Twisted Group Rings

Bodo Pareigis

Mathematisches Institut Universität München Theresienstraße 39 8000 München 2, Germany^{1}

1. Introduction

This paper deals with (twisted) Hopf algebra forms of group rings, i.e. with Hopf algebras H over a base ring K, such that for a suitable extension L of K the Hopf algebra L ⊗_{K} H is isomorphic to a group ring LG over L. These Hopf algebras arise from an adjoint situation as a solution of a universal problem in universal algebra.

One of the well known examples of a pair of adjoint functors is the units functor U: K-Alg ⟶ Gr associating the group of units or invertible elements U(A) with each K-algebra A, and its left adjoint functor K-: Gr ⟶ K-Alg associating with each group G its group ring KG. The group ring KG actually turns out to be a cocommutative Hopf algebra. We are going to generalize this situation.

Let F be a finite group. We consider the category F -Gr of groups on which F acts by automorphisms and group homomorphisms compatible with the action of F. Furthermore let L be a Galois extension of the commutative base ring K with Galois group F. Then there is a functor

Γ: K-Alg ⟶ F -Gr

defined by Γ(A) := U(L ⊗_{K} A). The action of F on Γ(A) is given by the Galois action of F on L. In this paper the Galois extension L of K

1May 8, 1989 is assumed to be a commutative ring and free as a K- module. Our first result will be that Γ possesses a left adjoint functor

K_{Γ}: F -Gr ⟶ K-Alg.

The K-algebras K_{Γ}(G) or simply K_{Γ}G will be called twisted group rings. They should not be confused with crossed product group rings, skew group rings, smash products, or similar constructions. These twisted group rings do in general not even allow a canonical map from G into K_{Γ}G. As we will see, there will, however, be a canonical map G ⟶ (K_{Γ}G)^{n} where n is the order of F.

The twisted group rings will turn out to be cocommutative Hopf algebras. They even have the property to coincide after base ring extension to L with the ordinary group ring L ⊗ K_{Γ}G ≅ L ⊗ KG. This is an isomorphism of Hopf algebras. Thus K_{Γ}G is a (twisted) L-form of the group ring KG.

In [2] we studied the construction and general theory of (twisted) forms of group rings KG for a fixed group G. We showed that the forms are in one to one correspondence with the F -Galois extensions of K where F = Aut(G). Using the specific knowledge of all quadratic extensions of K (under minor restrictions), we were able to describe all Hopf algebra forms of KZ, KC_{3}, KC_{4}, and KC_{6} by generators and relations. A form H of KG which corresponds to the F -Galois extension L under [2 Thm.5] is certainly split by L, i.e. L ⊗_{K} H ≅ L ⊗_{K} KG as Hopf algebras, but it may also be split by a different, even smaller Galois extension L′ of K. So we will not make use of the correspondence between the forms and Galois extensions. We will fix a Galois extension L of K and then study the Hopf algebra forms H of KG for all groups G which are split by the Galois extension L.

The group elements g in KG are connected with certain elements in K_{Γ}G by formulas which generalize the Euler formulas and functional equations for the trigonometric functions

exp = cos + i · sin, cos = \frac{1}{2}(exp + exp^{−1}), sin = \frac{1}{2i}(exp − exp^{−1}),

exp(x + y) = exp(x) + exp(y),

cos(x + y) = cos(x)cos(y) − sin(x)sin(y)

sin(x + y) = cos(x)sin(y) + sin(x)cos(y).

In fact these elements will be used in the construction of twisted group rings as canonical generators. These generators and the additional relations will be derived from certain generators and relations of the F -group G. A specific example is the group ring KZ = K⟨g⟩ and the twisted group ring K_{Γ}Z = K[c, s]/(c^{2} + s^{2} − 1). If 2 is invertible in K then these two Hopf algebras are isomorphic iff K contains a square root i of −1. Then the isomorphism is given by

g = c + is, c = \frac{1}{2}(g + g^{−1}), s = \frac{1}{2i}(g − g^{−1}).

The diagonal on g, c, and s satisfies the ’functional equations’

Δ(g) = g ⊗ g, Δ(c) = c ⊗ c − s ⊗ s, Δ(s) = c ⊗ s + s ⊗ c.

These generators and relations descibing K_{Γ}G will even be generators and relations of the group ring KG in case K contains a subring K′ over which it is F -Galois. Thus we obtain very interesting new generators and relations even for the classical case of group rings.

We will show that all L-forms of group rings for a connected Galois extension L are twisted group rings as described above. So, many new Hopf algebras can be constructed in this way. A number of questions in the representation theory of groups are answered by base field extension, i.e. the splitting of group rings. One can now try to find twisted group rings defined already over the base field, which also have the required properties. Furthermore the structure of twisted group rings seems to be quite unknown yet, e.g. their centers, radicals, representation type, etc. Many of our calculations could be simplified thanks to helpful comments in [0].

2. The Multiplication Coefficients of a Free Galois Extension.

Throughout this paper we shall work with a free Galois extension of commutative rings L : K with finite Galois group F in the sense of [1], i.e. an extension of commutative rings such that L is a finitely generated free K-module, K is the fixed ring under the operation of a finite group F of automorphisms of L and there is an isomorphism h : L ⊗_{K} L ⟶ L ⊗_{K} (KF)^{∗}, h(l′ ⊗ l) = ∑_{π∈F} l′π(l) ⊗ π^{∗}, where {π^{∗}} is a dual basis to the basis {π ∈ F } of KF.

