The projective class ring of basic and split Hopf algebras

Claude Cibils

Abstract

We describe the structure of the complexification of the projective class ring of a basic and split Hopf algebra using a positive integer determined by the composition series of the projective cover of the trivial module. If q is a root of unity of order n, the projective class algebra of u_{q}^{+}(sl_{2}) is the product of n − 1 copies of the dual numbers over C and two copies of C.

1991 Mathematics Subject Classification : 16E20 and 16W30

1 Introduction

We consider finite-dimensional Hopf algebras such that all the simple modules are one-dimensional over the ground field, that is basic and split associative algebras affording a Hopf algebra structure. Some of those algebras have been called half-quantum groups and are useful in knot theory and 3manifold invariants, see [7, 13]. Through the Drinfeld quantum double ([9]) they provide universal R-matrices and solutions to the Yang-Baxter equation. Notice that the relation between the K-theory of a Hopf algebra and the K-theory of its quantum double is not understood. Basic and split Hopf algebras have been studied in [6, 5, 10].

The purpose of this paper is to describe the structure of the projective class ring of an arbitrary basic and split finite-dimensional Hopf algebra, where the multiplicative structure is given by the tensor product of modules obtained through the comultiplication of the Hopf algebra. Actually we consider the ring provided by projective and semisimple modules, its unit element is the isomorphism class of the trivial simple module. This ring is a subring of the entire representation ring given by all the finitely generated modules.

We obtain for instance that for q a root of unity, the projective class ring of u_{q}^{+}(sl_{2}) has nilpotent elements. Notice that u_{q}^{+}(sl_{2}) has only a finite number of isomorphism classes of indecomposable modules (see [4]); by contrast Benson and Carlson proved in [2] that a group algebra having this property has no nilpotent elements in its representation ring.

The tensor product of simple modules is often commutative, this happens to be true for half-quantum groups at roots of unity. With this assumption, the structure of the complexification of the projective class ring can be made more precise than in the general case, see Theorem 3.13.

For split and basic Hopf algebras the isomorphism classes of simple modules affords a group structure and the projective cover of the trivial simple module provides a canonical element c in the center of the corresponding group algebra. The structure result relies on the positive integer determined by the number of irreducible characters which do not vanish on c. Its value for u_{q}^{+}(sl_{2}) is 1, and the complexification of the projective class ring of u_{q}^{+}(sl_{2}) is the product of two copies of the complex numbers with the product of n − 1 copies of the dual numbers where, n is the order of the root of unity q.

Notice that the projective class ring of half-quantum groups at roots of unity are commutative rings. Actually the entire representation ring of u_{q}^{+}(sl_{2}) is commutative, as a consequence of direct computations made in [4]; there is no theoretical result predicting this commutativity property since it is known that half-quantum groups are not quasi-triangular, and not even quasi-cocommutative (see [15]).

Acknowledgements: I thank each referee for an attentive reading and specially one of them who suggested a more direct way of describing the nilradical at Proposition 3.15.

2 Basic and split Hopf algebras

Let Λ be a finite-dimensional Hopf algebra over a field k, that is an associative algebra, provided with a comultiplication Δ : Λ → Λ ⊗ Λ, a counit ε : Λ → k and an antipode S : Λ → Λ (see [17]).

The tensor product M ⊗ N over k of two left Λ-modules M and N has a left Λ-module structure obtained through Δ. The ground field k_{ε} equipped with the module structure given by ε is the unit element; since M^{∗} = Hom_{k}(M, k) is a right Λ-module, it turns into a left module by composing the action with the antipode.

An associative algebra is split if the endomorphism rings of the simple modules are isomorphic to the ground field; this condition is always satisfied if k is algebraically closed. The algebra is basic if each simple module has multiplicity one in its maximal semisimple quotient algebra. The next result shows that each basic and split Hopf algebra has a structure group attached, which corresponds to the group of group-likes in the dual Hopf algebra as considered for instance in [14].

Lemma 2.1 Let S be the set of isomorphism classes of simple modules over a basic and split finite-dimensional Hopf algebra. Then S is a group.

