Automorphic equivalence of the representations of Lie algebras
In this paper we research the algebraic geometry of the representations of Lie algebras over fixed field k. We assume that this field is infinite and char (k) = 0. We consider the representations of Lie algebras as 2-sorted universal algebras. The representations of groups were considered by similar...
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| Zitieren: | Automorphic equivalence of the representations of Lie algebras / I. Shestakov, A. Tsurkov // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 1. — С. 96–126. — Бібліогр.: 5 назв. — англ. |
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| citation_txt | Automorphic equivalence of the representations of Lie algebras / I. Shestakov, A. Tsurkov // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 1. — С. 96–126. — Бібліогр.: 5 назв. — англ. |
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| description | In this paper we research the algebraic geometry of the representations of Lie algebras over fixed field k. We assume that this field is infinite and char (k) = 0. We consider the representations of Lie algebras as 2-sorted universal algebras. The representations of groups were considered by similar approach: as 2-sorted universal algebras - in [3] and [2]. The basic notions of the algebraic geometry of representations of Lie algebras we define similar to the basic notions of the algebraic geometry of representations of groups (see [2]). We prove that if a field k has not nontrivial automorphisms then automorphic equivalence of representations of Lie algebras coincide with geometric equivalence. This result is similar to the result of [4], which was achieved for representations of groups. But we achieve our result by another method: by consideration of 1-sorted objects. We suppose that our method can be more perspective in the further researches.
|
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Algebra and Discrete Mathematics
Volume 15 (2013). Number 1. pp. 96 – 126
c© Journal “Algebra and Discrete Mathematics”
Automorphic equivalence of the representations
of Lie algebras
I. Shestakov, A. Tsurkov
Abstract. In this paper we research the algebraic geom-
etry of the representations of Lie algebras over fixed field k. We
assume that this field is infinite and char (k) = 0. We consider the
representations of Lie algebras as 2-sorted universal algebras. The
representations of groups were considered by similar approach: as
2-sorted universal algebras - in [3] and [2]. The basic notions of the
algebraic geometry of representations of Lie algebras we define simi-
lar to the basic notions of the algebraic geometry of representations
of groups (see [2]). We prove that if a field k has not nontrivial
automorphisms then automorphic equivalence of representations
of Lie algebras coincide with geometric equivalence. This result is
similar to the result of [4], which was achieved for representations
of groups. But we achieve our result by another method: by con-
sideration of 1-sorted objects. We suppose that our method can be
more perspective in the further researches.
1. Introduction: representations of Lie algebras
as 2-sorted universal algebras
In this paper we research the algebraic geometry of the representations
of Lie algebras.
We consider the Lie algebras over the field k. And we say that we have
the representation of Lie algebra (L, V ) if the elements of the Lie algebra
L act on the vector space V over the field k as linear transformations
and the mapping f : L → Endk (V ) which we define by f (l) (v) = l ◦ v,
where l ∈ L, v ∈ V , ◦ is acting of the elements of the algebra L over
2010 MSC: 17B10.
Key words and phrases: universal algebraic geometry, representations of Lie
algebras, automorphic equivalence.
I . Shestakov, A. Tsurkov 97
elements of V , is a homomorphism from the Lie algebra L to the Lie
algebra End
(−)
k (V ) = endk (V ). Some time we will omit the symbol ◦. In
this paper we assume that k is infinite and char (k) = 0.
We consider a representation of Lie algebra as 2-sorted universal
algebra. Particularly the homomorphisms of representations we define by
this definition:
Definition 1.1. We say that we have a homomorphism (ϕ,ψ) from
the representation (L1, V1) to the representation (L2, V2) if we have a
homomorphism of Lie algebras ϕ : L1 → L2 and a linear map ψ : V1 → V2
such that
ϕ (l) ◦ ψ (v) = ψ (l ◦ v) (1.1)
holds for every l ∈ L1 and every v ∈ V1.
We denote (ϕ,ψ) : (L1, V1) → (L2, V2).
It means that the field k is fixed in our considerations. But algebras
Lie and theirs modules me can change and we can compare the algebraic
geometry of representations (L1, V1) and (L2, V2) such that L1 6= L2 and
V1 6= V2. Therefore the multiplication by scalars of the elements of the
algebra Lie L and the elements of its module V we can consider as unary
operations: for every scalar λ ∈ k we have two unary operations. But
the acting of the elements of the algebra Lie L over the elements of its
module V we must consider as one binary 2-sorted operation.
If (ϕ,ψ) : (L, V ) → (P,Q) is a homomorphism of the representations,
than kerϕ is an ideal of the Lie algebra L, kerψ is a L-submodule of the
L-module V , (kerϕ, kerψ) is a representation and a congruence in (L, V ).
If H = (L, V ) is a representation of Lie algebra and T1 ⊆ L, T2 ⊆ V
we will denote (T1, T2) ⊆ H. If also P1 ⊆ L, P2 ⊆ V we will denote
(T1, T2) ∩ (P1, P2) = (T1 ∩ P1, T2 ∩ P2).
2. Basic notions of the algebraic geometry
of representations of Lie algebras
We denote by Ξ the variety of the all representations of Lie algebras
over the fixed field k.
Definition 2.1. We say that the representation (L, V ) is a free represen-
tation with the pair of sets of the free generators (X,Y ) if X ⊂ L, Y ⊂ V
and for every representation (P,U) and every pair of mappings ϕ̃ : X → P ,
ψ̃ : Y → U there exists a homomorphism (ϕ,ψ) : (L, V ) → (P,U) such
that ϕ̃|X = ϕ|X , ψ̃|X = ψ|X .
98 Automorphic equivalence of the representations
From this one we will denote the mappings ϕ̃ and ϕ, ψ̃ and ψ by same
letters.
We will denote this representation by W = W (X,Y ). It is well known
that W (X,Y ) = (L (X) , A (X)Y ), where L (X) = L is the free Lie
algebra with the set X of free generators, A (X) is the free associative
algebra with unit which has the set X of free generators, A (X)Y =⊕
y∈Y
A (X) y = V is the free A (X) module with the basis Y . In this
notation the symbol ◦ of the action is omitted. In particular, if X = ∅
then L (X) = {0}, A (X) = k, if Y = ∅ then A (X)Y = {0}, if X = {x}
then L (X) = kx, A (X) = k [x].
X0, Y 0 will be infinite countable sets of symbols. We consider the
category Ξ0. ObΞ0 =
{
W (X,Y ) | |X| < ∞, |Y | < ∞, X ⊂ X0, Y ⊂ Y 0
}
.
Morphisms of this category are homomorphisms of its objects. The cat-
egory Ξ0 is a small category: ObΞ0 and MorΞ0 are sets. So we can tell
about elements and subsets of ObΞ0 and MorΞ0.
We will take our equations from the representations W = W (X,Y ) =
(L (X) , A (X)Y ) ∈ ObΞ0. We have two sorts of equations: the equations
in the Lie algebra - t1 ∈ L (X) and the action type equations - t2 ∈
A (X)Y . We can resolve our equations in arbitrary H = (L, V ) ∈ Ξ.
The homomorphism (ϕ,ψ) : W (X,Y ) → H will be the solution of the
equation t1 ∈ L (X) if ϕ (t1) = 0 and will be the solution of the equation
t2 ∈ A (X)Y if ψ (t2) = 0.
We can consider the system of equations T = (T1, T2), where T1 ⊆
L (X), T2 ⊆ A (X)Y . We can consider this system as a set T = T1 ∪ T2
but it is not natural because the subsets T1 and T2 have different origins:
T1 ⊆ L (X), T2 ⊆ A (X)Y . So it is natural to consider the system of
equations T = (T1, T2) as a pair of sets. However for the sake of brevity
we will some time write "the set (T1, T2)". The set of solutions of the
system (T1, T2) in the representation H = (L, V ) is
(T1, T2)′
H = {(ϕ,ψ) ∈ Hom (W (X,Y ) , H) | T1 ⊆ kerϕ, T2 ⊆ kerψ} .
Vice versa, for every set A ⊂ Hom (W (X,Y ) , H) we can consider the set
A′
H =
⋂
(ϕ,ψ)∈A
kerϕ,
⋂
(ϕ,ψ)∈A
kerψ
.