The choice of a free generating system {x_{i}} for L over K defines an isomorphism of free K-modules L ≅ K^{n}, x_{i} ↦ e_{i} = (0, …, 1, …, 0). This isomorphism defines a K-algebra structure on K^{n}, whose multiplication coefficients are determined by

x_{i}x_{j} = ∑_{k=1}^{n} α_{ij}^{k}x_{k}.

We determine these coefficients and those of the action of F and L solely in terms of a matrix C := (π(x_{i})|i = 1, …, n; π ∈ F). We avoid enumerating the group elements and thus the columns of C, but the reader can use any enumeration for F.

Lemma 1. Let L be a free F -Galois extension of K with basis {x_{i}|i = 1, …, n}. Let C = (γ_{i,π}|i = 1, …, n; π ∈ F) be the matrix with coefficients γ_{i,π} := π(x_{i}) in L. Then C is invertible with inverse matrix B = (β_{π,i}). The unit coefficients, the multiplication coefficients, and the F -action coefficients of L with respect to the basis {x_{i}} lie in K and are given by

ε_{i} = ∑_{π∈F} β_{π,i}  for  1 = ∑_{i=1}^{n} ε_{i}x_{i},  (1)

α_{ij}^{k} = ∑_{π∈F} γ_{i,π}γ_{j,π}β_{π,k}  for  x_{i}x_{j} = ∑_{k=1}^{n} α_{ij}^{k}x_{k},  (2)

ξ_{i,σ}^{k} = ∑_{π∈F} γ_{i,πσ}β_{π,k}  for  σ(x_{i}) = ∑_{k=1}^{n} ξ_{i,σ}^{k}x_{k}.  (3)

Proof: For the ”norm” (or ”integral”) e^{∗} ∈ (KF)^{∗} defined by e^{∗}(π) := δ_{e,π}, where e is the unit element of the group F, there is a uniquely determined element ∑ x_{i} ⊗y_{i} ∈ L⊗L such that h(∑_{i} x_{i} ⊗y_{i}) = ∑_{π,i} x_{i}π(y_{i})⊗ π^{∗} = 1 ⊗ e^{∗}. So by definition this element satisfies h(∑_{i} x_{i} ⊗ y_{i})(σ) = ∑_{i} x_{i}σ(y_{i}) = δ_{e,σ}, hence for all s ∈ L

s = ∑_{σ} σ(s)δ_{e,σ} = ∑_{i,σ} x_{i}σ(y_{i}s) = ∑_{i} x_{i}tr(y_{i}s).  (4)

We first show that the matrix B = (β_{π,i}) := (π(y_{i})|π ∈ F; i = 1, …, n) is the inverse matrix of C. We observe that ∑_{i} σ(x_{i})π(y_{i}) = σ(∑_{i} x_{i}σ^{−1}π(y_{i})) = σ(δ_{e,σ^{−1}π}) = δ_{e,σ^{−1}π} = δ_{σ,π}. Furthermore if we substitute s = x_{j} in (4) we get x_{j} = ∑_{i} x_{i} ∑_{π} π(y_{i})π(x_{j}). This implies ∑_{π} π(y_{i})π(x_{j}) = δ_{ij}.

If we substitute s = 1 in (4), we get the coefficients of the unit element of L from 1 = ∑_{i} tr(y_{i})x_{i} as

ε_{i} = tr(y_{i}) = ∑_{π∈F} β_{π,i}.

If we substitute s = x_{i}x_{j} in (4), we get the multiplication coefficients of L from x_{i}x_{j} = ∑_{k} x_{k}tr(y_{k}x_{i}x_{j}) = ∑_{k} x_{k} ∑_{π} π(x_{i})π(x_{j})π(y_{k}) as

α_{ij}^{k} = ∑_{π∈F} γ_{i,π}γ_{j,π}β_{π,k}.

Finally if we substitute s = σ(x_{i}) in (4), we get the coefficients for the F -action from σ(x_{i}) = ∑_{k} x_{k} ∑_{π} π(y_{k})πσ(x_{i}) as

ξ_{i,σ}^{k} = ∑_{π∈F} γ_{i,πσ}β_{π,k}. ⊔ ⊓

So we have described the algebra structure of L solely in terms of the matrix C (and its inverse B). The description of the F -action uses the right regular action of F on itself. The action is thus given by the induced permutations of the columns of C. Observe that the coefficients of C and B are in L, the coefficients of the unit, the multiplication and the F -action, however, are all elements in K (since they are obtained from the trace).

We consider the isomorphism

L^{n} ≅ L ⊗ K^{n} ≅ L ⊗ L ⟶^{h} L ⊗ (KF)^{∗} ≅ L ⊗ K^{F} ≅ L^{F}

where L^{F} is the set of n-tuples of elements from L, indexed by the set F, or the set of maps from F to L. This isomorphism sends (l_{i}|i = 1, …, n) to (∑_{i} l_{i}π(x_{i})|π ∈ F) = (∑_{i} l_{i}γ_{i,π}), i.e. it is multiplication by C on the right. The inverse map is multiplication by B. For a, b ∈ L^{F} we get (a · b)B = aB ⋆ bB, where ⋆ denotes the multiplication on L^{n} with multiplication coefficients (α_{ij}^{k}) and L^{F} has componentwise multiplication. Similarly the map is also compatible with the F -action, on L^{F} componentwise and on L^{n} with coefficients ξ_{i,σ}^{k}.

