Classification of Isomorphism Types of Hopf Algebras in a Class of Abelian Extensions

0.38 MB·English

by  Leonid Krop

CLASSIFICATION OF ISOMORPHISM TYPES OF
HOPF ALGEBRAS IN A CLASS OF ABELIAN
2 EXTENSIONS
1
0
2 LEONID KROP AND YEVGENIA KASHINA

v
o
Abstract. We study Hopf algebras which are extensions of the
N
groupalgebrakCpofacyclicgroupCpofprimeorderpbytheHopf
3 algebradualkG ofthegroupalgebrakGofafiniteabelianp-group
2 G over an algebraically closed field of characteristic 0. The main
resultsare: structuretheoremforthe2nd Hopfcohomologygroups
A] H2(⊳), a criterionforisomorphismoftwoextensions, enumeration
c
of isomorphism classes by certain orbits in H2(⊳), and a complete
Q c
classification of commutative extensions for any odd p.
.
h
t
a
m Keywords Hopf algebras, Abelian extensions, Crossed products,
[ Cohomology Groups

Mathematics Subject Classification (2000) 16W30 - 16G99
1
v
1 Introduction
2
6
The purpose of this article is to find a general criterion for isomor-
5
. phismoftwo Hopfalgebrasincertainclasses ofabelianextensions. The
1
classes in question consist of Hopf algebras which are extensions of the
1
2 group algebra kC by kG for a prime p, where C is a cyclic group of
p p
1
prime order p, G a finite abelian p-group, and k an algebraically closed
:
v field of characteristic zero. The customary definition of equivalence
i
X of extensions [16] gives rise to the set Ext(kC ,kG) of all equivalence
p
r classes of extensions.
a
Every algebra from any of our classes is semisimple by [18, 7.4.2].
By a fundamental result of D. Stefan [21] the number of isomorphism
types in any of our classes is finite. Our main results are similar to Ste-
fan’s inasmuch as both involve an orbit space. In our case this space
is constructed as follows. The group G is a right C -module under
p
the action denoted by ‘⊳’. The set Ext(kC ,kG) splits up into a dis-
p
joint union of groups Opext(kC ,kG,⊳) of classes of extensions with
p
the same C -action on G. Cohomology theory of abelian extensions
p
Date: 10/20/12.
Research by the first author partially supported by a grant from the College of
Liberal Arts and Sciences at DePaul University.
1
2 LEONID KROPANDYEVGENIAKASHINA
of finite-dimensional Hopf algebras [16] supplies us with the abelian
group H2 (kC ,kG,⊳) of Hopf 2-cocycles parametrizing algebras in
Hf p
Opext(kC ,kG,⊳). Two groups play a distinguished rˆole in the pa-
p
per. The first one A(⊳) = Aut (G) is the group of C -automorphisms
Cp p
of G. In the second place comes the group A = Aut(C ) of au-
p p
tomorphisms of C . A(⊳) acts on H2 (kC ,kG,⊳) turning it into an
p Hf p
A(⊳)-module, while A twists the action ‘⊳’ into p−1 actions ⊳α,α ∈
p
A . Let [G,⊳] denote the class of C -modules isomorphic to [G,⊳α]
p p
for some α ∈ A . Furthermore, we let Ext (kC ,kG) stand for
p [G,⊳] p
the set of equivalence classes of extensions whose C -module G lies
p
in [G,⊳]. The main result of the paper is the statement that two
Hopf algebras H,H′ ∈ Ext(kC ,kG) are isomorphic implies they lie
p
in Ext (kC ,kG) for some ‘⊳’, and moreover there exists a bijection
[G,⊳] p
between the orbits of A(⊳) in H2 (kC ,kG,⊳) generated by a single
Hf p
noncoboundary and the isomorphism classes of noncocommutative al-
gebras in Ext (kC ,kG). Under a stronger hypothesis on G, namely
[G,⊳] p
G elementary abelian, the theorem is strengthened to a bijection be-
tweenallorbitsgeneratedbyasingle2-cocycleandisomorphismclasses
of algebras.