This set will be the maximal system of equations, such that A is a subset
of the set of its solutions. Also we can consider the algebraic closer of the
system (T1, T2):
I . Shestakov, A. Tsurkov 99
(T1, T2)′′
H =
⋂
(ϕ,ψ)∈(T1,T2)′
H
kerϕ,
⋂
(ϕ,ψ)∈(T1,T2)′
H
kerψ
=
=
⋂
(ϕ,ψ)∈Hom(W,H),
T1⊆kerϕ, T2⊆kerψ
kerϕ,
⋂
(ϕ,ψ)∈Hom(W,H),
T1⊆kerϕ, T2⊆kerψ
kerψ
.
It will be the maximal system of equations which have the same solutions
as (T1, T2).
It is clear that (T1, T2) ⊆ (T1, T2)′′
H holds for every W (X,Y ) ∈ ObΞ0,
every (T1, T2) ⊆ W (X,Y ) and every H ∈ Ξ.
Definition 2.2. The set (T1, T2) ⊆ W (X,Y ) is H-closed if (T1, T2)′′
H =
(T1, T2).
It is clear that the closed sets are congruences. The family of the
all H-closed sets in the free representation W = W (X,Y ) ∈ ObΞ0 we
denote by ClH (W ).
Definition 2.3. H1, H2 ∈ Ξ. H1, H2 are called geometrically equivalent
if (T1, T2)′′
H1
= (T1, T2)′′
H2
holds for every W (X,Y ) ∈ ObΞ0 and every
(T1, T2) ⊆ W (X,Y ).
We consider W1 = W (X1, Y1) ,W2 = W (X2, Y2) ∈ ObΞ0 and (T1, T2)
some congruence in W2. We denote by β = βW1,W2 (T1, T2) the following
relation in Hom (W1,W2): ((ϕ1, ψ1) , (ϕ2, ψ2)) ∈ β if and only if ϕ1 (l) ≡
ϕ2 (l) (modT1) holds for every l ∈ L (X1) and ψ1 (v) ≡ ψ2 (v) (modT2)
holds for every v ∈ A (X1)Y1. This relation is a 2-sorted analog of the
relation β from [1, Subsection 3.3]. Now we define as in [1, Subsection 3.4]
Definition 2.4. H1, H2 ∈ Ξ. H1, H2 are called automorphically equivalent
if these 3 conditions hold:
1) There exists an automorphism Φ : Ξ0 → Ξ0.
2) There exists a function α = α (Φ) such that α (Φ)W : ClH1 (W ) →
ClH2 (Φ (W )) is a bijection for every W ∈ ObΞ0.
3) Φ (βW1,W2 (T1, T2)) = βΦ(W1),Φ(W2)
(
α (Φ)W2
(T1, T2)
)
holds for ev-
ery W1,W2 ∈ ObΞ0, and every (T1, T2) ∈ ClH1 (W2).
Here Φ ((ϕ1, ψ1) , (ϕ2, ψ2)) = (Φ (ϕ1, ψ1) ,Φ (ϕ2, ψ2)).
It can be proved as in [1, Proposition 8] that if H1 and H2 are auto-
morphically equivalent then function α is uniquely determined by auto-
morphism Φ.
100 Automorphic equivalence of the representations
3. Some facts about the closed congruences in the free
representations of Lie algebras
In this Section we assume that X1 ⊆ X2 ⊂ X0, Y1 ⊆ Y2 ⊂ Y 0,
(L, V ) = H ∈ Ξ. We denote (L (Xi) , A (Xi)Yi) = W (Xi, Yi) = Wi,
where i = 1, 2.
If (T1, T2) ⊆ W1, then, because W1 ⊆ W2, we can consider the sets
(T1, T2)′
Wi,H
= {(ϕ,ψ) : Wi → H | T1 ⊆ kerϕ, T2 ⊆ kerψ}
and we will denote
(
(T1, T2)′
Wi,H
)′
H
= (T1, T2)′′
Wi,H
, where i = 1, 2. We
say that (T1, T2) is H-closed in Wi if (T1, T2)′′
Wi,H
= (T1, T2). In all other
sections of this paper it is clear what kind of algebraic closer of the system
of equations we consider. But in this Section we must fine distinguish
between the different features.
Proposition 3.1. We assume that (T1, T2) ⊆ W2,
(µ, ν) ∈ (T1 ∩ L (X1) , T2 ∩A (X1)Y1)′
W1,H
. We denote
[(µ, ν)] =
{
(ϕ,ψ) ∈ (T1, T2)′
H | ϕ|X1
= µ|X1
, ψ|Y1
= ν|Y1
}
.
Then
⋂
(ϕ,ψ)∈[(µ,ν)]
kerϕ
∩ L (X1) ,
⋂
(ϕ,ψ)∈[(µ,ν)]
kerψ
∩A (X1)Y1
=
= (kerµ, ker ν)
holds.
Proof. If t1 ∈
(
⋂
(ϕ,ψ)∈[(µ,ν)]
kerϕ
)
∩ L (X1), then µ (t1) = ϕ (t1) = 0 for
every ϕ such that (ϕ,ψ) ∈ [(µ, ν)]. If t2 ∈
(
⋂
(ϕ,ψ)∈[(µ,ν)]
kerψ
)
∩A (X1)Y1,
then ν (t2) = ψ (t2) = 0 for every ψ such that (ϕ,ψ) ∈ [(µ, ν)].
If t1 ∈ kerµ, then t1 ∈ L (X1), so ϕ (t1) = µ (t1) = 0 holds for
every ϕ such that (ϕ,ψ) ∈ [(µ, ν)]. If t2 ∈ ker ν, then t2 ∈ A (X1)Y1, so
ψ (t2) = ν (t2) = 0 holds for every ψ such that (ϕ,ψ) ∈ [(µ, ν)].
Proposition 3.2. If (T1, T2) ⊆ W2 is H-closed, then
(T1, T2) ∩W1 = (T1 ∩ L (X1) , T2 ∩A (X1)Y1)
I . Shestakov, A. Tsurkov 101
is H-closed in W1.
Proof.
(T1, T2)′′
H =
⋂
(ϕ,ψ)∈(T1,T2)′
H
kerϕ,
⋂
(ϕ,ψ)∈(T1,T2)′
H
kerψ
= (T1, T2) .
(T1 ∩ L (X1) , T2 ∩A (X1)Y1)′′
W1,H
=
⋂
(µ,ν)∈(T1∩L(X1),T2∩A(X1)Y1)′
W1,H
kerµ,
⋂
(µ,ν)∈(T1∩L(X1),T2∩A(X1)Y1)′
W1,H
ker ν
.
We will consider (ϕ,ψ) ∈ (T1, T2)′
H . There exists only one (µ, ν) ∈
Hom (W1, H) such that ϕ|X1
= µ|X1
, ψ|Y1
= ν|Y1
. If t1 ∈ T1 ∩L (X1), then
µ (t1) = ϕ (t1) = 0, if t2 ∈ T2 ∩A (X1)Y1, then ν (t2) = ψ (t2) = 0. Hence
(µ, ν) ∈ (T1 ∩ L (X1) , T2 ∩A (X1)Y1)′
W1,H
. So by Proposition 3.1
⋂
(ϕ,ψ)∈[(µ,ν)]
kerϕ
∩ L (X1) ,
⋂
(ϕ,ψ)∈[(µ,ν)]
kerψ
∩A (X1)Y1
=
= (kerµ, ker ν) .
The set (T1, T2)′
H can by presented as union of the disjoint sets [(µ, ν)],
where (µ, ν) ∈ Hom (W1, H) such that exists (ϕ,ψ) ∈ (T1, T2)′
H , for which
(ϕ,ψ) ∈ [(µ, ν)] holds.
(T1 ∩ L (X1) , T2 ∩A (X1)Y1)′′
W1,H
⊆
⊆
⋂
(µ,ν)∈(T1∩L(X1),T2∩A(X1)Y1)′
H ,
∃(ϕ,ψ)∈(T1,T2)′
H |(ϕ,ψ)∈[(µ,ν)]
kerµ,
⋂
(µ,ν)∈(T1∩L(X1),T2∩A(X1)Y1)′
H ,
∃(ϕ,ψ)∈(T1,T2)′
H |(ϕ,ψ)∈[(µ,ν)]
ker ν
.