The coefficients as calculated in Lemma 1 can also be used to describe a K-algebra structure on S^{n} for any (non-commutative) K-algebra S and an action of F on S^{n} just by applying the isomorphism L ⊗_{K} S ≅ K^{n} ⊗_{K} S ≅ S^{n}. If S is an algebra of the form S ≅ L ⊗ T, then we can even use the isomorphism of F -algebras

(L ⊗ T)^{n} ≅ (L ⊗ T)^{F}

where (L⊗T)^{F} has componentwise operations and (L⊗T)^{n} has operations with coefficients from Lemma 1. The isomorphism is still given by the matrices B and C.

3. The Construction of a Twisted Group Ring

We want to construct the K-algebra K_{Γ}G for an F -group G by using defining generators and relations of G. We first have to look at free algebras.

Let X be a set and G_{f} be the free monoid on X. Then the monoid ring LG_{f} and the free (non-commutative) L-algebra L⟨X⟩ coincide. This is due to the fact that the underlying functor from the category Alg of non-commutative L-algebras to the category Set of sets factors through the category Mon of monoids. The free constructions are the corresponding left adjoint functors.

A similar argument can be used in a slightly more complicated setting. Let F be a finite group. Any of the categories Set, Mon, Alg named above can be used to construct a new category F -Set, F -Mon, F -Alg of objects on which F acts as automorphisms (actually pairs consisting of an object plus an action) and morphisms which are compatible with the F -action. It is easy to see that the left adjoint functors of the underlying functors Alg ⟶ Set, Alg ⟶ Mon, and Mon ⟶ Set can be used as left adjoint functors for the underlying functors F -Alg ⟶ F -Set, F -Alg ⟶ F -Mon, and F -Mon ⟶ F -Set, i.e. the diagrams

F -Cat1 \underrightarrow{free} F -Cat2
U ↓  U ↓
Cat1 \underrightarrow{free} Cat2

commute, since F acts by automorphisms. In particular we get that the monoid algebra LG_{f} of the free F -monoid G_{f} with generating set Y is the free F -algebra on the set Y and also the free algebra on the free F -set defined over Y. Now the free F -set on Y is just Y × F, so the algebra L⟨Y × F ⟩ is the monoid algebra LG_{f} over the free F -monoid generated by Y.

For a set Y = {g_{j}|j ∈ I} consider Y × F with elements π(g_{j}) := (g_{j}, π) and a set of ”variables” {S_{ij}|i = 1, …, n; j ∈ I} . Then the map Φ : LG_{f} ⟶ L⟨S_{ij}⟩ defined by

Φ(π(g_{j})) := ∑_{i=1}^{n} γ_{i,π}S_{ij}

is an L-algebra isomorphism with inverse map

Ψ(S_{ij}) := ∑_{π∈F} β_{π,i}π(g_{j}).

This follows immediately from the fact that B and C are inverse matrices of each other and that L-algebra homomorphisms are uniquely determined by the action on the free generating set. The algebra isomorphism Φ, however, is not an F -isomorphism, since no F -structure has been defined on L⟨S_{ij}⟩, but it could be used to define such a structure.

We want to use the isomorphism Φ to give a different representation of a group ring LG of an F -group G by generators and relations. For that purpose we first construct the monoid algebra LG_{f} of a free F -monoid G_{f} which has G as a quotient group. This algebra is isomorphic to L⟨S_{ij}⟩ by the isomorphism Φ. Then we import the relations of G written as pairs of elements of G_{f}. Their differences generate the ideal J of LG_{f} to be factored out to obtain (a representation of) LG. This ideal will be transferred to the ideal M of L⟨S_{ij}⟩ via the given isomorphism Φ. The central point of the calculation will be that the generators of M are already defined over K (instead of L), i.e. they are elements of K⟨S_{ij}⟩ which generate an ideal M′ ⊆ K⟨S_{ij}⟩. Thus we obtain a K-algebra K_{Γ}G := K⟨S_{ij}⟩/M′ with the property L ⊗_{K} K_{Γ}G ≅ LG.

More precisely let G be an F -group generated as an F -monoid by the set Y = {g_{i}|i ∈ I_{1}}. Then there is a surjective F -homomorphism ϕ: G_{f} ⟶ G, where G_{f} is the free F -monoid on Y. A relation for G is a pair of elements (r_{1}, r_{2}) ∈ G_{f} × G_{f} with ϕ(r_{1}) = ϕ(r_{2}). The set of all relations for G is an F -submonoid of G_{f} ×G_{f}. Let {r_{j} = (r_{j1}, r_{j2})|j ∈ I_{2}} be a generating set of the F -submonoid of relations. Then G can be considered as a quotient of G_{f} modulo the congruence relation generated by {r_{j}}.