Elements ofExt(kC ,kG)possess two special features. Every algebra
p
H there is equivalent to a smash product kG#kC , and kCp is central
p
in the dual Hopf algebra H∗. In consequence, it is only the coalge-
bra structure of H that is deformed by a 2-cococycle. These types of
abelian extensions have been studied by M. Mastnak [12]. We adopt
his notation H2(kC ,kG,⊳) for the group of Hopf 2-cocycles. A major
c p
result of the paper is structure theorem for H2(kC ,kG,⊳). It states
c p
that if G is any finite abelian p-group with p > 2, or a finite elementary
2-group then there is a C -isomorphism
p
(0.1) H2(kC ,kG,⊳) ≃ H2(C ,G)×H2(G,k•)
c p p N
where G is the dual group of G, H2(C ,bG) is the second cohomology
p
group of C over G with respect to the action ⊳, and H2(G,k•) is
p N
the kernbel of the norm mapping in the Schur multiplier of G. The for-
mula shows an interbesting interplay between Hopf algebras and groups.
For, in the case ⊳ is trivial, we have by Proposition 5.1 that the sub-
group of commutative extensions in Ext(kC ,kG) is identical with the
p
group of central extensions of G by C , so that H2(kC ,kG,triv) equals
p c p
H2(G,C ). Thus, (0.1) can be seen as a Hopf algebraic generalization
p
of a structure theorem for the 2nd cohomology group H2(G,A) of cen-
tral extensions of a finite abelian group by a group A [24, Thm. 2.2].
ISOMORPHISM TYPES OF HOPF ALGEBRAS 3
Another notable result is the computation of the number of isomor-
phism types of commutative algebras in Ext(kC ,kG) for G elementary
p
abelian and an odd p. We conclude the paper with an illustration of
our technique in the case G = C ×C completely described by
p p
A. Masuoka [15].
The paper is organized in six sections. In section 1 we review the
necessary facts of the theory of abelian extension with the emphasis on
duality inherent in the theory and we derive the formula for comulti-
plication in the dual of a crossed product from first principles. Each
abelian extension is associated with a pair of groups (G,F). In section
2 we narrow our focus to the so-called cocentral extensions [8] with
the additional property of the trivial 2-cohomology of F in kG. In sec-
tion 3 our setting is even more restrictive. There we study extensions
associated with pairs (G,F) where G is a p-group and F = C is a
p
cyclic group of order p and we calculate the second Hopf cohomology
groups. Section 4 contains the main results of the paper. In Section 5
we classify commutative extensions and compute their number up to
isomorphism. In the last section we rederive a key result of [15] using
our technique.
1. Background Review
1.1. Extensions of Hopf Algebras. Let k be a ground field and H
denote the category of Hopf algebras over k. For a Hopf algebra H we
let ǫ and u denote the augmentation H → k and the unit k → H
H H
mappings, respectively. k itself can be viewed as a Hopf algebra with
∆(1 ) = 1 ⊗ 1 , S(1 ) = 1 and ǫ(1 ) = 1 . It is easy to check that
k k k k k k k
for every H ∈ H, Hom (H,k) = {ǫ } and Hom (k,H) = {u }.
H H H H
From the point of view of the category theory k is the zero object in
H. Applying the categorical definition of kernel to H we say that a
Hopf algebra (K,ι) is the kernel of a Hopf morphism φ : F → G if
ι : K → F is a Hopf morphism and linear monomorphism fitting into
the commutative diagram
K −−ǫ−K→ k
(1.1) ι uG
 φ 
F −−−→ G
y y
and universal among all diagrams 1.1. That is, for every (K′,ι′) sat-
isfying 1.1 there is a unique λ : K′ → K with ιλ = ι′. Since (k,u )
F
satisfies 1.1, the set of subHopfalgebras of F satisfying 1.1 is nonempty.
Moreover, the subalgebra generated by two Hopf subalgebras satisfying
4 LEONID KROPANDYEVGENIAKASHINA
1.1 is a Hopf subalgebra satisfying 1.1 hence there exists a unique Hopf
subalgebra satisfying 1.1. Following [1] we denote that Hopf algebra
by HKerφ.
Dually, a Hopf morphism and a linear epimorphism π : G → Q is
called the Hopf cokernel of φ, notation Q = HCokerφ, if π fits into the
commutative diagram
φ
F −−−→ G
(1.2)
ǫF π
k −−u−Q→ Q
y y
and for every π′ : G → Q′ satisfying 1.2 there is a unique µ : Q → Q′
satisfying µπ = π′. It is immediate that π : G → G/Gι(F+)G is
HCokerφ.
We can make
Definition 1.1. A Hopf algebra C is called a weak extension of a Hopf
algebra B by a Hopf algebra A if there is a sequence of Hopf mappings
(S) A ֌ι C ։π B.
with A = HKerπ and B = HCokerι.