⋂
(µ,ν)∈(T1∩L(X1),T2∩A(X1)Y1)′
H ,
∃(ϕ,ψ)∈(T1,T2)′
H |(ϕ,ψ)∈[(µ,ν)]
kerµ =
=
⋂
(µ,ν)∈(T1∩L(X1),T2∩A(X1)Y1)′
H ,
∃(ϕ,ψ)∈(T1,T2)′
H |(ϕ,ψ)∈[(µ,ν)]
⋂
(ϕ,ψ)∈[(µ,ν)]
kerϕ
∩ L (X1)
=
102 Automorphic equivalence of the representations
=
⋂
(ϕ,ψ)∈(T1,T2)′
H
kerϕ
∩ L (X1) = T1 ∩ L (X1) .
⋂
(µ,ν)∈(T1∩L(X1),T2∩A(X1)Y1)′
H ,
∃(ϕ,ψ)∈(T1,T2)′
H |(ϕ,ψ)∈[(µ,ν)]
ker ν =
=
⋂
(µ,ν)∈(T1∩L(X1),T2∩A(X1)Y1)′
H ,
∃(ϕ,ψ)∈(T1,T2)′
H |(ϕ,ψ)∈[(µ,ν)]
⋂
(ϕ,ψ)∈[(µ,ν)]
kerψ
∩A (X1)Y1
=
=
⋂
(ϕ,ψ)∈(T1,T2)′
H
kerψ
∩A (X1)Y1 = T2 ∩A (X1)Y1.
Proposition 3.3. If (T1, T2) ⊆ W1 is H-closed in W1, then
(T1, T2) = (T1, T2)′′
W2,H
∩W1.
Proof. In (T1, T2)′
W2,H
= {(ϕ,ψ) ∈ Hom (W2, H) | kerϕ ⊇ T1, kerψ ⊇ T2}
we can define equivalence: (ϕ1, ψ1) ∼ (ϕ2, ψ2) if and only if ϕ1|X1
= ϕ2|X1
,
ψ1|Y1
= ψ2|Y1
. As in the proof of Proposition 3.2, for every class of this
equivalence there exist only one (µ, ν) ∈ (T1, T2)′
W1,H
such that this class
coincide with [(µ, ν)]. Vice versa, for every (µ, ν) ∈ (T1, T2)′
W1,H
there
exist only one class of elements of the set (T1, T2)′
W2,H
, which coincide
with [(µ, ν)].
(T1, T2)′′
W2,H
∩W (X1, Y1) =
⋂
(ϕ,ψ)∈(T1,T2)′
W2,H
kerϕ
∩ L (X1) ,
⋂
(ϕ,ψ)∈(T1,T2)′
W2,H
kerψ
∩A (X1)Y1
.
By Proposition 3.1 we have
⋂
(ϕ,ψ)∈(T1,T2)′
W2,H
kerϕ
∩ L (X1) =
=
⋂
(µ,ν)∈(T1,T2)′
W1,H
⋂
(ϕ,ψ)∈[(µ,ν)]
kerϕ
∩ L (X1)
=
I . Shestakov, A. Tsurkov 103
=
⋂
(µ,ν)∈(T1,T2)′
W1,H
kerµ = T1.
⋂
(ϕ,ψ)∈(T1,T2)′
W2,H
kerψ
∩A (X1)Y1 =
=
⋂
(µ,ν)∈(T1,T2)′
W1,H
⋂
(ϕ,ψ)∈[(µ,ν)]
kerψ
∩A (X1)Y1
=
=
⋂
(µ,ν)∈(T1,T2)′
W1,H
ker ν = T2.
Theorem 3.1. If (L1, V1) = H1, (L2, V2) = H2 ∈ Ξ and ClH1 (W2) =
ClH2 (W2), then ClH1 (W1) = ClH2 (W1).
Proof. We consider (T1, T2) ∈ ClH1 (W1). By Proposition 3.3 (T1, T2) =
(T1, T2)′′
W2,H1
∩ W1. (T1, T2)′′
W2,H1
∈ ClH1 (W2) = ClH2 (W2). Therefore,
by Proposition 3.2, (T1, T2)′′
W2,H1
∩W1 = (T1, T2) ∈ ClH2 (W1).
4. Representations of Lie algebras and Lie algebras with
projection-derivation
It is well known that if we have a representation of the Lie algebra
(L, V ) then in the k-linear space M = L⊕ V we can define the structure
of Lie algebra if we define the new Lie brackets [, ]M by this formula
[l1 + v1, l2 + v2]M = [l1, l2] + l1 ◦ v2 − l2 ◦ v1, (4.1)
where l1, l2 ∈ L, v1, v2 ∈ V .
We will denote by p the projection of M on the linear subspace V .
p (l + v) = v for every l ∈ L, v ∈ V . We have
p [l1 + v1, l2 + v2]M = p ([l1, l2] + l1 ◦ v2 − l2 ◦ v1) = l1 ◦ v2 − l2 ◦ v1,
[p (l1 + v1) , l2 + v2]M + [l1 + v1, p (l2 + v2)]M =
= [v1, l2 + v2]M + [l1 + v1, v2]M = −l2 ◦ v1 + l1 ◦ v2
for every l1, l2 ∈ L, v1, v2 ∈ V . Therefore in the new Lie algebra p will be a
derivation. The projection p we consider as an additional unary operation
104 Automorphic equivalence of the representations
defined on the Lie algebra M . We call these algebras: Lie algebras with
projection-derivation and denote (M,p).
Vice versa, if we assume that we have a Lie algebra with projection-
derivation (M,p) then we have the decomposition of the k-linear space
M = ker p ⊕ imp. If we denote ker p = L, imp = V , then we can prove
this proposition:
Proposition 4.1. If we consider L with the Lie brackets inducted from
M then L is a Lie algebra. If we define
l ◦ v = [l, v] (4.2)
for every l ∈ L and every v ∈ V then (L, V ) is a representations of the
Lie algebra L over the linear space V , for every v1, v2 ∈ V the [v1, v2] = 0
holds.
Proof. If l1, l2 ∈ ker p then p [l1, l2] = [p (l1) , l2] + [l1, p (l2)] = 0, so
L = ker p is a Lie algebra.
If l ∈ ker p, v ∈ imp then p [l, v] = [p (l) , v] + [l, p (v)] = [l, v], so
[l, v] = l ◦ v ∈ imp.
If l1, l2 ∈ ker p, v ∈ imp then
[l1, l2] ◦ v = [[l1, l2] , v] = − [[l2, v] , l1] − [[v, l1] , l2] =
= [l1, [l2, v]] − [l2, [l1, v]] = l1 ◦ (l2 ◦ v) − l2 ◦ (l1 ◦ v) .
Also we have for v1, v2 ∈ imp that p [v1, v2] = [p (v1) , v2] + [v1, p (v2)] =
[v1, v2] + [v1, v2]. char (k) 6= 2, so [v1, v2] ∈ imp, p [v1, v2] = [v1, v2] and
[v1, v2] = 0.
Proposition 4.2. We assume that (ϕ,ψ) : (L1, V1) → (L2, V2) is a
homomorphism of representations. Then f = ϕ ⊕ ψ : (L1 ⊕ V1, pV1) →
(L2 ⊕ V2, pV2), which define by formula f (l + v) = ϕ (l) + ψ (v) for every
l ∈ L1, v ∈ V1 is a homomorphism of the Lie algebras with projection-
derivation and ker f = kerϕ⊕kerψ. Vice versa, if f : (M1, p1) → (M2, p2)
is a homomorphism of the Lie algebras with projection-derivation then
(r2fκ1, p2fι1) : (ker p1, imp1) → (ker p2, imp2), where r2 = idM2 − p2 and
κ1 : ker p1 →֒ M1, ι1 : imp1 →֒ M1 are embeddings, is a homomorphism
of the representations of the Lie algebras and ker r2fκ1 = ker f ∩ ker p1,
ker p2fι1 = ker f ∩ imp1.
Proof. For the sake of brevity hear and in other proves we denote the
various Lie brackets, projections and embeddings by similar symbols. It
should not cause confusion because we not cause confusion when, for
I . Shestakov, A. Tsurkov 105
example, in the various groups denote multiplication, taking the inverse
element and unit by similar symbols.
If (ϕ,ψ) : (L1, V1) → (L2, V2) is a homomorphism of representations
then f = ϕ⊕ ψ is a linear mapping. If l1, l2 ∈ L1, v1, v2 ∈ V1 then
f [l1 + v1, l2 + v2] = f ([l1, l2] + l1 ◦ v2 − l2 ◦ v1) =
= ϕ [l1, l2] + ψ (l1 ◦ v2) − ψ (l2 ◦ v1) =
= [ϕ (l1) , ϕ (l2)] + ϕ (l1) ◦ ψ (v2) − ϕ (l2) ◦ ψ (v1) .