Proof: Let U and V be simple modules. Since they are one-dimensional, their tensor product is also one-dimensional, hence a simple module. The unit element in S is the isomorphism class of k_{ε}, while the inverse of U is U^{∗}.

Example 2.2 Let G be a finite group and k^{G} be the Hopf algebra of functions on it, that is the dual of the group algebra kG. The structure group of k^{G} is G.

Example 2.3 Let u_{q}^{+}(G) be the half-quantum group corresponding to a simple Lie algebra G with Cartan matrix of size t × t, and to a primitive root of unity of order n. This basic and split algebra has an abelian structure group, isomorphic to the product of t copies of the cyclic group of order n, see for instance [5].

If the algebra is not basic, the tensor product of simple modules is not semisimple in general. The structure group gives an easy proof of a particular case of a result due to P. Schauenburg. Recall that associative algebras are Morita equivalent if their module categories are equivalent. Hopf algebras are Morita-monoidally equivalent if their module categories are equivalent as monoidal categories, which means that there exist an equivalence of categories compatible with tensor products, dual operations and the unit object. In [16] Schauenburg shows that finite-dimensional Morita-monoidally equivalent Hopf algebras have isomorphic underlying associative algebras.

Proposition 2.4 Let Λ be a basic and split Hopf algebra, and let Λ′ be a Hopf algebra Morita-monoidally equivalent to Λ. Then Λ and Λ′ are isomorphic associative algebras.

Proof: Consider the structure group S of Λ and the set S′ of isomorphism classes of simple Λ′-modules. It follows from the definition that S′ is also a group with respect to the tensor product of modules, isomorphic to S. Since the dimension over the ground field is multiplicative on tensor products and since each simple module has an inverse, simple Λ′-modules are one-dimensional. Hence Λ′ is a basic and split algebra, and consequently isomorphic to Λ.

We recall now definitions which will be useful in the next section. The Grothendieck group of a small abelian category C is the free abelian group with basis the isomorphism classes [X] of objects X of C divided by the subgroup generated by elements [X_{2}] − [X_{1}] − [X_{3}] provided by each split exact sequence 0 → X_{1} → X_{2} → X_{3} → 0 in C. We denote the quotient group K_{0}(C), and K_{0}(Λ) denotes the Grothendieck group of the category of finitely generated modules over a ring Λ. In case C is a monoidal abelian category – for instance the category of modules over a Hopf algebra – the free abelian group above is a ring. Tensoring a split exact sequence yields to a split exact sequence, the quoted subgroup is a two-sided ideal and the Grothendieck group is a ring.

Remark 2.5 If Λ is a finite-dimensional algebra, K_{0}(Λ) is a free abelian group with a basis given by the isomorphism classes of indecomposable modules, as an immediate consequence of the Krull-Schmidt Theorem (see for instance [8]).

Consider the subgroup generated by elements given by non necessarily split exact sequences and denote the corresponding quotient group \overline{K}_{0}(C). For modules over a finite-dimensional algebra Λ, the Jordan-Hölder theorem insures that \overline{K}_{0}(Λ) is free abelian with a basis provided by the isomorphism classes of simple modules. Since Λ is a Hopf algebra over a field k, the functors X ⊗ − and − ⊗ X are exact for any Λ module X, and \overline{K}_{0}(Λ) is a ring. If Λ is a basic and split Hopf algebra, this ring is clearly isomorphic to the group ring ZS, where S is the structure group of Λ.

There is a canonical ring surjection K_{0}(Λ) → \overline{K}_{0}(Λ) called dimension vector which associates to a module M the sum \overline{M} of the quotients of one of its composition series. This map is an isomorphism if and only if Λ is a semisimple algebra.

3 Projective class rings

We study K_{0}^{p}(Λ), the subring of K_{0}(Λ) generated by the projective and semisimple modules. We recall the following standard results:

Lemma 3.1 Let Λ be a Hopf algebra, and let P and X be finitely generated modules such that P is projective. Then P ⊗ X is a projective module.