We consider two elements r and s of G_{f} (written with non-negative powers and F -multiples of the generators in Y). Let J be the ideal generated by {π(r) − π(s)|π ∈ F } in LG_{f}. Then M := Φ(J) is generated by {Φ(π(r) − π(s))|π ∈ F }. We want to change this generating set to a set of K-linear combinations of the S_{ij}. For this purpose we form

{∑_{π∈F} β_{π,i}Φ(π(r) − π(s))|i = 1, …, n}.

Lemma 2. The ideal M in L⟨S_{ij}⟩ generated by {Φ(π(r) − π(s))|π ∈ F } is also generated by the set {∑_{π∈F} β_{π,i}Φ(π(r) − π(s))|i = 1, …, n} ⊆ K⟨S_{ij}⟩.

Proof: Since the matrix (β_{π,i}) is invertible, this set generates the same ideal M in L⟨S_{ij}⟩. We show that the coefficients of the products of the S_{ij}’s are all in K. Let r = τ_{1}(g_{1}) · . . . · τ_{p}(g_{p}). Then we obtain

∑_{π∈F} β_{π,i}Φ(π(τ_{1}(g_{1}) · . . . · τ_{p}(g_{p}))

= ∑_{π∈F} β_{π,i}Φ(πτ_{1}(g_{1})) · . . . · Φ(πτ_{p}(g_{p}))

= ∑_{π∈F} ∑_{i_{1},...,i_{p}} β_{π,i}γ_{i_{1},πτ_{1}}S_{i_{1},1} · . . . · γ_{i_{p},πτ_{p}}S_{i_{p},p}

= ∑_{i_{1},...,i_{p}}(∑_{π∈F} β_{π,i}γ_{i_{1},πτ_{1}} · . . . · γ_{i_{p},πτ_{p}})S_{i_{1},1} · . . . · S_{i_{p},p}.

Since the coefficients

∑_{π∈F} β_{π,i}γ_{i_{1},πτ_{1}} · . . . · γ_{i_{p},πτ_{p}} = ∑_{π∈F} π(y_{i})πτ_{1}(x_{i_{1}}) … πτ_{p}(x_{i_{p}})

are the trace of certain elements they are in K. ⊔ ⊓

Theorem 3. Let G be an F -group with F -monoid generators {g_{i}|i ∈ I_{1}} and F -monoid relations {r_{i}|i ∈ I_{2}} generating all F -monoid relations of G. Let M′ be the ideal of the K-algebra K⟨S_{ij}|i = 1, …, n; j ∈ I_{1}⟩ generated by the set

{∑_{π∈F} β_{π,i}Φ(π(r_{j1}) − π(r_{j2}))|i = 1, …, n; j ∈ I_{2}}

where Φ(π(g_{j})) := ∑_{i} γ_{i,π}S_{ij}. Then K_{Γ}G := K⟨S_{ij}⟩/M′ is a Hopf algebra and an L-form of the Hopf algebra KG, i.e. Φ: L ⊗_{K} KG ≅ L ⊗_{K} K_{Γ}G.

Proof: By hypothesis the F -group G is represented as a quotient of a free F -monoid by certain relations. If r = (r_{1}, r_{2}) is a relation in G then r_{1} = r_{2} in G hence π(r_{1}) = π(r_{2}) for all π ∈ F. It is easy to see that the group algebra KG is the quotient KG_{f}/J′ where G_{f} is the free F - monoid generated by {g_{i}|i ∈ I_{1}} and J′ is the ideal of KG_{f} generated by {π(r_{i1}) − π(r_{i2})|i ∈ I_{2}, π ∈ F }. If we identify L ⊗_{K} KG and LG, L⊗_{K}K⟨S_{ij}⟩ and L⟨S_{ij}⟩, more generally the tensor product with L with the multiplication with L, then we get an algebra isomorphism Φ: L ⊗ KG ≅ L ⊗ K⟨S_{ij}⟩/M′, since L is faithfully flat.

The diagonal on L⟨S_{ij}⟩/L·M′ is induced by Φ: LG ⟶ L⟨S_{ij}⟩/L·M′. We have

Δ(s_{kl}) = Δ(∑_{π∈F} β_{π,k}Φ(π(g_{l})))

= ∑_{π∈F} β_{π,k}Φ(π(g_{l})) ⊗ Φ(π(g_{l})) = ∑_{i,j,π} γ_{i,π}γ_{j,π}β_{π,k}s_{il} ⊗ s_{jl}

= ∑_{i,j} α_{ij}^{k}s_{il} ⊗ s_{jl}

where s_{ij} is the residue class of S_{ij}. Since the coefficients α_{ij}^{k} are in K the diagonal is already defined in K⟨s_{ij}⟩ = K⟨S_{ij}⟩/M′. In a similar way we obtain ε(s_{ij}) = ∑_{π} β_{π,i} = ε_{i} (independent of j!) also defined on K⟨s_{ij}⟩. Since L⟨s_{ij}⟩ is a bialgebra (isomorphic to LG) and L/K is faithfully flat, we get that K⟨s_{ij}⟩ is also a bialgebra. It is a bialgebra form of KG. By [2 Remark following Thm.1] a bialgebra form of a Hopf algebra is already a Hopf algebra. ⊔ ⊓

The Hopf algebra K_{Γ}G := K⟨s_{ij}⟩ is called a Γ-twisted group ring. We have constructed the Γ-twisted group ring by generators s_{ij} and relations using the formula

s_{ij} = ∑_{π∈F} π(y_{i})π(g_{j})

in LG. Each relation r for G induces n relations in K_{Γ}G

∑_{π∈F} β_{π,i}Φ(π(r_{1}) − π(r_{2})) = 0.

