b The correspondence takes a subgroup H ⊂ G to and the subgroups of G. b H ⊥ = {λ ∈ G|λ(h) = 1 for all h ∈ H}.

bb = G and we note The inverse is defined similarly using the identification G that H ⊥⊥ = H. We have (H ∩ K)⊥ = H ⊥ + K ⊥ and (H + K)⊥ = H ⊥ ∩ K ⊥ [16]. Now considering the Hopf group-algebra kG we extend the natural pairing

to a bilinear pairing

b → k• , hg, λi → λ(g) G×G

X X X b → k, h kG × kG ri gi , ρ i λi i → ri ρj λj (gi ).

b we associate a minimal idempotent To every λ ∈ G 1 X λ(g−1 )g eλ = |G| g∈G


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of kG. The subspaces keλ afford a one-dimensional representation of G with character λ, i.e. geλ = λ(g)eλ . Therefore 1 X ( λ(g−1 )µ(g))eµ eλ eµ = |G| g∈G

= µ(eλ )eµ .

By orthogonality of idempotents eλ we have (1.4)

heλ , µi = µ(eλ ) = δλ,µ .

b of kG is Equality (1.4) can be interpreted as saying that basis {eλ |λ ∈ G} b of kG. b That is to say eλ maps to the dual to the standard basis {µ|µ ∈ G} b ∗. characteristic function pλ under the Hopf algebra isomorphism kG ∼ = (kG) Therefore X ∆(eλ ) = eµ ⊗ eν . λ=µν

2. General Theory of Liftings 2.1. Types of Liftings. The general lifting H of V is given by a lifting datum D = {G, a, b, ǫ1 , ǫ2 , χ1 , χ2 , γ} b and ǫi ∈ {0, 1}, γ ∈ k. H is generated by G and x, y where a, b ∈ G, χi ∈ G subject to the relations of G and the following (2.1)

(2.2) (2.3)

xn1 = ǫ1 (an1 − 1) xn2 = ǫ2 (bn2 − 1) gx = χ1 (g)xg

(2.4)

gy = χ2 (g)yg

(2.5)

xy − χ2 (a)yx = γ(ab − 1)

The coalgebra structure of H is given by (2.6) (2.7) (2.8)

∆(x) = a ⊗ x + x ⊗ 1 ∆(y) = b ⊗ y + y ⊗ 1 ∆(g) = g ⊗ g

for all g ∈ G. The datum satisfies the conditions (2.9)

(2.10) (2.11) (2.12) (2.13)

1 < n1 = |χ1 (a)| 1 < n2 = |χ2 (b)|

χ1 (b)χ2 (a) = 1

χni i = ǫ if ǫi = 1 χ1 · χ2 = ǫ and ab 6= 1 if γ 6= 0

We adopt the following terminology • A lifting is linked if γ 6= 0

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• A lifting is nilpotent if ǫi = 0 for all i • A lifting is seminipotent if ǫi = 0 for exactly one i • A lifting is unipotent if ǫi = 1 for all i We comment on existence of data. Assume G is a product of several cyclic groups, viz. G = hg1 i × hg2 i × · · · . Suppose |gi | = mi , i = 1, 2 and b i = 1, 2 by let θi be a primitive root of 1 of order mi . Define χi ∈ G, χi (gj ) = δij θi . The tuple {G, g1 , g2 , χ1 , χ2 , 0} is a lifting datum. If G is a cyclic p-group generated by g, pick an integer s such that s2 6≡ 0 mod |g|. Set a = g, b = gs , and χ1 (a) = θ, χ2 (a) = θ −s . Then {G, a, b, χ1 , χ2 , 0} is a lifting datum. Following [4] we call a datum linkable if χ1 χ2 = ǫ and ab 6= 1. A linkable datum can be constructed as follows. Let G 6= Z2 . Pick a subgroup L of G such that G/L is cyclic of order N , generated by g = gL with g ∈ G. Define a character φ : G/L → k• by sending g to θ, where θ is a root of 1. Let χ be the pull-back of φ to G. We claim that there exists a, b ∈ G such that b = al, l ∈ G with ab 6= 1. If so, the tuple {G, a, b, χ, χ−1 , γ} is a linkable datum for every γ ∈ k• . It remains to justify the claim about the elements a, b. If L = 1, then G is a cyclic group of order > 2. Hence g2 6= 1. Thus a = g = b will do. Suppose L 6= 1. If g2 ∈ / L, then a = g, b = gl, l ∈ L will do. In case 2 g ∈ L, say, g2 = l0 , then either l0 6= 1, and we can set a = g = b, or g2 = 1 and then a = g, b = gl, l 6= 1 will do. 2.2. Class idempotents. b we associate the idempotent eλ as above Definition 2.1. For each λ ∈ G and let X = hχ1 , χ2 i. Let X eλX = eµ µ∈λX

for each coset λX.