In what follows we assume the antipode of C is invertible. We
point out that our definition of Hopf kernels coincides with the one
in [19] and [1]. Let Fcoφ and coφF be the subalgebras of coinvari-
ants [18, 3.4], also denoted by RKerφ and LKerφ, respectively, in [3].
A characterization of HKerφ in [19] and [1] yields immediately inclu-
sions HKerφ ⊂ LKerφ ∩ RKerφ. On the other hand, the equalities
HKerφ = RKerφ = LKerφ, hold provided RKerφ or LKerφ are Hopf
subalgebras of F by the arguments [3, 4.19] or [1, 1.1.4].
A major drawback of Definition 1.1 is that it does not guarantee,
in general, that ι(A) coincides with either one of subalgebras of coin-
variants. That is, HKerφ is, in general, not equal to RKerφ. This is
already easy to see for any Taft Hopf algebra [22] with respect to the
standard epimorphism π : A → kG, G a cyclic group generated by T
in the notation of [19, Ex.1.2]. In this case HKerπ is zero, i.e. equals
to k, while RKerπ is |G|-dimensional. An example of a weak extension
(S) with ι(A) $ RKerπ ∩ LKerπ is harder and can be found in [19,
Ex.1.2]. We adopt the following definition of a Hopf algebra extension.
Definition 1.2. A Hopf algebra C is said to be an extension of B by
A if C is a weak extension of B by A and HKerφ = RKerφ.
ISOMORPHISM TYPES OF HOPF ALGEBRAS 5
We add some comments about the definition. The equality HKerφ =
RKerφ is equivalent to HKerφ = LKerφ and thanks to [3, 4.13] we have
that ι(A) is normal in C. It then follows from [19] that HCokerι =
ι(A)+C = Cι(A)+. In consequence our definition coincides with the
definition of extension in [2]. If A is finite-dimensional, it is equivalent
to H.-J. Schneider definition of strict extension in [19]. To see the
latter,use[19,L.2.1(2)]toobtainthatC isfaithfullyflatA. Conversely,
assuming A is normal andC is faithfully flat over A, apply [20, Remark
1.2] or [18, 3.4.3] to derive ι(A) = RKerπ.
Inthecaseofafinite-dimensionalC weakerconditionssuffice. Namely,
a sequence (S) with ι linear monomorphism, π linear epimorphism is
an extension if either ι(A) = Ccoπ or Kerπ = ι(A)+C. For details see
[2, 3.3.1].
1.2. Abelian Extensions. We assume in what follows the ground
field k to be an algebraically closed field of characteristic 0 and C to be
a finite-dimensional Hopf algebra. An extension (S) is called abelian
if A is commutative and B is cocommutative. It is well-known [10,
Theorem 1] and [18, 2.3.1] that in this case A = kG and B = kF for
some finite groups G and F. Below we consider only extensions of this
kind and we use the notation
(A) kG ֌ι H ։π kF.
Of the crucial importance to the sequel is a theorem [20, 2.4] and [14,
3.5]asserting H isa kF-crossed product over kG 1. Thetheorem entails
the existence of a mapping called section ( see, e.g. [2, 3.1.13])
(1.3) χ : kF → H
giving rise to the crossed product structure on H. Thus H = kGχ(F)
with the multiplication
(1.4) (αχ(x))(α′χ(y)) = α(χ(x)α′χ−1(x))χ(x)χ(y)
(1.5) = α(χ(x)α′χ−1(x))[χ(x)χ(y)χ−1(xy)]χ(xy)
forα,α′ ∈ kG, x,y ∈ F
The mapping x ⊗ α 7→ x.α := χ(x)αχ−1(x) defines a module-algebra
action of F on kG and the function σ : F × F → kG,σ(x,y) =
χ(x)χ(y)χ−1(xy) is a left, normalized 2-cocycle for that action [18,
7.2.3]. The definition of action is independent of a choice of section.
This can be made more precise. Let us write Sec(kF,H) for the set
of all sections of kF in H. Next define the group R = Reg (kF,kG)
1,ǫ
of all convolution invertible mapping preserving the unit and counit.
1 A short independent proof for abelian extension is given in the Appendix
6 LEONID KROPANDYEVGENIAKASHINA
Identifying ι(A) with A, Sec(B,C) becomes an R-set under multiplica-
tion by convolution, viz. f.χ = f ∗χ, f ∈ R,χ ∈ Sec(kF,kG), and by
[18, 7.3.5] Sec(kF,H) is a transitive R-set. The claim follows, for f ∗χ
evidently induces the same action as χ.