[f (l1 + v1) , f (l2 + v2)] = [ϕ (l1) + ψ (v1) , ϕ (l2) + ψ (v2)] =
= [ϕ (l1) , ϕ (l2)] + ϕ (l1) ◦ ψ (v2) − ϕ (l2) ◦ ψ (v1) .
If l ∈ L1, v ∈ V1 then
fp (l + v) = f (v) = ψ (v) ,
pf (l + v) = p (ϕ (l) + ψ (v)) = ψ (v) .
So f is a homomorphism of the Lie algebras with projection-derivation.
It is clear that ker f ⊇ kerϕ⊕ kerψ. If l ∈ L1, v ∈ V1 and f (l + v) =
ϕ (l)+ψ (v) = 0, then, because ϕ (l) ∈ L2, ψ (v) ∈ V2, ϕ (l) = 0, ψ (v) = 0.
So ker f = kerϕ⊕ kerψ.
Now we assume that f : (M1, p1) → (M2, p2) is a homomorphism of
the Lie algebras with projection-derivation. pr = p (id− p) = p− p2 = 0,
so rfκ : ker p → ker p. Also is clear that pfι : imp → imp.
It is clear that rfκ and pfι are linear mappings. For every l ∈ ker p
we have pf (l) = fp (l) = 0. So we have for every l1, l2 ∈ ker p
rfκ [l1, l2] = r [f (l1) , f (l2)] = (id− p) [f (l1) , f (l2)] =
= [f (l1) , f (l2)] − [pf (l1) , f (l2)] − [f (l1) , pf (l2)] = [f (l1) , f (l2)] .
[rfκ (l1) , rfκ (l2)] = [(id− p) f (l1) , (id− p) f (l2)] =
= [f (l1) , f (l2)] − [pf (l1) , f (l2)] − [f (l1) , pf (l2)] + [pf (l1) , pf (l2)] =
= [f (l1) , f (l2)] .
So rfκ is a homomorphism of the Lie algebras. If l ∈ ker p, v ∈ imp, then
rfκ (l) ◦ pfι (v) = [rf (l) , pf (v)] =
= [f (l) , pf (v)] − [pf (l) , pf (v)] = [f (l) , pf (v)] .
106 Automorphic equivalence of the representations
pfι (l ◦ v) = pf [l, v] = p [f (l) , f (v)] =
= [pf (l) , f (v)] + [f (l) , pf (v)] = [f (l) , pf (v)] .
So (rfκ, pfι) is a homomorphism of the representations of the Lie algebras.
It is clear that ker f ∩ ker p ⊆ ker rfκ. If l ∈ ker p and rfκ (l) = 0
then l = r (l) and f (l) = fr (l) = rfκ (l) = 0. So ker rfκ = ker f ∩ ker p.
Analogously ker pfι = ker f ∩ imp.
We denote by Θ the variety of all Lie algebras with projection-
derivation. The elements of this variety are Lie algebras with all operations
and axioms of Lie algebras and with one additional unary operation: pro-
jection p, which fulfills two axioms of linear map and two additional
axioms:
1) p (p (m)) = p (m) holds for every m ∈ M ,
2) p [m1,m2] = [p (m1) ,m2]+[m1, p (m2)] holds for every m1,m2 ∈ M ,
where M ∈ Θ.
We can consider the varieties Ξ and Θ as categories. The objects of
these categories are universal algebras from these varieties and morphisms
are homomorphisms. We have a functor F : Ξ → Θ, such that
F (L, V ) = (L⊕ V, pV )
for (L, V ) ∈ ObΞ,
F ((ϕ,ψ) : (L1, V1) → (L2, V2)) = ϕ⊕ψ : (L1 ⊕ V1, pV1) → (L2 ⊕ V2, pV2)
for (ϕ,ψ) ∈ MorΞ.
Also we have a functor F−1 : Θ → Ξ, such that
F−1 (M,p) = (ker p, imp)
for (M,p) ∈ ObΘ,
F−1 (f : (M1, p1) → (M2, p2)) = (rfκ, pfι) : (ker p1, imp1) → (ker p2, imp2)
for f ∈ MorΘ.
By Propositions 4.1 and 4.2 FF−1 = idΘ, F−1F = idΞ so these
functors are isomorphisms of categories.
Theorem 4.1. If (F, p) = F (m1, . . . ,mn) is a free Lie algebras with
projection-derivation with free generators {m1, . . . ,mn} then F−1 (F, p) =
(L, V ) is a free representation with the pair of sets of the free generators
(X,Y ), where X = {r (m1) , . . . , r (mn)} and Y = {p (m1) , . . . , p (mn)}.
I . Shestakov, A. Tsurkov 107
Proof. It is clear that X ⊂ ker p, Y ⊂ imp. We will consider an arbitrary
(Q,U) ∈ Ξ. We assume that we have 2 mappings:
ϕ : {r (m1) , . . . , r (mn)} ∋ r (mi) → qi ∈ Q
and
ψ : {p (m1) , . . . , p (mn)} ∋ p (mi) → ui ∈ U.
So we have a mapping
f : {m1, . . . ,mn} ∋ mi → qi + ui = ϕr (mi) + ψp (mi) ∈ Q⊕ U.
Hence, by our assumption about (F, p), this mapping can be extended to
the homomorphism
f : (F, p) → F (Q,U) = (Q⊕ U, pU ) .
So there is a homomorphism
F−1 (f) = (rfκ, pfι) : F−1 (F, p) = (ker p, imp) → (Q,U) .
rfκ (r (mi)) = (r)2 f (mi) = r (ϕr (mi) + ψp (mi)) = ϕr (mi) ,
pfι (p (mi)) = p (ϕr (mi) + ψp (mi)) = ψp (mi)
holds for 1 ≤ i ≤ n, because ϕr (mi) ∈ Q, ψp (mi) ∈ U .
We will denote
Ξ′ =
{
W (X,Y ) ∈ ObΞ0 | |X| = |Y |
}
.
Theorem 4.2. If W = (L, V ) =
(
L (x1, . . . , xn) ,
n⊕
i=1
A (x1, . . . , xn) yi
)
∈
Ξ′ then F (W ) = (F, p) is a free Lie algebra with projection-derivation
which has n free generators mi = xi + yi, 1 ≤ i ≤ n.
Proof. For 1 ≤ i ≤ n we have that xi ∈ L, yi ∈ V , so mi = xi + yi ∈ F =
L⊕V . We will consider an arbitrary (N, p) ∈ Θ. We assume that we have
a mapping
f : {m1, . . . ,mn} ∋ mi → ni ∈ N.
We will construct two other mappings
ϕ : {x1, . . . , xn} ∋ xi → r (ni) ∈ ker p ⊂ N
and
ψ : {y1, . . . , yn} ∋ yi → p (ni) ∈ imp ⊂ N.
108 Automorphic equivalence of the representations
By our assumption about (L, V ), these mappings can be extended to the
homomorphism
(ϕ,ψ) : (L, V ) → F−1 (N, p) = (ker p, imp) .
So there is a homomorphism
F (ϕ,ψ) = (ϕ⊕ ψ) : F (L, V ) = (F, p) → (N, p) .
(ϕ⊕ ψ) (mi) = (ϕ⊕ ψ) (xi + yi) = ϕ (xi) + ψ (yi) =
= r (ni) + p (ni) = ni = f (mi)
holds for 1 ≤ i ≤ n, because xi ∈ L, yi ∈ V .
5. Automorphisms of the category Ξ
0
and of the category Θ
0
If we have a category K, which objects are universal algebras and
morphisms are homomorphism, then automorphism Φ of this category
transform the homomorphism idA ∈ MorK, where A ∈ ObK, to homo-
morphism idΦ(A), because homomorphism idA uniquely defined by its
"algebraic" property: idA is a unit of the semigroup EndA. Therefore we
have a
Proposition 5.1. If A,B ∈ ObK, A ∼= B, Φ ∈ AutK then Φ (A) ∼= Φ (B).
Theorem 5.1. The category Ξ0 has 2-sorted IBN propriety: if W (X1, Y1),
W (X2, Y2) ∈ ObΞ0 and W (X1, Y1) ∼= W (X2, Y2), then |X1| = |X2| and
|Y1| = |Y2|.
Proof. We consider W (X,Y ) = (L (X) , A (X)Y ) = (L, V ) ∈ ObΞ0.