For h ∈ HeλX we shall write h for its image heλX ∈ HeλX . Let T be a transversal for X ⊥ in G. For each subgroup L ⊂ G, we shall write λL for the restriction of λ to L and put X λ(g−1 )g ∈ kL. eλL = |L|−1 g∈L

b Lemma 2.2. eλX is a central idempotent in H for all λ ∈ G. Proof: Observe that

xeλ = |G|−1

X

λ(g−1 )xg

g∈G

= (|G|−1

X

g∈G

= eλχ1 x

λ(g−1 )χ1 (g−1 )g)x


and similarly yeλ = eλχ2 y. This yields the assertion.

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Proposition 2.3. The following sets (a) {txj1 y j2 |0 ≤ ji < ni , t ∈ T } and (b) {eλχ xj1 yj2 |0 ≤ ji < ni , χ ∈ X} are bases of HeλX . Proof: We have eλ g = λ(g)eλ for all λ ∈ X and g ∈ G. This yields eλX g = λ(g)eλX if g ∈ X ⊥ . It follows directly that the set in (a) in the statement spans HeλX . The cardinality of the set in (a) is [G : X ⊥ ]n1 n2 , and since [G : X ⊥ ] = |X|, the set has cardinality |X|n1 n2 . Summing over b gives a spanning set for ⊕HλX having cardinality a transversal for X in G b [G : X]|X|n1 n2 = |G|n1 n2 . This is just the dimension of H, so our spanning set is a basis of H. This trivially implies that the spanning set in (a) is a basis for HeλX . The set in (b) is clearly a spanning set for HeλX , and as in the proof of (a), it is a basis because it has the requisite cardinality |X|n1 n2 . Lemma 2.4. eλX = eλX ⊥

Proof: Since eλX ⊥ ∈ kX ⊥ ⊂ kG, we can express X eλX ⊥ = eµ µ∈Y

b Multiplying in terms of the basis {eµ } for kG, indexed over some set Y ⊂ G. ⊥ this expression by h ∈ X we conclude that X λ(h)eλX ⊥ = µ(h)eµ . µ∈Y

This implies that µX ⊥ = λX ⊥ , i.e. µ ∈ λX. Therefore the number of µ’s occuring in the sum equals the dimension of eλX ⊥ kG, which is |G/X ⊥ | = |X|. It follows that the sum runs over all of λX, as desired. 3. Representations of Unlinked Liftings 3.1. Nilpotent unlinked liftings. Theorem 3.1. Let H be an unlinked lifting and suppose ǫ1 = 0 = ǫ2 . Then (a) Every indecomposable projective module has the form Pλ = Heλ for b some λ ∈ G (b) The central idempotents of H are the eλ indexed by the cosets λX ∈ b G/X. (c) The Gabriel quiver for each block HeλX has vertices corresponding to elements of λX and a pair of arrows µ → µχ1 µ → µχ2

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for every µ ∈ λX. (d) H is of wild representation type, unless n1 = n2 = 2 in which case it is of tame representation type. Proof: Set J = xH + yH. By the hypothesis, J is a nilpotent ideal, and so is the Jacobson radical of H. Therefore H has |G| indecomposable projective b account for all of them. modules. Thus the projective modules Heλ , λ ∈ G This proves (a). We let Lλ = Heλ /Jeλ denote the corresponding simple H- module. It is b easy to see that for all r ∈ N, λ ∈ G, X J r eλ = kxi y j eλ , i+j≥r

whence each Lλ is one-dimensional, and Jeλ /J 2 eλ is two-dimensional, being spanned by the images of x and y. Note that the respective simple modules have weights λχ1 and λχ2 . This says that Ext1H (Lλ , Lλχi ), i = 1, 2 are one b is arbitrary, this accounts for the arrows labelled - dimensional. As λ ∈ G as in the statement. Thus the blocks are precisely indexed by the cosets as claimed. This completes the proof of (c). Since H is a basic algebra, the central idempotents are exactly sum of the primitive idempotents corresponding to the vertices in each connected component of the quiver. This yields (b). H is a skew group ring over the subalgebra generated by x and y. One can easily see that we can replace x by a−1 x and assume that x and y commute. Thus H is a skew group ring over a truncated polynomial ring, say A. Since |G| is invertible in k, the arguments [5, 6.3] show that A and H = AG have the same representation type. The assertion for the truncated polynomial rings follows from [9], whence (d).