Proof: We prove that Λ ⊗ X is free. The module Λ ⊗ X is isomorphic to Λ ⊗ X_{ε}, where X_{ε} is the vector space X equipped with the trivial action given by the counit ε. The result follows since Λ ⊗ X_{ε} is free, of rank the dimension of X. An isomorphism of left Λ-modules φ : Λ ⊗ X → Λ ⊗ X_{ε} is given by φ(a⊗x) = ∑ a_{(1)}⊗S(a_{(2)})x, where Δa = ∑ a_{(1)}⊗a_{(2)}.

Proposition 3.2 Let Λ be a finite-dimensional basic and split Hopf algebra. The ring K_{0}^{p}(Λ) is a free abelian group with basis the set of isomorphism classes of simple and indecomposable projective modules.

Proof: K_{0}(Λ) is free abelian with basis the set of isomorphism classes of indecomposable modules. The described subgroup is the smallest containing semisimple and projective modules. Moreover this subgroup is a subring, due to Lemma 3.1 and to the fact that isomorphism classes of simple modules forms a group.

Let P be the set of isomorphism classes of indecomposable projective modules. The ring \overline{K}_{0}(Λ) is the integral group ring of the structure group S, and ZP is a ZS-bimodule. It is well known that S and P are canonically bijective sets (see for instance [8]) : each simple module U has a unique projective cover P_{U}, a projective module such that there exist an essential surjection ψ : P_{U} → U (for any strict submodule X of P_{U}, the restriction of ψ to X is not surjective).

Lemma 3.3 For a basic and split finite-dimensional Hopf algebra, ZP is isomorphic to ZS as a ZS-bimodule.

Proof: Consider the above canonical bijection : we have to prove that V ⊗P_{U} is isomorphic to P_{V ⊗U} for V and U simple modules. Since ψ : P_{U} → U is an essential surjection, we obtain that V ⊗ P_{U} → V ⊗ U is also essential since simple modules are one-dimensional. Moreover, V ⊗ P_{U} is a projective module, and the uniqueness of projective covers insures that V ⊗ P_{U} is the projective cover of V ⊗ U.

Lemma 3.4 Let Λ be a finite-dimensional basic and split Hopf algebra which is not semisimple. Then K_{0}^{p}(Λ) is the direct sum of the subring ZS and the ZS-bimodule ZP.

Proof: Since \overline{K}_{0}(Λ) is the subring ZS, we need to prove that S ∩P is empty; indeed if there exist a simple and projective module, we infer that any simple module is projective by Lemma 3.1 and the fact that S is a group. If all the simple modules are projective modules, there are no non-trivial extensions between them, and the algebra is semisimple.

Remark 3.5 Let G be a finite group and let k^{G} be the commutative algebra of functions from G to k, which is a Hopf algebra with comultiplication Δf(x, y) = f(xy). Any semisimple basic and split Hopf algebra Λ is of this form. In this case K_{0}^{p}(Λ) is the group algebra ZG since the structure group of k^{G} is G.

The next purpose is to describe the multiplicative structure of ZP, in other words to describe the ZS-bimodule map

ZP ⊗_{ZS} ZP ⟶ ZP

induced by the tensor product of modules over a basic and split Hopf algebra.

Proposition 3.6 Let Λ be any finite-dimensional Hopf algebra, and let M and P be finitely generated left Λ-modules where P is projective. Then M ⊗P is isomorphic to \overline{M} ⊗ P.

Proof: Recall that \overline{M} is the semisimple module obtained by adding the quotients of a composition series of M. It suffices to prove that for an exact sequence

0 → M_{1} → M → M_{2} → 0

of Λ-modules we have that M ⊗ P is isomorphic to M_{1} ⊗ P ⊕ M_{2} ⊗ P. By tensoring, we get an exact sequence

0 → M_{1} ⊗ P → M ⊗ P → M_{2} ⊗ P → 0

of Λ-modules. Since M_{2} ⊗P is projective by Lemma 3.1, the sequence splits.

Corollary 3.7 If the structure group of a basic and split finite-dimensional Hopf algebra is abelian, the projective modules are central in K_{0}(Λ).