L/L2 is a k-linear space and dimL/L2 = |X|.
In the associative algebra A (X) we will consider 〈L〉 - two-sided ideal
generated by the set L = L (X) ⊂ A (X). This ideal coincide with 〈X〉 -
two-sided ideal generated by the set X, because every element of L can
be generated by elements of X. In the A (X)-module V = A (X)Y we
will consider submodule 〈L〉V = Spank (av | a ∈ 〈L〉 , v ∈ V ) = 〈X〉V .
dimV/ 〈X〉V = dimV/ 〈L〉V = |Y |.
We assume that we have two objects of Ξ0: W (X1, Y1) =
(L (X1) , A (X1)Y1) = (L1, V1) and W (X2, Y2) = (L (X2) , A (X2)Y2) =
(L2, V2) - and there is an isomorphism (ϕ,ψ) : (L1, V1) → (L2, V2). It
means that ϕ : L1 → L2 is an isomorphism and L1/L
2
1
∼= L2/L
2
2, so
|X1| = |X2|.
I . Shestakov, A. Tsurkov 109
By (1.1) we have that ψ : V1 → V2 is an isomorphism of the L1-
modules, when acting of L1 over V2 defined by l ◦ v = ϕ (l) v, where
l ∈ L1, v ∈ V2. A (X1) and A (X2) are universal enveloped algebras of
L1 and L2 respectively. Therefore the isomorphism ϕ : L1 → L2 can be
extended to the isomorphism of algebras with unit ϕ : A (X1) → A (X2).
A (X1) is generated as algebra with unit by elements of L1, so ψ is also an
isomorphism of the A (X1)-modules. Therefore there is an isomorphism of
A (X1)-modules V1/ 〈L1〉V1
∼= V2/ 〈L2〉V2, because ψ (〈L1〉V1) = 〈L2〉V2.
So dimV1/ 〈L1〉V1 = dimV2/ 〈L2〉V2 and |Y1| = |Y2|.
This is a well-known
Definition 5.1. We consider a category K and the family of objects
{Ai}i∈I ⊆ ObK. The pair
(
Q ∈ ObK, {ηi : Ai → Q}i∈I ⊆ MorK
)
called
coproduct of {Ai}i∈I if for every B ∈ ObK and every {αi : Ai → B}i∈I ⊆
MorK there exists only one α : Q → B ∈ MorK such that αi = αηi.
The coproduct is defined up to isomorphism. It is clear that if Φ ∈
AutK then
Φ
(
∐
i∈I
Ai
)
∼=
∐
i∈I
Φ (Ai) . (5.1)
It is easy to check that if W (X1, Y1) ,W (X2, Y2) ∈ ObΞ0 then
W (X1, Y1) ⊔W (X2, Y2) ∼= W (X3, Y3), where
|X3| = |X1| + |X2| , |Y3| = |Y1| + |Y2| . (5.2)
Similar to [3] we define
Definition 5.2. We say that the free representation W (X,Y ) ∈ ObΞ0
is a cyclic if |X| = 1 and |Y | = 1.
Proposition 5.2. For every Φ ∈ AutΞ0 the Φ (W (∅,∅)) = W (∅,∅)
holds and if W (x, y) ∈ ObΞ0 is a cyclic representation then Φ (W (x, y))
is also a free cyclic representation.
Proof. We consider the factor set ObΞ0/ ∼= (skeleton of the set ObΞ0).
The elements of this set we denote by
[W (X,Y )] =
{
W (X1, Y1) ∈ ObΞ0 | W (X,Y ) ∼= W (X1, Y1)
}
.
By Theorem 5.1 we can define the mapping
g : ObΞ0/ ∼=∋ [W (X,Y )] → (|X| , |Y |) ∈ N ⊕ N
110 Automorphic equivalence of the representations
and this mapping is bijection. N⊕N we consider as a set with the operation
of the addition according the components. N ⊕ N will be a commutative
semigroup with this operation. ObΞ0/ ∼= we consider as a set with the
operation of the coproduct. ObΞ0/ ∼= with this operation also will be a
commutative semigroup. By (5.2) g is an isomorphism.
We consider an arbitrary Φ ∈ AutΞ0. By Proposition 5.1 in the set
ObΞ0/ ∼= we can define the factor mapping
Φ̃ : ObΞ0/ ∼=∋ [W (X,Y )] → [Φ (W (X,Y ))] ∈ ObΞ0/ ∼= .
Φ̃ is a bijection, because Φ ∈ AutΞ0. By (5.1) Φ̃ is an isomorphism.
Therefore gΦ̃g−1 is an automorphism of N ⊕ N. (0, 0) is a unit of
the semigroup N ⊕ N, so gΦ̃g−1 (0, 0) = (0, 0) or Φ̃g−1 (0, 0) = g−1 (0, 0).
g−1 (0, 0) = {W (∅,∅)}, therefore Φ (W (∅,∅)) = W (∅,∅).
The semigroup N⊕N has only one minimal set of generators: {(1, 0) , (0, 1)}.
So the automorphism gΦ̃g−1 must preserve this set. Therefore we have
two cases: or Φ̃g−1 (1, 0) = g−1 (0, 1) and Φ̃g−1 (0, 1) = g−1 (1, 0), or
Φ̃g−1 (1, 0) = g−1 (1, 0) and Φ̃g−1 (0, 1) = g−1 (0, 1).
g−1 (1, 0) =
{
W (x,∅) | x ∈ X0
}
, g−1 (0, 1) =
{
W (∅, y) | y ∈ Y 0
}
. In
the first case we have that Φ (W (x,∅)) = W (∅, y1), Φ (W (∅, y)) =
W (x1,∅), where x, x1 ∈ X0, y, y1 ∈ Y 0. Therefore
Φ (W (x, y)) ∼= Φ (W (x,∅) ⊔W (∅, y)) ∼=
∼= W (∅, y1) ⊔W (x1,∅) ∼= W (x1, y1) .
So Φ (W (x, y)) = W (x2, y2), where x2 ∈ X0, y2 ∈ Y 0. In the second case
we achieve the similar result.
Corollary 1. If Φ ∈ AutΞ0, then Φ (Ξ′) = Ξ′.
Proof. IfX ⊂ X0, Y ⊂ Y 0 and |X| = |Y | = n > 1 then X = {x1, . . . , xn},
Y = {y1, . . . , yn} and we have that
Φ (W (X,Y )) ∼= Φ
(
n∐
i=1
W (xi, yi)
)
∼=
∼=
n∐
i=1
Φ (W (xi, yi)) ∼=
n∐
i=1
W
(
x′
i, y
′
i
) ∼= W
(
X ′, Y ′) ,
where X ′ = {x′
1, . . . , x
′
n} ⊂ X0, Y ′ = {y′
1, . . . , y
′
n} ⊂ Y 0 and |X ′| =
|Y ′| = n.
I . Shestakov, A. Tsurkov 111
In the category Θ we can consider subcategory Θ0. We take a infinite
countable sets of symbols M0. The objects of Θ0 will be the free algebras
in Θ with the set of free generators M such that M ⊂ M0, |M | < ∞. We
will denote these algebras by F (M). The morphisms of Θ0 will be the
homomorphisms of these algebras.
By using of the Theorems 4.1 and 4.2 F (Ξ′) = ObΘ0 and
F−1
(
ObΘ0
)
= Ξ′, so, by Corollary 1 from the Proposition 5.2 we prove
the
Theorem 5.2. If Φ ∈ AutΞ0 then FΦ|Ξ′F−1 ∈ AutΘ0.
6. Automorphic equivalence in the variety Ξ
and in the variety Θ
Proposition 6.1. If (T1, T2) ⊂ W = W (X,Y ) ∈ Ξ′, H = (L, V ) ∈ Ξ,
(T1, T2) is an H-closed congruence, then T1 ⊕ T2 ⊂ F (W (X,Y )) is an
F (H)-closed congruence. If T ⊂ F (M) ∈ ObΘ0, N = (N, p) ∈ Θ, T is
an N -closed congruence, then (T ∩ ker p, T ∩ imp) ⊂ F−1 (F (M)) is an
F−1 (N)-closed congruence. The mappings
FW,H : ClH (W ) ∋ (T1, T2) → T1 ⊕ T2 ∈ ClF(H) (F (W ))
and
F−1
F (M),N : ClN (F (M)) ∋ T → (T ∩ ker p, T ∩ imp) ∈
∈ ClF−1(N)
(
F−1 (F (M))
)
are bijections.