3.2. Seminilpotent unlinked liftings. Next we describe indecomposable projective modules for unlinked data with exactly one ǫi = 0. Let Xi denote b generated by χi , and put X = X1 X2 as before. We the subgroup of G b in this setting as follows. Let N denote introduce another subgroup of G n ⊥ b Note that X ⊂ N . This can be argued as follows. the subgroup ha 1 i of G. n1 Since χ1 = ǫ and χ2 (a) = χ1 (b−1 ), we obtain χ2 (an1 ) = 1, which proves the inclusion. We let N/X1 denote the cosets modulo X1 represented by elements of N b \ N )/X1 for the complementary set of cosets. Note as usual, and write (G b \ N )/X1 are stable under the action by that the disjoint sets N/X1 and (G multiplication by elements of X.

Theorem 3.2. Suppose H is an unlinked seminilpotent lifting with ǫ1 = 1 and ǫ2 = 0. Then (a) Let λ ∈ N . Then Heλ is the projective cover of a one- dimensional H- module.


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b \ N . Then Heµ is the projective cover of a simple n1 (b) Let µ ∈ G dimensional H- module. (c) The isotypic component of Heµ is given by equivalence modulo X1 , consisting of the projectives Heλ , λ ∈ µX1 . b are a complete set of primitive orthogonal idempotents (d) The eλ , λ ∈ G in H (e) Let λ ∈ N . Then HeλX is a block. The Gabriel quiver of this block has vertices corresponding to elements of λX and a pair of arrows µ → µχ1 µ → µχ2 for every µ ∈ λX (with doubled arrows if χ1 = χ2 ). The block HeλX is of wild representation type, unless n1 = n2 = 2, in which case it is of tame representation type. b \ N . Then HeµX is a block. The Gabriel quiver of this (f) Let µ ∈ G block is cyclic and has vertices corresponding to cosets in µX/X1 and arrows µX1 → χ2 µX1

corresponding to multiplication by χ2 . The block HeµX is a Nakayama algebra. b (g) The eλX indexed by the cosets λX ∈ G/X are a complete set of block idempotents of H. Proof: As noted above the subgroup N contains X, so that the equivalence b \ N instead, we classes λX, λ ∈ N form a partition of N ; taking λ ∈ G b \ N . By Lemma similarly see that the classes mod X form a partition of G 2.2 the idempotents eλX are all central. The two partitions combine to form b and we let a partition of G X HN = Heλ λ∈N

= HG\N b

X

Heλ

b λ∈G\N

denote the complementary two-sided summands of H. Further let J = , which is a two-sided ideal of H since HN and HG\N xHN + yHN + yHG\N b b can be written as the sum of class subalgebras, each of the form HeλX . First suppose that λ ∈ N . Then xn1 eλ = 0 as λ(an1 ) = 1. We also have n 2 y = 0, hence the image J in HN is nilpotent. Thus J lies in the Jacobson radical of HN . Next consider the projective modules Heλ . It is plain that Jeλ = ⊕i+j>0 kxi y j eλ , so Heλ /Jeλ is one-dimensional, spanned by a vector of weight λ. This finishes the proof of (a). b Secondly, suppose that µ ∈ G/N and consider the factor Heµ /Jeµ . Since is nilpotent. In this case y n2 = 0, it is clear that the image of J in HG\N b

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xn1 eµ is a nonzero scalar and Heµ /Jeµ = ⊕0≤i≤n1 −1 kxi eµ + Jeµ Observe that the action of x cyclically permutes the basis {xi eµ + Jeµ }, the vector xi eµ has weight µχi for all i = 0, . . . , n1 − 1 and that these weights are pairwise distinct. It now follows easily that Heµ /Jeµ is a simple Hmodule. For future reference let us denote this simple module by Lµ . This finishes the proof of (b). We show next that the isotypic component of Lµ consists of the simple modules {Lµχ |χ ∈ X1 }. Since the multiplicity of Lµ equals dim Lµ = n1 , we need to find n1 modules isomorphic to Lµ . If µ = µ′ χj1 , then the Hmodule map specified by eµ + Jeµ 7→ xj eµ′ + Jeµ′ gives an isomorphism Lµ → Lµ′ . This accounts for the isotypic component consisting of the n1 mutually isomorphic projective (or simple) modules, which demonstrates (c). b are a complete set of orthogTo prove (d), first observe that the eλ , λ ∈ G onal idempotents in kG by construction (§1.2). The proofs of parts (a) and (b) show the projective modules of Heλ are all simple modulo the nilpotent ideal J. Thus J is the radical of H, so the eλ are all primitive as well. The proof of (e) is similar to the nilpotent case addressed in the previous theorem, and will be omitted. We prove (f). Observe that Jeµ /J 2 eµ = ⊕0≤i 0, then HeλX vs is a nonzero proper submodule of Z ′ (ρ).