Proof: We prove that any indecomposable projective module is in the center of K_{0}(Λ), more precisely we have that M ⊗ P_{U} and P_{U} ⊗ M are isomorphic modules for any finitely generated module M and any simple module U. From the proof of the preceding result we have that P_{U} ⊗ M and P_{U} ⊗ \overline{M} are isomorphic modules. We need to prove that P_{U} ⊗ V and V ⊗ P_{U} are isomorphic for any simple module V. The module P_{U} ⊗ V is the projective cover of U ⊗V (see the proof of Lemma 3.3) and V ⊗P_{U} the projective cover of V ⊗ U. Since S is commutative, and projective covers are unique up to isomorphism, we obtain the result.

Definition. Let A be a ring; a multiplicative A-bimodule is an A-bimodule M provided with an associative A-bimodule map φ : M ⊗_{A} M → M.

If M is a multiplicative A-bimodule, then A ⊕ M has a natural ring structure where M is a two-sided ideal. Conversely, if R is a ring with a subring A and a decomposition R = A ⊕ I where I is a two-sided ideal of R, we obtain that I is a multiplicative A-bimodule. The ring K_{0}^{p}(Λ) corresponds to this situation, since K_{0}^{p}(Λ) = ZS ⊕ ZP. Moreover, ZP is a ZS-bimodule isomorphic to ZS; we need to study the multiplicative bimodule structures of a ring having underlying bimodule the ring itself.

Proposition 3.8 The multiplicative bimodule structures on a ring A are in bijection with its center ZA. The isomorphism classes of bimodule structures on A are in one-to-one correspondence with ZA/U(ZA), where U(ZA) is the group of units of ZA.

Proof: Notice that an A-bimodule map φ : A ⊗_{A} A → A is an A-bimodule endomorphism of A which is completely determined by the image of 1, an element of the center of A. The associativity always holds since

φ (φ(a_{1}⊗a_{2})⊗a_{3}) = a_{1}a_{2}a_{3} φ(1)^{2} = φ (a⊗φ(b⊗c)) .

Finally an A-bimodule automorphism of A is given by a unit of ZA, we infer that two multiplicative bimodule structures φ and φ′ are isomorphic if and only if there exist a unit u ∈ U(ZA) such that φ(1) = uφ′(1).

Remark 3.9 According to this proof, the corresponding multiplications on A ⊕ A are given by the formula

(a_{1}, b_{1}).(a_{2}, b_{2}) = (a_{1}a_{2}, b_{1}a_{2} + a_{2}b_{1} + b_{1}cb_{2})

where a_{1}, a_{2}, b_{1}, b_{2} are elements of A and c = (0, 1)(0, 1) = φ(1) is a central element of A.

We define now a canonical element c of the integral group ring ZS as follows: let P_{1} be the projective cover of the trivial module k_{ε} (according to the preceding notation we have P_{1} = P_{k_{ε}}). The element c is provided by a composition series of P_{1}, more precisely c = \overline{P_{1}}. Notice that for each U ∈ S, we have \overline{P_{U}} = Uc in the group ring.

Proposition 3.10 Let Λ be a basic and split Hopf algebra, with structure group S (non necessarily abelian) and canonical element c. Then K_{0}^{p}(Λ) is a ring isomorphic to ZS ⊕ ZS, where the first factor is the group ring and multiplication is given by :

(s_{1}, t_{1}).(s_{2}, t_{2}) = (s_{1}s_{2}, s_{1}t_{2} + t_{1}s_{2} + t_{1}ct_{2}).

Proof: The canonical ZS-bimodule isomorphism between ZS and ZP sends the unit element to ZP. Hence the multiplication on ZP is determined by P_{1} ⊗ P_{1} = \overline{P}_{1} ⊗ P_{1} = cP_{1} ∈ ZP. Through the canonical isomorphism the corresponding element in ZS is c. The remark 3.9 provides the formula.

Example 3.11 Let q be a primitive n-th root of unity in C, and let u_{q}^{+}(sl_{2}) be the corresponding finite-dimensional half-quantum group, whose structure group S is isomorphic to the cyclic group of order n (see [6]). The element c is the trace element in ZS, i.e. the sum of all the group elements (the structure of u_{q}^{+}(sl_{2})-modules is given in [4]).