These generators and relations are obtained from the defining F -genera-tors and relations for G. Each generator g_{j} (in fact each element) of G and its F -orbit in G ⊆ KG corresponds to a family (s_{ij}|i = 1, …, n) in K_{Γ}G by the given formula. The diagonal of K_{Γ}G applied to an element of such a family can be expressed solely by the elements of the same family (and coefficients in K) so we define a Γ-group-like sequence of a Hopf algebra H to be an n-tuple (s_{i}) in H such that

Δ(s_{k}) = ∑_{i,j} α_{ij}^{k}s_{i} ⊗ s_{j} and ε(s_{i}) = ε_{i}

holds.

Corollary 4. Let G be an F -group. Let L be a commutative ring which contains subrings K ⊆ L′ ⊆ L such that L′ : K is an F -Galois extension. Then the group algebra LG has generators {s_{il}|i = 1, …, n; l ∈ I} such that for all l ∈ I

Δ(s_{kl}) = ∑ α_{ij}^{k}s_{il} ⊗ s_{jl}  and  ε(s_{il}) = ε_{i}.

The relations for the {s_{ij}} are those of Theorem 3.

Proof: follows immediately from the isomorphism LG ≅ L ⊗_{L′} L′ ⊗_{K} KG ≅ L ⊗_{L′} L′ ⊗_{K} K_{Γ}G. ⊔ ⊓

4. Twisted Group Rings as Adjoint Functors

In this paragraph we shall identify LG and L ⊗ K_{Γ}G via the isomorphism Φ as constructed above. Furthermore we view KG and K_{Γ}G as subalgebras in LG. In particular we have the equations

π(g_{j}) = ∑_{i=1}^{n} γ_{i,π}s_{ij}, s_{ij} = ∑_{π∈F} β_{π,i}π(g_{j}),

where {g_{i}|i ∈ I_{1}} is a generating set of G as an F -monoid.

Theorem 5. Let G be an F -group. Then there exists a cocommutative Hopf algebra K_{Γ}G and a homomorphism of F -groups ι: G ⟶ Γ(K_{Γ}G) such that for each K-algebra S and homomorphism of F -groups f: G ⟶ Γ(S) there exists precisely one K-algebra homomorphism f̃: K_{Γ}G ⟶ S, such that the diagram

G ⟶^{ι} Γ(K_{Γ}G)
f ↘ ↓ _{Γ(_{f})}
Γ(S)

commutes.

Proof: We define ι: G ⟶ Γ(K_{Γ}G) = U(LG) by ι(g) := g ∈ LG. This is obviously an F -homomorphism.

We choose a set of F -monoid generators {g_{i}|i ∈ I_{1}} of G and a generating set of relations {r_{i}|i ∈ I_{2}} . Construct K_{Γ}G as in Theorem 3. For the generating elements of G let f(g_{j}) = ∑_{i} x_{i} ⊗ t_{ij} ∈ U(L ⊗ S). The coefficients t_{ij} ∈ S are uniquely determined. Observe that

f(π(g_{j})) = πf(g_{j}) = ∑_{i=1}^{n} π(x_{i}) ⊗ t_{ij}.

Define f̃(s_{ij}) := t_{ij}. In a first step this is only defined on K⟨S_{ij}⟩. In order to show that this gives a well-defined algebra homomorphism f̃: K_{Γ}G ⟶ S we have to check, that it preserves the relations in K_{Γ}G. Let r = (r_{1}, r_{2}) be a generating relation for G and ∑_{π∈F} β_{π,i}(π(r_{1}) − π(r_{2})) for some i ∈ {1, …, n} be one of the defining relations for the ideal M′. We calculate the action of f̃ only on the first part of this term

f̃(∑_{π∈F} β_{π,i}π(r_{1})) = f̃(∑_{π∈F} β_{π,i}π(τ_{1}(g_{1}) … τ_{p}(g_{p}))

= f̃( ∑_{i_{1},...,i_{p}} ∑_{π∈F} β_{π,i}γ_{i_{1},πτ_{1}} … γ_{i_{p},πτ_{p}}s_{i_{1}1} … s_{i_{p}p})

= f̃( ∑_{i_{1},...,i_{p}}(∑_{π∈F} π(y_{i})π(τ_{1}(x_{i_{1}}) … τ_{p}(x_{i_{p}})))s_{i_{1}1} … s_{i_{p}p})

= ∑_{i_{1},...,i_{p}} tr(y_{i}τ_{1}(x_{i_{1}}) … τ_{p}(x_{i_{p}}))f̃(s_{i_{1}1} … s_{i_{p}p})

= ∑_{i_{1},...,i_{p}}(∑_{π∈F} π(y_{i}τ_{1}(x_{i_{1}}) … τ_{p}(x_{i_{p}})) ⊗ t_{i_{1}1} … t_{i_{p}p})

= ∑_{π∈F} β_{π,i}(∑_{i_{1}} πτ_{1}(x_{i_{1}}) ⊗ t_{i_{1}1}) … (∑_{i_{p}} πτ_{p}(x_{i_{p}}) ⊗ t_{i_{p}p})

= ∑_{π∈F} β_{π,i}πτ_{1}f(g_{1}) … πτ_{p}f(g_{p})

= ∑_{π∈F} β_{π,i}f(πτ_{1}(g_{1}) … πτ_{p}(g_{p}))

= ∑_{π∈F} β_{π,i}f(π(r_{1})).