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Using Lemma 4.11(2) we quickly deduce that (for s > 0) F.vs = 0 if and only if ρ2m = q 1−s . Setting e′ = s − 1, we arrive at the desired equivalence. It follows from ρ2m = q 1−s that HeλX vs contains no F -trivial vectors other than vs . Thus HeλX vs is the unique proper nonzero submodule of Z ′ (ρ). The proof of (b) is similar using Lemma 4.11 (1). We dispose easily of the case where κ2 6= 1. Theorem 4.13. Assume that κ2 6= 1. then HeλX ∼ = Mn (k)N .

Proof: Suppose that Z(ρ) is not simple. Then the result above says that ρ2m = q e ,which implies κ2 = ρ2N = ρ2mn = q en = 1. We conclude that every Z(ρ) is simple. Furthermore, since each Z(ρ) contains a unique Etrivial vector of weight ρ, we see that the Z(ρ) are pairwise nonisomorphic. As dim HeλX = N n2 , the proof is complete. 4.7. κ2 = 1. Let rad Z(ρ) (resp. rad Z ′ (ρ)) denote the proper (possibly zero) submodule of Z(ρ) (resp. Z ′ (ρ)). Further let L(ρ) = Z(ρ)/rad Z(ρ). L′ (ρ) = Z ′ (ρ)/rad Z ′ (ρ). Proposition 4.14. Assume κ2 = 1. (a) The simple modules L(ρ), ρ ∈ Rκ,N are a full set of representatives of simple HeλX - modules. (b) The simple modules L′ (ρ), ρ ∈ Rκ,N are another set of representatives of simple HeλX - modules. (c) If Z(ρ) is not simple, Z(ρ) is a nonsplit extension of L(θ −(e(ρ)+1) ρ) by L(ρ). (d) If Z ′ (ρ) is not simple, Z ′ (ρ) is a nonsplit extension of L(θ n−1 ρ) by ′ L(θ e (ρ) ρ). (e) dim L(ρ) = e(ρ) + 1. (f) dim L′ (ρ) = e′ (ρ) + 1. Proof: (a)-(b). Every simple HeλX - module contains an E- trivial vector v, say of weight ρ ∈ Rκ,N . The map given by 1ρ 7→ v induces a nontrivial homomorphism Z(ρ) → M . Since L(ρ) contains a unique line of E-trivial vectors of weight ρ , L(ρ) ≇ L(ρ′ ) for ρ 6= ρ′ . This proves (a). The proof of (b) is similar. (c) and (e). By Proposition 4.12 (b) rad Z(ρ) is generated by an Eprimitive we(ρ)+1 , hence is the span of {wi |e(ρ) + 1 ≤ i ≤ n − 1}. This proves (e). Further, the weight of we(ρ)+1 is θ −(e(ρ)+1) ρ, so that rad Z(ρ) ∼ = L(θ −(e(ρ)+1) ρ). This proves (c). (d) and (f). By Proposition 4.12 (a) rad Z ′ (ρ) is the span of {vi |e′ (ρ) + 1 ≤ i ≤ n − 1}. Now, up to a scalar multiple, the only F - trivial weight vector of rad Z ′ (ρ) is vn−1 and the only F -trivial weight vector in L′ (ρ) is