Corollary 3.12 The canonical element c of a basic and split non semisimple Hopf algebra lies in the center of the integral group ring of its structure group.

In case the structure group is commutative there is a more explicit description of the algebra C ⊗ K_{0}^{p}(Λ) . Recall that CS is a semisimple commutative algebra isomorphic to C^{S} = ×_{U∈S}δ_{U}, where {δ_{U}}_{U∈S} is the complete set of primitive orthogonal idempotents provided by the Dirac masses on elements of S. An explicit isomorphism is provided by the Fourier transform once generators of S and corresponding roots of unity are given.

Definition: Let Λ be a basic and split Hopf algebra with abelian structure group S and canonical element c. The positive integer r is the number of non-zero coordinates of c in the above decomposition of the group algebra CS. Equivalently, r is the number of irreducible characters of S which do not vanish at c.

Theorem 3.13 Let Λ be a basic and split Hopf algebra with abelian structure group of order s. Then C⊗K_{0}^{p}(Λ) is isomorphic to the algebra C^{2r} ×C[ϵ]^{s−r} where C[ϵ] is the algebra of dual numbers C[X]/X^{2}.

Proof: We know that a multiplicative bimodule structure on a ring A having underlying bimodule A is determined by a central element up to multiplication by a unit of the center. After normalization we have to describe the algebra structure of C^{s} ⊕C^{s} where the first factor affords its natural algebra structure and the second is considered as a C^{s}-bimodule with multiplicative structure provided by an element c′ having r entries with value 1, the others entries having zero value. Clearly this algebra is isomorphic to

[C[X]/ (X^{2} − X)]^{r} × [C[X]/X^{2}]^{s−r}

and observe that C[X]/ (X^{2} − X) = C^{2}.

Remark 3.14 For u_{q}^{+}(sl_{2}), we notice that c is the sum of all the elements of the cyclic group of order n, hence r = 1. Then

C ⊗ K_{0}^{p} (u_{q}^{+}(sl_{2})) ≅ C^{2} × C[ϵ]^{n−1}.

Finally we describe the sums of projective modules with complex coefficients which are nilpotent elements.

Let Λ be a basic and split Hopf algebra with abelian structure group S and canonical element c. We already know that nilpotent elements in C⊗K_{0}^{p}(Λ) have zero square and that C⊗K_{0}^{p}(Λ) = CS ⊕(CS)_{c} where (CS)_{c} is an ideal with multiplicative structure given by t_{1}.t_{2} = t_{1}ct_{2}. Consider an isomorphism of Hopf algebras α : CS → C^{S} and let β : CS × CS → C be the corresponding non-degenerate symmetric bilinear form. Let B_{c} = {U ∈ S | β(U, c) ≠ 0} = {U ∈ S | α(c)_{(U)} ≠ 0}.

Proposition 3.15 The nilradical of C ⊗ K_{0}^{p}(Λ) is the orthogonal vector space to B_{c} in CS.

Proof : Consider the product in (C^{S})_{c} corresponding to the product in (CS)_{c} through the isomorphism α, we have f.g = fα(c)g. Notice that two functions are orthogonal for the natural product of C^{S} if and only if their supports are disjoint. An element f has zero square in (C^{S})_{c} if 0 = f.f = ffα(c) which only holds if the supports of f and α(c) are disjoint.

This means that for each U ∈ S such that 0 ≠ α(c)_{(U)} = β(c, U) we have f(U) = 0. The corresponding element to f in CS is then orthogonal to each element of B_{c}. The reverse inclusion is also clear.

Example 3.16 For u_{q}^{+}(sl_{2}) the element c is the sum of all the elements of the cyclic group of order n, then B_{c} = {1}. The orthogonal space is the augmentation ideal and any difference of two indecomposable projective modules has zero square.

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Dép. de Mathématiques, Université de Montpellier 2, F-34095 Montpellier cedex 5.

cibils@math.univ-montp2.fr

September 1998