The theory of extensions depends fundamentally on the fact that for
each sequence (S) its companion sequence
(S∗) B∗ ֌π∗ C∗ ։ι∗ A∗
isalsoanextension, see[4, 4.1]or[2, 3.3.1]. Intheabeliancase(kF)∗ =
kF and also (kG)∗ = kG via the Hopf isomorphism
ev : kG → kG,ev(g)(α) = α(g) for allα ∈ kG
Thus every diagram (A) induces a diagram
(A∗) kF ֌ H∗ ։ kG
A crossed product structure on H∗ is effected by a section
(1.6) ω : kG → H∗
We choose to write H∗ = ω(G)kF with the multiplication
(1.7) (ω(a)β)(ω(b)β′) = ω(ab)τ(a,b)(β.b)β′,
where for a,b ∈ G,β,β′ ∈ kF
(1.8) β.b = ω−1(b)βω(b),and
(1.9) τ(a,b) = ω−1(ab)ω(a)ω(b).
We note that τ : G×G → kF is a right, normalized 2-cocycle for the
action β ⊗b 7→ β.b. Below we write x for χ(x) and a for ω(a).
The left action of F on kG induces a right action of F on G by set
permutations of G. Namely, we define the element a⊳x by the equality
(1.10) (x.f)(a) = f(a⊳x) for allf ∈ kG
Noting that an algebra action permutes minimal central idempotents,
the action ‘⊳’ is related to the action of F on kG in the basis {p |a ∈ G}
a
[18] by
(1.11) x.pa = pa⊳x−1
For, x.p = p iff (x.p )(b) = 1. Now, (x.p )(b) = p (b ⊳ x) = 1 iff
a b a a a
b⊳x = a, hence b = a⊳x−1.
Similarly, the right action of G on kF induces a left action of G on
F by permutations denoted by a⊲x satisfying the property
(1.12) px.a = pa−1⊲x
ISOMORPHISM TYPES OF HOPF ALGEBRAS 7
We fuse both actions into the definition of a product on F ×G via
(1.13) (xa)(yb) = x(a⊲y)(a⊳y)b
It was noted by M.Takeuchi [23] that composition 1.13 defines a group
structure on F ×G provided the actions ⊲,⊳ satisfy the conditions
(1.14) ab⊳x = (a⊳(b⊲x))(b⊳x)
(1.15) a⊲xy = (a⊲x)((a⊳x)⊲y)
We use the standard notation F ⊲⊳ G for the set F ×G endowed with
multiplication (1.13).
1.3. Co-crossed Coproducts. Fundamental to the theory of abelian
extensions is an explicit description of the coalgebra structure of the
dual coalgebra of a crossed product algebra H = A# B [18], A and
σ
B finite-dimensional. The coalgebra H∗ is an example of a co-crossed
co-product of the coalgebra A∗ with bialgebra B∗. Coalgebras of this
type were introduced in [11], generalizing an earlier construction in [17]
and studied in [1]. The formula below is [11, p.9] and [1, 2.15], though
we derive it rather than postulate. We need some preliminaries. We
identify H∗ with A∗⊗B∗ via thepairing ha∗ ⊗b∗,a⊗bi = ha∗,aihb∗,bi.
Aweak actionr : B⊗A → Ainduces a weak coactionρ : A∗ → B∗⊗A∗
via
(1.16) ρ(a∗) = (a∗)(1) ⊗(a∗)(2) ⇔ a∗(b.a) = (a∗)(1)(b)(a∗)(2)(a).
Next, the 2-cocycle σ : B ⊗B → A induces a 2-co-cocycle
θ : A∗ → B∗ ⊗B∗ as follows
(1.17) θ(a∗) = (a∗)h1i ⊗(a∗)h2i ⇔ a∗(σ(b,b′)) = (a∗)h1i(b)(a∗)h2i(b′).
Finally, we use notation (a∗) ⊗ (a∗) for the coproduct in A∗ and a
1 2
similar notation in B∗. We agree to write a∗θ♯ b∗ for a∗⊗b∗ viewed as
an element of the co-crossed co-product coalgebra H∗.
Lemma 1.3. Let ∆H∗ and ǫH∗ be the structure maps of H∗.
There hold the formulas:
∆H∗(a∗θ♯ b∗) = (a∗)1θ♯ (a∗)(21)(a∗)h31i(b∗)1 ⊗(a∗)(22)θ♯ (a∗)h32i(b∗)2,
ǫH∗(a∗θ♯ b∗) = a∗(1A)b∗(1B).