23

′

ve′ + rad Z ′ (ρ). These vectors have weights θ e (ρ) ρ and θ n−1 ρ, respectively. This establishes (d) and (f). We define a mapping σ : Rκ,N → Rκ,N by ( θ −(e(ρ)+1) ρ if ρ is nonexceptional σ(ρ) = σ(ρ) = ρ otherwise. Lemma 4.15. (a) σ is a permutation of order 2m. (b) The orbits of Rκ,N under σ are of size 2m or 1. (c) σ −1 (ρ) = θ n−e(ρ)−1 ρ. Proof: (a)-(b). Suppose ρ is nonexceptional. By the two previous propositions e(σ(ρ)) + 1 + e(ρ) + 1 = n. This equality gives σ 2 (ρ) = θ −(e(σ(ρ))+1) σ(ρ) = θ −n ρ. We conclude that σ is a bijection and σ 2s (ρ) = θ −sn ρ for every integer s ≥ 0 hence σ 2m = 1. Suppose that for some odd number 1 + 2s < 2m σ 1+2s (ρ) = ρ. Then ρ = σ(σ 2s (ρ)) = σ(θ −sn ρ) = θ −(e(ρ)+1) θ −sn ρ because e(θ −sn ρ) = e(ρ). This implies e(ρ)+1 ≡ 0 mod n which contradicts the fact that e(ρ) ≤ n − 2. (c) To prove (c), notice that σ −2 (ρ) = θ n ρ. So σ −1 (ρ) = σ(θ n ρ) = n−e(ρ)−1 θ ρ as asserted. Proposition 4.16. For every nonexceptional root ρ, Z ′ (θ −e(ρ) ρ) is a nonsplit extension of L(σ −1 (ρ)) by L(ρ). When σ = σ −1 , Z ′ (θ −e(ρ) ρ) ≇ Z(ρ). Proof: Set ζ = θ −e(ρ) ρ and consider Z ′ (ζ). By Proposition 4.14 Z ′ (ζ) is ′ a nonsplit extension of L(θ n−1 ζ) by L(θ e (ζ) ζ). Now ζ 2m = q −2e(ρ) q e(ρ) = ′ q −e(ρ) so by the definition of e′ (ζ) we have e′ (ζ) = e(ρ). Therefore θ e (ζ) ζ = ρ while θ n−1 ζ = σ −1 (ρ) by part (c) of the preceding Lemma.. In the case where σ = σ −1 (i.e. m = 1) we have two modules with the same composition series as in the preceding result, namely Z ′ (θ −e(ρ) ρ) and Z ′ (ρ). These modules are nonisomorphic. For, by definition, Z(ρ) has an E-trivial generator, while Z ′ (θ −e(ρ) ρ) does not. We need one more result before the main theorem. This is Lemma 4.17. (1) If n is odd, then for every d, 1 ≤ d ≤ n there are exactly m nonisomorphic simple modules of dimension d. (2) If n is even and κ = 1, then for every odd d, 1 ≤ d ≤ n there are exactly 2m nonisomorphic simple modules of dimension d. (3) If n is even and κ = −1, then for every even d, 1 ≤ d ≤ n there are exactly 2m nonisomorphic simple modules of dimension d. Proof: By Proposition 4.14 dim L(ρ) = d if and only if e(ρ) = d − 1. So the asserted results are a question of the number of solutions in Rκ,N of the equation ρ2m = q e . Recall the mapping p : Rκ,N → Rn , p(ρ) = ρ2m . For ρ, ζ ∈ Rκ,N p(ρ) = p(ζ) if and only if p(ρζ −1 ) = 1. Now ρζ −1 ∈ RN ,

24


and as RN is generated by θ with θ m = q, the image of p restricted to RN is the subgroup of Rn generated by q 2 . This proves parts (1) and (2) as p(Rκ,N ) = p(RN ) in both cases. Assume conditions of part (3). Pick an element π ∈ k such that π 2 = θ. Clearly π ∈ R−1,N and moreover every ρ ∈ R−1,N is of the form θ r π for some 0 ≤ r < N . Now p(θ r π) = q 2r+1 which shows that e(ρ) runs over all odd integers between 1 and n. This proves (3). We let P (ρ) denote the projective cover of L(ρ). The ith term of the radical series of P (ρ) shall be denoted by radi P (ρ), and the socle of P (ρ) shall be denoted by soc P (ρ). Theorem 4.18. (1) For every nonexceptional root ρ (a) dim P (ρ) = 2n. (b) The radical series of P (ρ) is P (ρ) ⊃ rad P (ρ) ⊃ soc P (ρ) ⊃ 0

with rad P (ρ)/rad 2 P (ρ) ∼ = L(σ(ρ)) ⊕ L(σ −1 (ρ)). (2) If ρ is exceptional, then P (ρ) = L(ρ). Proof: Since H is symmetric algebra by Theorem 4.2 soc P (ρ) = L(ρ). Let ρ be a nonexceptional root. By Propositions 4.14 and 4.16 there are epimorphisms of P (ρ) on Z(ρ), Z ′ (θ −e(ρ) ρ) carrying rad P (ρ) to L(σ(ρ)) and L(σ −1 (ρ)), respectively. Thus rad P (ρ)/rad2 P (ρ) has a summand L(σ(ρ) ⊕ L(σ −1 (ρ)). Therefore dim P (ρ) ≥ 2 dim L(ρ) + dim L(σ −1 (ρ)) + dim L(σ(ρ)) = 2n where the last equality holds since Z(ρ)/L(σ(ρ)) = L(ρ) and Z ′ (θ −e(ρ) ρ)/L(σ −1 (ρ)) = L(ρ). By general principles M (4.20) HeλX = dim P (ρ)dim L(ρ) ρ∈Rκ,N

If n is odd, Lemma 4.17 (1) gives X dim P (ρ) · dim L(ρ) =

X

ρ nonexceptional

ρ∈Rκ,N

+

X

ρ exceptional

dim P (ρ) · dim L(ρ)

dim P (ρ) · dim L(ρ)

≥ 2n(1 + 2 + . . . + n − 1)m + n2 m = N n2 . Since dim HeλX = N n2 that expression forces dim P (ρ) = 2n for all nonexceptional roots ρ and dim P (ρ) = n2 for the exceptional roots. This completes the proof. The even cases are treated similarly. We denote by B(ρ) the block of HeλX containing P (ρ). We let W denote group generated by σ.