Now f(r_{1}) = f(r_{2}) implies f̃(∑_{π∈F} β_{π,i}(π(r_{1})−π(r_{2}))) = 0. Thus we get a well-defined K-algebra homomorphism f̃: K_{Γ}G ⟶ S. For g_{j} ∈ G we then get Γ(f̃)ι(g_{j}) = Γ(f̃)(∑_{i} γ_{i,e}s_{ij}) = ∑_{i} x_{i} ⊗ (f̃(s_{ij})) = ∑_{i} x_{i} ⊗ t_{ij} = f(g_{j}).

To show the uniqueness of f̃ let f̂ be another extension of f so that Γ(f̃)ι = Γ(f̂)ι = f. f̂ induces an L-algebra homomorphism from LG to L ⊗ S which we also denote by f̂. Then f̃(s_{ij}) = f̃(∑_{π} β_{π,i}π(g_{j})) = ∑_{π} β_{π,i}f̃(π(g_{j})) = ∑_{π} β_{π,i}Γ(f̃)ι(π(g_{j})) = ∑_{π} β_{π,i}f(π(g_{j})) = f̂(∑_{π} β_{π,i}π(g_{j})) = f̂(s_{ij}) so that f̃ = f̂. ⊔ ⊓

The universal property shown above implies immediately

Corollary 6. The functor K_{Γ}: F -Gr ⟶ K-Alg is left adjoint to Γ: K-Alg ⟶ F -Gr. ⊔ ⊓

One could also have shown that Γ is an algebraic functor in the sense of [4]. Then the adjointness property would have resulted from the general theory, but the explicit construction as K_{Γ} would not have followed. Furthermore the Hopf algebra property of K_{Γ}G would have required additional investigations.

5. The Structure of Forms of Group Rings

Up to now we have constructed a specific version of (twisted) Lforms of group rings (Theorem 3), namely the twisted group rings. We have studied their structure in terms of generators and relations and their property as a left adjoint functor. Now we want to prove that all forms of group rings are twisted group rings under only minor restrictions. For this purpose we consider those forms H of group rings which are split by a free (F -)Galois extension L of K, i.e. L⊗_{K} H ≅ L⊗_{K} KG and we assume that L is connected, i.e. has only the idempotents 0 and 1.

Theorem 7. Let L : K be a free Galois extension with finite Galois group F and let L be connected. Let G be a group and H be a K-Hopf algebra which is an L-form of the group ring KG. Then there is an F -structure on G such that H is isomorphic to a Γ-twisted group ring K_{Γ}G.

Proof: Since L : K is faithfully flat we can identify H and KG with specific K-subalgebras of LG. In particular the H-module LG(≅ L ⊗_{K} H) is freely generated over the basis {x_{i}}. Each element g ∈ G has a representation g = ∑_{i=1}^{n} x_{i}s_{i}(g) with uniquely determined coefficients s_{i}(g) ∈ H. This defines maps s_{i} : G ⟶ H. These maps correspond to the generators s_{ij} coming from the group generators g_{j} in the preceding paragraphs.

We define an F -action on G by π(g) := ∑_{i} π(x_{i})s_{i}(g). First we show that π(g) is an element of G. Since L is connected it suffices to prove that π(g) is a group-like element in LG. By Lemma 1 the diagonal of the elements s_{i}(g) can be derived from ∑_{k} x_{k}Δ(s_{k}(g)) = Δ(g) = g ⊗ g = ∑_{ij} x_{i}x_{j}s_{i}(g) ⊗ s_{j}(g) = ∑_{k} ∑_{ij} α_{ij}^{k}x_{k}s_{i}(g) ⊗ s_{j}(g) as Δ(s_{k}(g)) = ∑_{ij} α_{ij}^{k}s_{i}(g) ⊗ s_{j}(g). The augmentation is ∑_{k} x_{k}ε(s_{k}(g)) = ε(g) = 1 = ∑ x_{k}ε_{k} hence ε(s_{i}(g) = ε_{i}. So (s_{i}(g)|i = 1, …, n) is a Γgroup-like sequence in H. Then

\begin{aligned}Δ(π(g)) & = ∑_{k} π(x_{k})Δ(s_{k}(g)) = ∑_{k} ∑_{ij} α_{ij}^{k}π(x_{k})s_{i}(g) ⊗ s_{j}(g) \\ & = ∑_{ij} π(x_{i}x_{j})s_{i}(g) ⊗ s_{j}(g) = π(g) ⊗ π(g) \end{aligned}

and ε(π(g)) = ∑_{i} π(x_{i})ε(s_{i}(g)) = π(∑_{i} ε_{i}x_{i}) = π(1) = 1 imply that π(g) is group-like, hence π(g) ∈ G.

Now it is easy to verify that this is an action of F on G by automorphisms. From Lemma 1 we get

τ(σ(g)) = τ(∑_{i} σ(x_{i})s_{i}(g)) = τ(∑_{i} ∑_{k} ξ_{i,σ}^{k}x_{k}s_{i}(g))

= ∑_{i,k,π} γ_{i,πσ}β_{π,k}γ_{k,τ} s_{i}(g) = ∑_{i} γ_{i,τσ}s_{i}(g) = (τσ)(g).