Proof:By definition of ∆H∗ and multiplication in H [18, 7.1.1]
∆H∗(a∗θ♯ b∗)(a#σb⊗a′#σb′) = ha∗ ⊗b∗,(a#σb)(a′#σb′)i
= a∗(a(b .a′)σ(b ,b′))b∗(b b′).
1 2 1 3 2
8 LEONID KROPANDYEVGENIAKASHINA
Expanding the right-hand side of the last equality using (1.16) and
(1.17) and reshuffling factors we get
(a∗) (a)(a∗)(1)(b )(a∗)(2)(a′)(a∗)h1i(b )(a∗)h2i(b′)(b∗) (b )(b∗) (b′)
1 2 1 2 3 2 3 1 1 3 2 2
= [(a∗) (a)((a∗)(1)(b )(a∗)h1i(b )(b∗) (b )][(a∗)(2)(a′)((a∗)h2i(b′)(b∗) (b′)]
1 2 1 3 2 1 3 2 3 1 2 2
= (a∗) (a)[(a∗)(1)(a∗)h1i(b∗) ](b)(a∗)(2)(a′)[(a∗)h2i(b∗) ](b′),
1 2 3 1 2 3 2
which is the the first formula. The formula for the counit is just
ha∗ ⊗b∗,1 ⊗1 i. (cid:3)
A B
We return to abelian extensions. We introduce some notation. For
a group Γ we let Γn be the direct product of n copies of Γ. Let F and
G be groups. We abbreviate (x ,...,x ) ∈ Fn and (a ,...,a ) ∈ Gm
1 n 1 m
to x and a, respectively. We note that every α : Fn → (kGm)• can be
uniquely represented in the standard basis {p |a ∈ Gm} as
a
(1.18) α(x) = α(x)(a)p .
a
a
X
There α(x,a) := α(x)(a) is a mapping Fn×Gm → k•. Conversely, ev-
ery φ : Fn×Gm → k• gives rise to two mappings φ′ : Fn → (kGm)• and
φ′′ : Gm → (kFn)• via(1.18)forφ′ andtheformulaφ′′(a) = φ(x,a)p
x
for φ′′. Let Fun(X,Y) be the set of mappings from a set X to a
set Y. We arrive at the conclusion that the groups Fun(FPn,(kGm)•),
Fun(Gm,(kFn)•) and Fun(Fn ×Gm,k•) can be identified.
Proposition 1.4. Suppose H = kG# kF. The structure mappings of
σ
coalgebra H∗ are given by the formulas
(1.19)
∆H∗(a⊗β) = σ(x,y,a)a⊗pxβ1 ⊗(a⊳x)⊗pyβ2, a ∈ G,β ∈ kF,
x,y∈F
X
ǫ(a⊗β) = β(1 )
F
Proof:By the preceeding Lemma we need to know the coaction ρ :
kG → kF ⊗ kG dual to the action of F on kG or, equivalently, to
the action of F on G (see (1.11)). As well, we need a formula for the
co-cocycle θ : kG → kF ⊗kF. These formulas are as follows:
(1.20) ρ(a) = p ⊗a⊳x
x
x∈F
X
(1.21) θ(a) = σ(x,y,a)p ⊗p
x y
x,y∈F
X
ISOMORPHISM TYPES OF HOPF ALGEBRAS 9
To show the first formula, begin with the expression ρ(a) = p ⊗
y∈F y
r with r ∈ kG. Apply (1.16) and calculate
y y
P
hρ(a),x⊗pbi = ev(a)(x.pb) = (x.pb)(a) = pb⊳x−1(a) = δb,a⊳x
On the other hand hρ(a),x⊗p i = p (x)p (r ) = p (r ). Say r =
b y b y b x x
kc c,kc ∈ k,c ∈ G. From p (r ) = kb = δ we get the equality
x x b x x b,a⊳x
P
r = a⊳x, which is the result.
x
P
To see the second formula, we use (1.17) to compute
hθ(a),x⊗yi = ev(a)(σ(x,y)) = σ(x,y,a)
which gives the result.