25

Corollary 4.19. The blocks of HeλX correspond to W- orbits in Rκ,N . The correspondence is given by the formula X B(ρ) = P (ζ)dim L(ζ) ζ∈Wρ

Proof: Recall that two projective indecomposable modules P and P ′ are linked if they share a composition factor. Linkage gives rise to an equivalence relation on the set of projective indecomposable modules. The block of P is the sum of all projective indecomposable modules equivalent to P . By the preceding Theorem P (ρ) is linked to P (ζ) if and only if ζ = σ ± (ρ). The formula follows. Theorem 4.20. Assume ρ is a nonexceptional root. Then (a) The Gabriel quiver Qρ of the block containing P (ρ) is the quiver with vertices being the isomorphism classes of simples [L(σ i (ρ))], i ∈ Z/2mZ with arrows a

i i+1 [L(σ i (ρ))]−→ ←−[L(σ (ρ))]

bi

corresponding to translation by σ. (b) The basic algebra of the block containing P (ρ) is the quotient of the path algebra of Qρ with relations ai bi − bi+1 ai+1

and all other paths of length ≥ 2. (c) The block of Qρ is a special biserial algebra and therefore of tame representation type. Proof: For generalities on basic algebras and their presentations by quivers with relations, the reader may consult [5]. The present situation is similar to [11]; however, the argument there is not compatible with the theory developed here. We proceed as follows. As a first case assume m > 1. By Theorem 4.18 rad P (ρ)/rad 2 P (ρ) equals L(σ(ρ)) ⊕ L(σ −1 (ρ)), therefore the quiver has arrows as in the statement. Let kQρ be the path algebra of the quiver and B be the basic algebra of B(ρ). We want to construct an epimorphism of kQρ onto B (cf. [5, III 1.9]) whose kernel is generated by relations in (b). Thanks to Propositions 4.14 and 4.16 there are nonsplit extensions Ei and Ei′ of L(σ i (ρ)) by L(σ i−1 (ρ)) and L(σ i (ρ)) by L(σ i+1 (ρ)), respectively. Since P (σ i (ρ)) is injective both embed in P (σ i (ρ)). Now we assign to ai an epimorphism P (σ i (ρ)) → Ei+1 and to bi an epimorphism P (σ i+1 (ρ)) → Ei′ which we still denote by ai and bi . Since σ i−1 (ρ) 6= σ i (ρ), σ i+1 (ρ) we have ai (Ei ) = 0, hence ai ai−1 = 0, and likewise bi−1 bi = 0. Further ai bi and bi+1 ai+1 are endomorphisms of P (σ i+1 (ρ)) into its socle L(σ i+1 (ρ)). Therefore γi ai bi = bi+1 ai+1

26


for some γi ∈ k and all i ∈ Z/2mZ. We wish to change the basis to get rid of the coefficients. We will follow argumentation of [11]. Replacing ai by a′i := γi · · · γ2m−1 ai , i = 1, 2, . . . , 2m − 1, we obtain the relations a′i bi = bi+1 a′i+1 for i = 1, . . . , 2m − 2

a′2m−1 b2m−1 = b0 a0 γ0 · · · γ2m−1 a0 b0 = b1 a′1

We know by Theorem 4.2 that H is a symmetric algebra and then so are the blocks of HeλX and their basic algebras. Let ψ : B → k be a symmetrizing linear form. Then ψ(γ0 · · · γ2m−1 a0 b0 ) = ψ(b1 a′1 ) = ψ(a′1 b1 )

= · · · = ψ(a′2m−1 b2m−1 ) = ψ(b0 a0 ) = ψ(a0 b0 )