Furthermore we have

π(gh) = π(∑_{i,j} x_{i}x_{j}s_{i}(g)s_{j}(h)) = π(∑_{k} x_{k} ∑_{i,j} α_{ij}^{k}s_{i}(g)s_{j}(h))

= ∑_{k} π(x_{k}) ∑_{i,j} α_{ij}^{k}s_{i}(g)s_{j}(h)) = ∑_{i,j} π(∑_{k} α_{ij}^{k}x_{k})s_{i}(g)s_{j}(h))

= ∑_{i,j} π(x_{i})π(x_{j})s_{i}(g)s_{j}(h) = π(g)π(h).

By definition we also have e(g) = g so that G is an F -group.

The choice of F -monoid generators {g_{j}} for G defines K-algebra generators s_{i}(g_{j}) for H with the formulas g_{j} = ∑_{i} x_{i}s_{i}(g_{j}) and s_{i}(g_{j}) = ∑_{π} β_{π,i}π(g_{j}). The relations are transformed as in Theorem 3. ⊔ ⊓

The connection with some known results should be mentioned here. In [6 Thm.6.4.] the category of finite etale group schemes over a field K was found to be equivalent to the category of finite G-groups, where G is the profinite Galois group of the separable closure of K, acting continuously on the finite groups. These group schemes, represented by commutative Hopf algebras, become constant group schemes already after a finite Galois field extension L, i.e. L ⊗ H^{∗} ≅ LG for a finite group G. The action of G on G is given by the action of F = Aut(L/K) on G via the isomorphism

K-Alg(H, L) ≅ L-Alg(L ⊗_{K} H, L) ≅ L-Coalg(L, L ⊗_{K} H^{∗})
≅ Group-Likes(L ⊗ H^{∗}) ≅ Group-Likes(LG) ≅ G.

Theorem 7 is a generalization of this theorem to infinite groups and away from the field requirements for L and K. On the other hand our theorem also generalizes part of the known antiequivalence between group schemes of multiplicative type and abelian G-groups [6 Thm.7.3.] to noncommutative Hopf algebras. In fact our Theorem 3 gives a description of the representing algebras of group schemes of multiplicative type by generators and relations of the corresponding character group.

6. Two Examples

We want to illustrate our results with two examples. Let C_{2} = {π_{1}, π_{2}} = {ϵ, σ} be the cyclic group with two elements. Assume that 2 is invertible in K. Consider the C_{2}-Galois extension L = K(i) of K with i^{2} = −1 and basis x_{1} = 1, x_{2} = i. We form the matrix

(γ_{iπ}) = ( \begin{matrix}π_{1}(x_{1}) & π_{2}(x_{1}) \\ π_{1}(x_{2}) & π_{2}(x_{2}) \end{matrix} ) = ( \begin{matrix}1 & 1 \\ i & −i \end{matrix} ) .

with the inverse matrix

(β_{ij}) = ( \frac{1}{2} −\frac{1}{2i}^{\frac{1}{2} \frac{1}{2i}} )

Then the multiplication coefficients are

(α_{ij}^{k}) = (∑_{l=1}^{2} γ_{kl}β_{li}β_{lj}) = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} | \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

where the two adjacent matices are (α_{ij}^{1}) and (α_{ij}^{2}). The coefficients of the action of F on L are

(ξ_{i,σ}^{k}) = (∑_{π∈F} γ_{i,πσ}β_{π,k}) = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )

and the coefficients of the unit are (ε_{1}, ε_{2}) = (1, 0).

For a Γ-twisted group ring K_{Γ}G we use the isomorphism (LG)^{n} ≅ (LG)^{F} as described at the end of paragraph 2 given by multiplication by B resp. C. We denote the multiplication in the algebra (LG)^{n} by ⋆, the action of F with no special notation. We still consider K_{Γ}G and KG embedded in LG. Each element g ∈ G induces an element (π(g)|π ∈ F) ∈ (LG)^{F} and the fact that F acts by automorphisms induces (π(g))·(π(h)) = (π(gh)). Each relation in G thus induces a relation in (LG)^{F}, hence n relations in LG.

We construct the twisted group ring K_{Γ}Z. The group G := Z has one (F -monoid) generator g and C_{2} acts on G by σ(g) = g^{−1}. So there is a relation σ(g)g = 1 (if we consider G as a multiplicative group). We define

(c, s) := (g, σ(g)) ( \frac{1}{2i} −\frac{1}{2i}^{\frac{1}{2} \frac{1}{2}} ) ,

i.e. c = \frac{1}{2}(g + g^{−1}) and s = \frac{1}{2i}(g − g^{−1}) in LG. These elements generate a K-subalgebra K[c, s] ⊆ LG. There is a relation c^{2} + s^{2} = 1, in fact the subalgebra is isomorphic to K_{Γ}G = K⟨C, S⟩/(C^{2} + S^{2} − 1, CS − SC) where K⟨C, S⟩ is the free algebra on C, S (the polynomial ring in noncommuting variables C, S). The relations arise from a translation of the relation σ(g)g = 1 to

σ(c, s) ⋆ (c, s) = (ε_{1}, ε_{2})

or using the definition of ⋆ and the operation of F on (LG)^{n}

(c, −s)(α_{ij}^{k})(c, s)^{t} = (1, 0)

hence c^{2} + s^{2} = 1 and −sc + cs = 0.