Substituting those expressions of ρ and θ in Lemma 1.3 gives the
first statement. To compute the counit use the fact that 1 = ǫ . (cid:3)
kG G
Remark 1.5. Starting off with the crossed product H∗ = kG# kF,
τ
an argument similar to the one in Proposition 1.4 gives the coalgebra
structure mappings for H [16, p.7], namely
(1.22) ∆ (α⊗x) = τ(x,a,b)α p ⊗b⊲x⊗α p ⊗x
H 1 a 2 b
a,b∈G
X
ǫ (α⊗a) = α(1 )
H G
1.4. Compatible Cocycles and Equivalence Relation. The dis-
cussion in §1.2 enables us to associate to every Hopf algebra H in a
diagram (A) a datum {σ,τ,⊳,⊲} and we write H = H(σ,τ,⊳,⊲). Two
data σ,⊳ and τ,⊲ appearing in the crossed products H and H∗ are
compatible if the coalgebra structure induced in H by multiplication
in H∗ (or equivalently in H∗ by multiplication in H) turns H (or H∗)
into a bialgebra, hence a Hopf algebra [2, 3.1.12]. The compatibility
conditions are derived using (1.19) or (1.22). They can be found in [16,
4.7] as follows.
Theorem 1.6. (1) F ⊲⊳ G is a group,
(2) σ(x,y,1) = 1 = τ(a,b,1),
σ(x,y,ab)σ−1(b⊲x,(b⊳x)⊲y,a)σ−1(x,y,b)
(3)
= τ(a,b,x)τ(a⊳(b⊲x),b⊳x,y)τ−1(a,b,xy)
The second key concept of the theory is equivalence of extensions.
Following [6, 3.10] we call two extensions E = (A ֌ι C ։π B) and
E′ = (A ֌ι′ C ։π′ B) equivalent, and write E ∼ E′, if there is a Hopf
10 LEONID KROPANDYEVGENIAKASHINA
algebra map ψ : H → H′ making the following diagram commute
ι π
A −−−→ H −−−→ B
(1.23)
ψ
A(cid:13)(cid:13) −−ι−′→ H′ −−π−′→ B(cid:13)(cid:13)
(cid:13) y (cid:13)
Welet Ext(kF,kG) stand fortheset ofequivalence classes ofextensions
oftype(A).WedesignateOpext(kF,kG,⊳,⊲))fortheset ofequivalence
classes of extensions with the fixed actions ⊳ and ⊲. We proceed to
describe equivalent extensions precisely. As a preliminary we introduce
groups Map(Fn × Gm,k•) of all normalized mappings, i.e. f : Fn ×
Gm → k• such that f(x ,...,x ,a ,...,a ) = 1 if some of x ,...,x
1 n 1 m 1 n
or a ...,a are equal to 1 or 1 , respectively, under the pointwise
1 m F G
multiplication. We define actions of F and G on Map(Fn×Gm,k•) as
follows. For any f : Fn ×Gm → k•,
(1.24)
y.f(x ,...,x ,a ,...,a ) = f(x ,...,x ,a ⊳(a ···a ⊲y),...,a ⊳y)
1 n 1 m 1 n 1 2 m m
(1.25)
f(x ,...,x ,a ,...,a ).b = f(b⊲x ,...,(b⊳x ···x )⊲x ,a ,...,a ),
1 n 1 m 1 1 n−1 n 1 n
where all x ,y ∈ F and a ,b ∈ G.
i j
The formulas below reflect the fact, standard in the theory of exten-
sions, that an equivalence is just a diagonal change of basis. They are
particular cases of [6, 3.11] and also [16, 5.2]. Combining [16, 3.4]or
[18, 7.3.4] with (1.19) or (1.22) one can show
Lemma 1.7. Two extensions defined by data {σ,τ,⊳,⊲} and
{σ′,τ′,⊳′,′⊲} are equivalent if and only if ⊳ = ⊳′, ⊲ = ′⊲ and there exists
ζ ∈ Map(F ×G,k•) satisfying
(1.26) σ′(x,y,a) = σ(x,y,a)(x.ζ−1(y,a))ζ(xy,a)ζ−1(x,a),
(1.27) τ′(x,a,b) = τ(x,a,b)(ζ(x,a).b)ζ(x,ab)−1ζ(x,b)
The mapping ψ of (1.23) is given by ψ(αx) = αζ(x)x for all α ∈
kG,x ∈ F.
We shall need the dual version of the above Lemma. We agree to
write H∗(σ,τ,⊳,⊲) for H(σ,τ,⊳,⊲)∗.
Lemma 1.8. H∗(σ,τ,⊳,⊲) and H∗(σ′,τ′,⊳′,′⊲) are equivalent exten-
sions if and only if ⊳ = ⊳′, ⊲ = ′⊲ and there exists η ∈ Map(F ×G,k•)