whence (γ0 · · · γ2m−1 − 1)a0 b0 ∈ ker ψ and this element spans a one- dimensional left ideal of H. Thus γ0 · · · γ2m−1 = 1 as desired. Moving to the case m = 1 we let i ∈ Z/2Z. Now we have just two projective indecomposable modules in B(ρ), call them P0 and P1 , and let Li = Pi /rad Pi+1 . In contrast to the previous case, there doesn’t exist a uniform choice of length two indecomposable submodules in rad Pi . We argue as follows. As above by Proposition 4.16 there are noisomorphic submodules E1 and E1′ in rad P1 with the top composition factor L0 . Consequently, there are epimorphisms a0 : P0 → E1 and b1 : P0 → E1′ . Put F0 = ker a0 and F0′ = ker b1 . Both F0 and F0′ lie in rad P0 hence by Theorem 4.18 they are extensions of L0 by L1 . Therefore there are epimorphisms a1 : P1 → F0 and b0 : P1 → F0′ ; hence a0 a1 = 0 = b1 b0 holds by construction. However, since ai , bi are nonisomorphisms they send the socle of Pj to zero for all i. Therefore every path of length ≥ 2 is either zero or spans a 1dimensional ideal of B. Since B is symmetric a1 a0 = 0 = b0 b1 as well, whence E1 ⊂ ker a1 and E1′ ⊂ ker b0 . As dim ker a1 = n = dim E1 we see that E1 = ker a1 and similarly E1′ = ker b0 . From this we conclude that F0 6= F0′ . Indeed, since dim Hom(P1 , F0 ) = [F0 : L1 ] = 1 were F0 = F0′ we would have a1 = αb0 , α ∈ k• , which forces E1 = E1′ , a contradiction. It follows that a0 b0 and b1 a1 are nonzero homomorphisms P1 → soc P1 , and similar conclusions hold for b0 a0 and a1 b1 . Therefore there are γi ∈ k• such that γ1 a0 b0 = b1 a1 γ2 b0 a0 = a1 b1 But then the calculation ψ(b1 a1 ) = γ1 ψ(a0 b0 ) = ψ(a1 b1 ) = γ2 ψ(b0 a0 ) = γ2 ψ(a0 b0 ) gives γ1 = γ2 . Replacing a0 by γ1 a0 we obtain the desired result.


27

The proof of (c) follows directly from the definition and theory of special biserial algebras, see [10]. Lastly, we want to describe embeddings of blocks and projective indecomposable modules in H. First, we take the subalgebra Aλ of HeλX of section 4.2. From its definition we have Aλ ∼ = k[t]/(tN − κ), hence Aλ = ⊕keζ where eζ is a primitive idempotent uniquely determined by the property Keζ = ζeζ . Secondly, we use primitive idempotents ǫµ associated to roots µ ∈ Rκ,n defined in section 4.5. For every ρ ∈ Rκ,N , ρm ∈ Rκ,n hence gives rise to the idempotent ǫρm . Abusing notation we write the latter as ǫρ . Proposition 4.21. Suppose ρ is a nonexceptional root. Then (1) B(ρ) = HeλX ǫρ . (2) Every nonsimple projective indecomposable module of H is isomorphic to HeλX ǫρ eζ for some λ, ρ, ζ. Proof: It is elementary to see that for a root µ of the minimal polynomial fλ (t), ǫρ (µ) = 1 if µ = −η ′ D(ρ), and zero, otherwise. For every simple HeλX - module L(ζ) Cλ acts on L(ζ) via multiplication by −η ′ D(ζ). Therefore ǫρ acts on L(ζ) via multiplication by ǫρ (−η ′ D(ζ)). Now D(ζ) = D(ρ) if and only if ζ m = ρm or ζ m ρm = q −1 . Since ρ is nonexceptional, q −1 ρ−m 6= ρm ; hence both cases occur. Next observe that for every fixed element µ ∈ Rκ,n the set of solutions to the equation xm = µ in Rκ,N has m elements. Therefore Y = {ζ|D(ζ) = D(ρ)} has 2m elements. On the other hand by Lemma 4.15, ζ ∈ W · ρ if and only if ζ = θ −npρ or ζ = θ −np−e(ρ)−1 ρ. In the first case ζ m = ρm and in the second ζ m ρm = q −1 . Thus W · ρ ⊂ Y , hence by Lemma 4.15 (b) they are equal. It follows that ǫρ acts as the identity on B(ρ) and annihilates every other block, whence (1). The number P of projective indecomposable modules in a decomposition of B(ρ) equals ζ∈W·ρ dim L(ζ). For every ζ ∈ W · ρ, σ(ζ) 6= ζ, hence there are m distinct pairs {ζ, σ(ζ)) in W · ρ. For each such pair Proposition 4.14 (c) gives dim L(ζ) + dim L(σ(ζ)) = n. Thus B(ρ) is the direct sum N indecomposable summands. Since we have N idempotents eζ , the result follows. We turn to the case of a projective simple module P (ρ). Proposition 4.22. Suppose ρ is an exceptional root. Let f = eρ E n−1 F n−1 . Then HeλX f ∼ = P (ρ) and HeλX f is a direct summand of

HeλX .

Proof: EvidentlyY f has weight ρ. Further, it is elementary to derive the ρ (K−ζ) and then a simple calculation gives Eeρ = eθρ E, formula eρ = Nκ ζ6=ρ

hence f is E-trivial, which proves the first assertion.