Proof. If (ϕ,ψ) ∈ (T1, T2)′
H , then by Proposition 4.2
ker F (ϕ,ψ) = ker (ϕ⊕ ψ) = kerϕ⊕ kerψ ⊇ T1 ⊕ T2,
so
F
(
(T1, T2)′
H
)
=
{
f = ϕ⊕ ψ | (ϕ,ψ) ∈ (T1, T2)′
H
}
⊆ (T1 ⊕ T2)′
F(H) .
We will consider l + v ∈ (T1 ⊕ T2)′′
F(H) =
⋂
f∈(T1⊕T2)′
F(H)
ker f , where
l ∈ ker p, v ∈ imp. l + v ∈
⋂
(ϕ,ψ)∈(T1,T2)′
H
ker (ϕ⊕ ψ) holds, so for every
(ϕ,ψ) ∈ (T1, T2)′
H we have (ϕ⊕ ψ) (l + v) = ϕ (l) + ψ (v) = 0. ϕ (l) ∈
ker p, ψ (v) ∈ imp, so ϕ (l) = 0, ψ (v) = 0. Hence l ∈
⋂
(ϕ,ψ)∈(T1,T2)′
H
kerϕ =
112 Automorphic equivalence of the representations
T1, v ∈
⋂
(ϕ,ψ)∈(T1,T2)′
H
kerψ = T2. Therefore l + v ∈ T1 ⊕ T2 and
(T1 ⊕ T2)′′
F(H) ⊆ T1 ⊕ T2. It means that T1 ⊕ T2 is an F (H)-closed con-
gruence.
If f ∈ T ′
N then F−1 (f) = (rfκ, pfι) and by Proposition 4.2
ker rfκ = ker f ∩ ker p ⊇ T ∩ ker p, ker pfι = ker f ∩ imp ⊇ T ∩ imp
holds. So (rfκ, pfι) ∈ (T ∩ ker p, T ∩ imp)′
F−1(N) and F−1 (T ′
N ) ⊆
(T ∩ ker p, T ∩ imp)′
F−1(N). Therefore
(T ∩ ker p, T ∩ imp)′′
F−1(N) ⊆
⋂
f∈T ′
N
ker rfκ,
⋂
f∈T ′
N
ker pfι
=
=
⋂
f∈T ′
N
(ker f ∩ ker p) ,
⋂
f∈T ′
N
(ker f ∩ imp)
=
=
⋂
f∈T ′
N
ker f
∩ ker p,
⋂
f∈T ′
N
ker f
∩ imp
=
= (T ∩ ker p, T ∩ imp) ,
so (T ∩ ker p, T ∩ imp) is an F−1 (N)-closed congruence.
If (T1, T2) ∈ ClH (W ) and T = T1 ⊕ T2 ⊂ F (W ), then T1 = T ∩ ker p,
T2 = T ∩ imp, so F−1
F(W ),F(H)FW,H = idClH(W ).
If N1 = (N1, p1) , N2 = (N2, p2) ∈ Θ and f : N1 → N2 is a homomor-
phism then ker f is p-invariant. So, T ∈ ClN (F (M)) is also p-invariant
and T = (T ∩ ker p)⊕ (T ∩ imp). Therefore FF−1(F (M)),F−1(N)F
−1
F (M),N =
idClN (F (M)).
If F1 = F (M1) , F2 = F (M2) ∈ ObΘ0 and T is a congruence in F2
then βF1,F2 (T ) will be a relation in Hom (F1, F2) which we define as in [1,
Subsection 3.3]: (f1, f2) ∈ βF1,F2 (T ) if and only if f1 (n) ≡ f2 (n) (modT )
holds for every n ∈ F1.
Proposition 6.2. If W1 = W (X1, Y1) ,W2 = W (X2, Y2) ∈
Ξ′, H ∈ Ξ, (T1, T2) ∈ ClH (W2) then F (βW1,W2 (T1, T2)) =
βF(W1),F(W2) (FW2,H (T1, T2)). If F1 = F (M1) , F2 = F (M2) ∈
ObΘ0, N = (N, p) ∈ Θ, T ∈ ClN (F2) then F−1 (βF1,F2 (T )) =
βF−1(F1),F−1(F2)
(
F−1
F2,N
(T )
)
.
Proof. F (Wi) =
(
L (Xi) ⊕A (Xi)Yi, p|A(Xi)Yi
)
where i = 1, 2.
FW2,H (T1, T2) = T1 ⊕ T2 ⊆ L (X2) ⊕ A (X2)Y2 = F (W2). If
I . Shestakov, A. Tsurkov 113
((ϕ1, ψ1) , (ϕ2, ψ2)) ∈ βW1,W2 (T1, T2) then ϕ1 (l) ≡ ϕ2 (l) (modT1) holds
for every l ∈ L (X1) and ψ1 (v) ≡ ψ2 (v) (modT2) holds for every v ∈
A (X1)Y1. F (ϕi, ψi) = ϕi ⊕ ψi ∈ Hom (F (W1) ,F (W2)) where i = 1, 2.
For every n ∈ F (W1) we have n = l+ v, where l ∈ L (X1), v ∈ A (X1)Y1.
So (ϕ1 ⊕ ψ1) (n) = ϕ1 (l) + ψ1 (v) ≡ ϕ2 (l) + ψ2 (v) (modT1 ⊕ T2),
ϕ2 (l) + ψ2 (v) = (ϕ2 ⊕ ψ2) (n) and
(F (ϕ1, ψ1) ,F (ϕ2, ψ2)) ∈ βF(W1),F(W2) (T1 ⊕ T2) .
We assume that (f1, f2) ∈ βF(W1),F(W2) (T1 ⊕ T2). F−1 (fi) =
(rfiκ, pfiι) ∈ Hom (W1,W2) where i = 1, 2. If l ∈ L (X1) then
f1 (l) − f2 (l) ∈ T1 ⊕ T2 and rf1κ (l) − rf2κ (l) ∈ T1. Analogously we
have pf1ι (v) − pf2ι (v) ∈ T2 for every v ∈ A (X2)Y2, so
(
F−1 (f1) ,F−1 (f2)
)
= ((rf1κ, pf1ι) , (rf2κ, pf2ι)) ∈ βW1,W2 (T1, T2) .
Therefore
F (βW1,W2 (T1, T2)) = βF(W1),F(W2) (FW2,H (T1, T2)) .
From this fact and from proving of Proposition 6.1 we can conclude
that
F−1 (βF1,F2 (T )) = βF−1(F1),F−1(F2)
(
F−1
F2,N
(T )
)
.
Theorem 6.1. If H1 = (L1, V1) , H2 = (L2, V2) ∈ Ξ are automorphi-
cally equivalent then N1 = F (H1) , N2 = F (H2) are automorphically
equivalent.
Proof. We have an automorphism Φ ∈ AutΞ0 and the system of bijections
α (Φ)W : ClH1 (W ) → ClH2 (Φ (W )) for every W ∈ ObΞ0. Also the
equation
Φ (βW1,W2 (T1, T2)) = βΦ(W1),Φ(W2)
(
α (Φ)W2
(T1, T2)
)
holds for every W1,W2 ∈ ObΞ0, and every (T1, T2) ∈ ClH1 (W2).
By Proposition 5.2 there is an automorphism Ψ = FΦ|Ξ′F−1 ∈ AutΘ0.
By Proposition 6.1 the mapping:
α (Ψ)F = FΦF−1(F ),H2
α (Φ)F−1(F ) F−1
F,N1
: ClN1 (F ) → ClN2 (Ψ (F ))
is a bijection for every F ∈ ObΘ0. By Proposition 6.2 we have for every
F1, F2 ∈ ObΘ0, and every T ∈ ClN1 (F2) that
114 Automorphic equivalence of the representations
Ψ (βF1,F2 (T )) = FΦF−1 (βF1,F2 (T )) =
= FΦ
(
βF−1(F1),F−1(F2)
(
F−1
F2,N1
(T )
))
=
= F
(
βΦF−1(F1),ΦF−1(F2)
(
α (Φ)F−1(F2)
(
F−1
F2,N1
(T )
)))
=
= βFΦF−1(F1),FΦF−1(F2)
(
FΦF−1(F2),H2
α (Φ)F−1(F2)
(
F−1
F2,N1
(T )
))
=
= βΨ(F1),Ψ(F2)
(
α (Ψ)F2
(T )
)
.