It is obvious that this algebra K_{Γ}G represents the circle group functor which associates with each K-algebra S the ”circle” {(c, s)|c, s ∈ S; c^{2} + s^{2} = 1} with multiplication (c, s) · (c′, s′) = (cc′ − ss′, cs′ + sc′) and unit (1, 0) which we studied in [2].

The second example uses again the Galois extension L = K(i) over K with Galois group C_{2}, but this time we use the symmetric group G = S_{3}. The Γ-twisted group ring K_{Γ}S_{3} is a specific form of KS_{3}. In the theory of [2] we were able to describe all forms of KS_{3} by all S_{3} = Aut(S_{3})-Galois extensions of K, which are difficult to describe. Here we need only a C_{2}- Galois extension of K, but we obtain only one specific form of KS_{3}. The Galois group C_{2} acts on S_{3} in the following way. We represent S_{3} by the generators g_{1}, g_{2} and the relations g_{1}^{2} = 1, g_{2}^{3} = 1, and g_{1}g_{2} = g_{2}^{2}g_{1}. The action of C_{2} then is given by σ(g_{1}) = g_{1} and σ(g_{2}) = g_{2}^{2}. Consider the equations

(c, s) := (g_{1}, σ(g_{1})) ( \frac{1}{2i} −\frac{1}{2i}^{\frac{1}{2} \frac{1}{2}} ) ,

and

(d, t) := (g_{2}, σ(g_{2})) ( \frac{1}{2i} −\frac{1}{2i}^{\frac{1}{2} \frac{1}{2}} ) .

They define elements c = \frac{1}{2}(g_{1} + g_{1}) = g_{1}, s = \frac{1}{2i}(g_{1} − g_{1}) = 0, d = \frac{1}{2}(g_{2} + g_{2}^{2}), and t = \frac{1}{2i}(g_{2}−g_{2}^{2}) in LG. The twisted group ring K_{Γ}S_{3} is the subring K[c, s, d, t] ⊆ LG which can be represented as K_{Γ}S_{3} = K⟨C, S, D, T ⟩/I, where I is the ideal generated by the following expressions (simplified by s = 0):

C^{2} − 1,  S,
D^{3} − T^{2}D − T DT − DT^{2} − 1, T^{3} − D^{2}T − DT D − T D^{2},
D^{2}C − T^{2}C − CD,  DT C + T DC − CT,
D^{2} − T^{2} − D,  DT + T D + T.

These expressions follow from the translation of the group relations

(c, s) ⋆ (c, s) = (1, 0),
((d, t) ⋆ (d, t)) ⋆ (d, t) = (1, 0),
(c, s) ⋆ (d, t) = ((d, t) ⋆ (d, t)) ⋆ (c, s),
σ(d, t) = (d, t) ⋆ (d, t),

or from

(C^{2} − S^{2}, SC + CS) = (1, 0),

(D^{3} − T^{2}D − DT^{2} − T DT, D^{2}T − T^{3} + DT D + T D^{2}) = (1, 0), (CD − ST, CT + SD)

= (D^{2}C − T^{2}C − DT S − T DS, D^{2}S − T^{2}S + DT C + T DC),

(D, −T) = (D^{2} − T^{2}, DT + T D).

The ⋆-multiplication is still defined by the matrix (α_{ij}^{k}) and the action of C_{2} is as in Lemma 1.

As C. Greither pointed out, the particularly nice multiplication and operation matrices for the C_{2}-extension K(i)/K as calculated at the beginning of this paragraph are due to the fact that K(i)/K is a Kummer extension and hold for general Kummer extensions. Consider a Kummer extension K(α)/K with α^{n} = b ∈ K, n invertible in K and ζ a primitive n-th root of unity in K. Choose 1, α, …, α^{n−1} as a basis of K(α) over K. Then by a simple computation the matrices obtained for this basis are

C = (π(α^{i})) = ((ζ^{j}α)^{i}),

B = (β_{j,π}) = \frac{1}{n}((ζ^{i}α)^{−j}),

A = (α_{ij}^{k}) with α_{ij}^{k} = \{ \begin{matrix}1 & for i = j + k, \\ b^{−1} & for i + n = j + k, \\ 0 & else, \end{matrix}

and E = ^{t}(1, 0, …, 0).

References

[0] Anonymous: Referees Report (unpublished).
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Galois Cohomology of Commutative Rings. Mem. Am. Math. Soc. 52 (1965).

[2] R. Haggenmüller, B. Pareigis: Hopf Algebra Forms of the

Multiplicative Group and other Groups. manuscr. math. 55, 121-136 (1986).

[3] B. Pareigis: Descent Theory applied to Galois Theory. Technical

Report Univ. California, San Diego, 1986.

[4] B. Pareigis: Categories and Functors. Academic Press 1970.
[5] M. Sweedler: Hopf Algebras. Benjamin – New York 1969.
[6] W.C. Waterhouse: Introduction to Affine Group Schemes.

Springer Verlag – New York, Heidelberg, Berlin 1979