28


Next we adjust Kac’s formula [8, (1.3.1)] to our context obtaining min(r,s)

(4.21)

s

r

[E , F ] = η(

X

F r−i His,r E s−i )

i=1

Qj=i −m (K − q i+j−r−sK m ). We now where His,r = (r)q · · · (r − i + 1)q si q j=1 compute f 2 using (4.21). Since E n = 0 we deduce n−1,n−1 f 2 = eρ Hn−1 f = cf

Qn−1 −m where c = η(n − 1)q ! j=1 (ρ − q j−n+1 ρm ) and since ρ is exceptional, every factor in c is nonzero, hence c 6= 0. Thus c−1 f is an idempotent, and the proof is complete. References [1] N. Andruskiewitsch and M. Beattie, The coradical of the dual of a lifting of a quantum plane. Hopf algebras, 47–63, Lecture Notes in Pure and Appl. Math., 237, Dekker, New York, 2004. [2] N. Andruskiewitsch and M. Beattie, Irreducible representations of liftings of quantum planes, Lie theory and its applications in physics V, 414–423, World Sci. Publ., River Edge, NJ, 2004. [3] N. Andruskiewitsch and H.-J. Schneider, Lifting of quantum linear spaces and pointed Hopf algebras of order p3 , J. Algebra 209 (1998), 658-691. [4] N. Andruskiewitsch and H.-J. Schneider, Pointed Hopf algebras. New directions in Hopf algebras, 1-68, Math. Sci. Res. Ins. Publ. 43, Cambridge Univ. Press, Cambridge, 2002. [5] M. Auslander, I. Reiten and S. O. Smalo, Representation Theory of Artin Algebras, Cambridge University Press, 1995. [6] M. Beattie, S. D˘ asc˘ alescu, and L. Gr¨ unefelder, Constructing pointed Hopf algebras by Ore extensions, J. Algebra 225 (2000), 743-770. [7] V. Chari and A. Premet, Indecomposable restricted representations of Quantum sl2 , Publ. RIMS, Kyoto Univ. 30(1994), 335-352. [8] C. De Concini and V. G. Kac, Representations of quantum groups at roots of 1, in Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, (Paris 1989), Birkh¨ auser, Boston, 471-506. [9] Y. Drozd, Representations of commutative algebras, Funk. Anal. Prilozh., 6(4)(1972), 41-43. English translation: Functional Anal. Appl. 6(1972), 286-288 (1973). [10] K. Erdmann, Blocks of Tame Representation Type and Related Algebras, Lecture Notes in Mathematics, 1428, Springer-Verlag, Berlin,1990. [11] K. Erdmann, E.L. Green, N. Snashall and R. Taillefer, Representation theory of the Drinfeld doubles of a family of Hopf algebras. J. Pure Appl. Algebra 204 (2006), no. 2, 413–454. [12] C. Kassel, Quantum Groups, Springer- Verlag, 1995. [13] L. Krop and D. Radford, Finite-dimensional Hopf algebras of rank one in characteristic zero. J. Algebra 302 (2006), no. 1, 214–230. [14] G. Lusztig, Introduction to Quantum Groups, Birkh¨ auser, 1993. [15] S. Majid, Crossed products by braided groups and bosonization, J. Algebra 163 (1994), 165-190. [16] Marshall Hall, Jr. The Theory of Groups. The Macmillan Co., New York, N.Y. 1959.


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¨ [17] U. Oberst and H. J. Schneider, Uber Untergruppen endlicher algebraischer Gruppen. Manuscripta Math. 8 (1973), 217–241. [18] D. E. Radford, On the coradical of a finite-dimensional Hopf algebra, Proc. Amer.. math. Soc. 53, No. 1(1975), 9-15. [19] D.E. Radford and J. Towber, Yetter-Drinfel’d categories associated to an arbitrary bialgebra, J. Pure and Appl. Algebra 87(1993), 259-279. [20] D.E. Radford, Hopf algebras with projection, J. Algebra 92 (1985), 322-347. [21] R. Suter, Modules for Uq (sl2 ), Comm. Math. Phys. 163(1994), 359-393. [22] D. N. Yetter, Quantum groups and representations of monoidal categories, Math. Proc. Cambridge Philos. Soc. 108(1990), 261-290. [23] J. Xiao, Finite dimensional representations of Ut (sl(2)) at roots of unity, Can. J. Math. 49(4)(1997), 772-787. DePaul University, Department of Mathematical Sciences, 2320 N. Kenmore, Chicago, IL 60614 E-mail address: [email protected] DePaul University, Department of Mathematical Sciences, 2320 N. Kenmore, Chicago, IL 60614 E-mail address: [email protected]