7. Automorphisms of the category of the finitely
generated free algebras of the some variety
of 1-sorted algebras
In this Section we explain the method of verbal operations which
we will use for the studying of the relation between the automorphic
equivalence and geometric equivalence in the our variety Θ. We use
results of the [4] and [5].
In this Section the word "algebra" means "universal algebra". Also so
on in this Section Θ will be an arbitrary variety of 1-sorted algebras. As
in the Section 5 we define the category Θ0 of the finitely generated free
algebras of our variety Θ. The infinite countable sets of symbols which
will be the generators of our free algebras we will denote in this Section
by X0.
Definition 7.1. An automorphism Υ of a category K is inner, if it is
isomorphic as a functor to the identity automorphism of the category K.
It means that for every A ∈ ObK there exists an isomorphism sΥ
A :
A → Υ (A) such that for every α ∈ MorK (A,B) the diagram
A
sΥ
A
//
α
��
Υ (A)
Υ(α)
��
B
sΥ
B
// Υ (B)
commutes. The group of the all automorphisms of the category Θ0 we
denote by A. The subgroup of the all inner automorphisms of Θ0 we
denote by Y. This is a normal subgroup of A: Y ⊳ A.
We know from [1] that if automorphic equivalence of algebras H1, H2 ∈
Θ provided by inner automorphism then H1 and H2 are geometrically
equivalent. Hear variety Θ can by even a variety of many-sorted algebras.
I . Shestakov, A. Tsurkov 115
So for studying of the difference between the automorphic equivalence
and geometric equivalence of the algebras from Θ, we must calculate the
quotient group A/Y.
In the 1-sorted case there is a reason to define
Definition 7.2. An automorphism Φ of the category Θ0 is called strongly
stable if it satisfies the conditions:
A1) Φ preserves all objects of Θ0,
A2) there exists a system of bijections
{
sΦ
F : F → F | F ∈ ObΘ0
}
such
that Φ acts on the morphisms α : D → F of Θ0 by this way:
Φ (α) = sΦ
Fα
(
sΦ
D
)−1
, (7.1)
A3) sΦ
F |X= idX , for every free algebra F = F (X) ∈ ObΘ0.
The subgroup of the all strongly stable automorphisms of Θ0 we
denote by S.
We say that the variety Θ has IBN propriety if for every F (X) , F (Y ) ∈
ObΘ0 we have F (X) ∼= F (Y ) only if |X| = |Y |. In this case we have the
decomposition
A = YS (7.2)
so A/Y = S/S ∩ Y.
The system of bijections
{
sΦ
F = sF : F → F | F ∈ ObΘ0
}
mentioned
in definition of the strongly stable automorphism fulfills these two condi-
tions:
B1) for every homomorphism α : A → B ∈ MorΘ0 the mappings sBαs
−1
A
and s−1
B αsA are homomorphisms;
B2) sF |X= idX for every free algebra F ∈ ObΘ0.
These bijections uniquely defined by the strongly stable automorphism
Φ, because for every F ∈ ObΘ0 and every f ∈ F we have
sΦ
F (f) = sΦ
Fα (x) =
(
sΦ
Fα
(
sΦ
D
)−1
)
(x) = (Φ (α)) (x) , (7.3)
where D = D (x) ∈ ObΘ0 is a 1-generated free algebra and α : D → F
homomorphism such that α (x) = f .
On the other side by system of bijections
{
sF : F → F | F ∈ ObΘ0
}
which fulfills conditions B1) and B2) we can define the strongly stable
116 Automorphic equivalence of the representations
automorphism Φ, which preserves all objects of Θ0 and acts on the
morphisms α : D → F of Θ0 by formula (7.1) with sΦ
F = sF . By this way
we construct an one-to-one and onto correspondence between the set of
the all strongly stable automorphisms of the category Θ0 and the set of
the all systems of bijections which fulfill conditions B1) and B2).
We denote the signature of the algebras from the variety Θ by Ω.
The arity of the operation ω ∈ Ω we denote by nω and by Fω we denote
F (x1, . . . , xnω ) ∈ ObΘ0. ω (x1, . . . , xnω ) ∈ Fω. If we have system of
bijections
{
sF : F → F | F ∈ ObΘ0
}
which fulfills conditions B1) and
B2) then
wω (x1, . . . , xnω ) = sFω (ω (x1, . . . , xnω )) ∈ Fω. (7.4)
We will consider the system of words W = {wω | ω ∈ Ω}. In every H ∈ Θ
we can define new operations {ω∗ | ω ∈ Ω} by using of the system of
words W :
ω∗ (h1, . . . , hnω ) = wω (h1, . . . , hnω ) (7.5)
for every h1, . . . , hnω ∈ H. We denote by H∗
W the new algebra which
coincide as set with H but has other operations: {ω∗ | ω ∈ Ω} instead
{ω | ω ∈ Ω}. The system of words W = {wω | ω ∈ Ω} fulfills these two
conditions:
Op1) wω (x1, . . . , xnω ) ∈ Fω for every ω ∈ Ω,
Op2) for every F = F (X) ∈ ObΘ0 there exists an isomorphism σF : F →
F ∗
W such that σF |X= idX because the bijections
{
sF | F ∈ ObΘ0
}
will be isomorphisms σF : F → F ∗
W .
On the other side if we have a system of words W = {wω | ω ∈ Ω}
which fulfills conditions Op1) and Op2), then we have that F ∗
W ∈ Θ, so
the isomorphisms σF : F → F ∗
W are uniquely determined by the system
of words W . This system of isomorphisms
{
σF : F → F ∗
W | F ∈ ObΘ0
}
is
a system of bijections which fulfills conditions B1) and B2) with sF = σF .
By this way we construct an one-to-one and onto correspondence between
the set of the all system of bijections which fulfills conditions B1) and
B2) and the set of the all system of words which fulfills conditions Op1)
and Op2).
Therefore we can calculate the group S if we can find the all system
of words which fulfill conditions Op1) and Op2). For calculation of the
group S ∩ Y we also have a
Criterion 7.1. The strongly stable automorphism Φ of the category Θ0
which corresponds to the system of words W is inner if and only if for
I . Shestakov, A. Tsurkov 117
every F ∈ ObΘ0 there exists an isomorphism cF : F → F ∗
W such that
cFα = αcD fulfills for every (α : D → F ) ∈ MorΘ0.
8. Strongly stable automorphisms of the category Θ
0
The variety Θ, which was defined in the Section 4, is a variety of
1-sorted universal algebras. If F (M) ∈ ObΘ0 then by Theorem 4.1 |M | =
dim
(
ker p/ (ker p)2
)
, so variety Θ possesses the IBN property: for free
algebras F (M1) , F (M2) ∈ Θ we have F (M1) ∼= F (M2) if and only if
|M1| = |M2|. So we have for our variety Θ0 the decomposition (7.2) and
for calculation of the group A/Y = S/S ∩ Y we can use the method
described in the Section 7.
The signature of our variety Θ is Ω = {0, λ (λ ∈ k) ,+, [, ] , p}, where
0 is 0-ary operation of the taking 0, λ for every λ ∈ k is the 1-nary
operation of the multiplication by this scalar, p is the 1-nary operation of
projection, + is the addition and [, ] are the Lie brackets. We must find
for the calculation of the group S all the system of words
W =
{
w0, wλ (λ ∈ k) , w+, w[,], wp
}
(8.1)
which fulfill conditions Op1) and Op2) and after use the Criterion 7.1 for
the calculation of the group S ∩ Y. By this way we will prove the
Theorem 8.1. If Autk = {idk} then the group A/Y is a trivial.
Proof. If (F, p) = F (m1, . . . ,mn) ∈ ObΘ0 then by Theorems 4.1 and 4.2
(F, p) = FF−1 (F, p) = (L⊕ V, pV ), where p = pV ,
L = ker p = L (r (m1) , . . . , r (mn))
is a free Lie algebra with the free generators r (m1) , . . . , r (mn),
V = imp =
n⊕
i=1
A (r (m1) , . . . , r (mn)) p (mi)
is a free module with the basis {p (m1) , . . . , p (mn)} over algebra
A (r (m1) , . . . , r (mn)), which is an associative algebra with unit gener-
ated by the free generators r (m1) , . . . , r (mn). Hear we must understand
that by formula (4.2)
r (mi1) . . . r (mis) v = [r (mi1) , [. . . , [r (mis) , v]]] ,
118 Automorphic equivalence of the representations
where v ∈ V , 1 ≤ i1, . . . , is ≤ n, s ∈ N, if s = 0 then 1v = v. So
by linearity we can understand what means f (r (m1) , . . . , r (mn)) v, for
every associative polynomial from n variables f ∈ A (x1, . . . , xn).
We assume that Ψ ∈ S corresponds to the system of bijections{
sΨ
F = sF : F → F | F ∈ ObΘ0
}
and to the system of words (8.1) and
the words of this system correspond to the operations from Ω by formula
(7.4) with sFω = sΨ
Fω
. W fulfills conditions Op1) and Op2). In particular
by condition Op2) all axioms of the variety Θ must fulfill for operations
defined by system of words W . In this proof we have more convenient to
denote by an other symbols than the symbols of Ω the operations defined
by the words from W according the (7.5).
w0 = 0 because w0 ∈ F (∅) and F (∅) = {0}.
We denote by λ∗ the operation defined by the word wλ ∈ F (m)
(λ ∈ k), where F (m) is a 1-generated object of the category Θ0.
(F (m) , p) = (L⊕ V, pV ), where L = L (r (m)) = spk {r (m)}, V =
A (r (m)) p (m) = k [r (m)] p (m), so
λ ∗m = wλ (m) = ϕ (λ) r (m) + qλ (r (m)) p (m) ,
where ϕ (λ) ∈ k, qλ ∈ k [x]. If λ 6= 0 then λ−1 ∗ (λ ∗m) = m must fulfill.
λ−1 ∗ (λ ∗m) = λ−1 ∗ (ϕ (λ) r (m) + qλ (r (m)) p (m)) =
= ϕ
(
λ−1
)
r (ϕ (λ) r (m) + qλ (r (m)) p (m)) +
+qλ−1 (r (ϕ (λ) r (m) + qλ (r (m)) p (m))) p (ϕ (λ) r (m) + qλ (r (m)) p (m)) =
= ϕ
(
λ−1
)
ϕ (λ) r (m) + qλ−1 (ϕ (λ) r (m)) (qλ (r (m)) p (m)) .
On the other side m = r (m) + p (m). So ϕ
(
λ−1
)
ϕ (λ) = 1 and ϕ (λ) 6= 0.
qλ−1 (ϕ (λ) r (m)) (qλ (r (m)) p (m)) = s (r (m)) p (m) ,
where s ∈ k [x]. If deg qλ−1 = n, deg qλ = t then deg s = n + t, but
deg s = 0 must hold, so n = 0, t = 0. Therefore qλ = ψ (λ) ∈ k,
λ ∗m = ϕ (λ) r (m) + ψ (λ) p (m) . (8.2)
For λ = 0 it also fulfills with ϕ (0) = ψ (0) = 0, because 0 ∗m = 0.
µ ∗ (λ ∗m) = (µλ) ∗m must fulfill for every µ, λ ∈ k so
µ ∗ (λ ∗m) = µ ∗ (ϕ (λ) r (m) + ψ (λ) p (m)) =
= ϕ (µ) r (ϕ (λ) r (m) + ψ (λ) p (m))+ψ (µ) p (ϕ (λ) r (m) + ψ (λ) p (m)) =
= ϕ (µ)ϕ (λ) r (m) + ψ (µ)ψ (λ) p (m) .
I . Shestakov, A. Tsurkov 119
On the other side
(µλ) ∗m = ϕ (µλ) r (m) + ψ (µλ) p (m) .
Hence
ϕ (µ)ϕ (λ) = ϕ (µλ) , ψ (µ)ψ (λ) = ψ (µλ) . (8.3)
We denote by ⊥ the operation defined by the word w+ ∈ F (m1,m2),
where F (m1,m2) is a 2-generated object of the category Θ0.
m1 ⊥ m2 = l (r (m1) , r (m2)) + q1 (r (m1) , r (m2)) p (m1) +
+ q2 (r (m1) , r (m2)) p (m2) ,
where l ∈ L (x1, x2), q1, q2 ∈ A (x1, x2). We can write
l (r (m1) , r (m2)) = α1r (m1) + α2r (m2) + l̃ (r (m1) , r (m2)) ,
where l̃ ∈ L2 (x1, x2), α1, α2 ∈ k. And
qi (r (m1) , r (m2)) p (mi) = q̃i (r (m1) , r (m2)) p (mi) + βip (mi) ,
where q̃i is a polynomial from A (x1, x2) such that all its monomials have
entries of x1 or x2, βi ∈ k, i = 1, 2.
m1 ⊥ 0 = m1 must fulfill. m1 = r (m1) + p (m1). But
m1 ⊥ 0 = α1r (m1) + q̃1 (r (m1) , 0) p (m1) + β1p (m1) .
Therefore α1 = β1 = 1. From 0 ⊥ m2 = m2 we conclude that α2 = β2 = 1.
In F (m) the (λ+ µ) ∗ m = (λ ∗m) ⊥ (µ ∗m) must fulfill for every
µ, λ ∈ k.
(λ+ µ) ∗m = ϕ (λ+ µ) r (m) + ψ (λ+ µ) p (m) .
Also we have that
(λ ∗m) ⊥ (µ ∗m) = r (λ ∗m) + r (µ ∗m) + l̃ (r (λ ∗m) , r (µ ∗m)) +
+ q̃1 (r (λ ∗m) , r (µ ∗m)) p (λ ∗m) + p (λ ∗m) +
+ q̃2 (r (λ ∗m) , r (µ ∗m)) p (µ ∗m) + p (µ ∗m) .
r (λ ∗m) = r (ϕ (λ) r (m) + ψ (λ) p (m)) = ϕ (λ) r (m) ,
p (λ ∗m) = p (ϕ (λ) r (m) + ψ (λ) p (m)) = ψ (λ) p (m) .
So
120 Automorphic equivalence of the representations
(λ ∗m) ⊥ (µ ∗m) =
= ϕ (λ) r (m) + ϕ (µ) r (m) + l̃ (ϕ (λ) r (m) , ϕ (µ) r (m)) +
+ q̃1 (ϕ (λ) r (m) , ϕ (µ) r (m))ψ (λ) p (m) + ψ (λ) p (m) +
+ q̃2 (ϕ (λ) r (m) , ϕ (µ) r (m))ψ (µ) p (m) + ψ (µ) p (m) .
Hence
ϕ (λ+ µ) = ϕ (λ) + ϕ (µ) , ψ (λ+ µ) = ψ (λ) + ψ (µ) . (8.4)
Therefore ϕ,ψ are homomorphisms k → k.
ϕ (1) = ψ (1) = 1, (8.5)
because 1 ∗ m = m, so kerϕ = kerψ = 0 and Imϕ ∼= Imψ ∼= k. Hence
Imϕ, Imψ are infinite sets.
λ ∗ (m1 ⊥ m2) = (λ ∗m1) ⊥ (λ ∗m2) must fulfill for every λ ∈ k.
λ ∗ (m1 ⊥ m2) = ϕ (λ)
(
r (m1) + r (m2) + l̃ (r (m1) , r (m2))
)
+
+ ψ (λ) (q̃1 (r (m1) , r (m2)) p (m1) + p (m1)) +
+ ψ (λ) (q̃2 (r (m1) , r (m2)) p (m2) + p (m2)) . (8.6)
(λ ∗m1) ⊥ (λ ∗m2) =
= r (λ ∗m1) + r (λ ∗m2) + l̃ (r (λ ∗m1) , r (λ ∗m2)) +
+ q̃1 (r (λ ∗m1) , r (λ ∗m2)) p (λ ∗m1) + p (λ ∗m1) +
+ q̃2 (r (λ ∗m1) , r (λ ∗m2)) p (λ ∗m2) + p (λ ∗m2) =
= ϕ (λ) r (m1) + ϕ (λ) r (m2) + l̃ (ϕ (λ) r (m1) , ϕ (λ) r (m2)) +
+ q̃1 (ϕ (λ) r (m1) , ϕ (λ) r (m2))ψ (λ) p (m1) + ψ (λ) p (m1) +
+ q̃2 (ϕ (λ) r (m1) , ϕ (λ) r (m2))ψ (λ) p (m2) + ψ (λ) p (m2) . (8.7)
We decompose l̃, q̃1 and q̃2 to the homogeneous components according
the degrees (sum of degrees of variables x1 and x2) of monomials: l̃ =
l2 + . . . + ln0 , q̃i = qi,1 + . . . + qi,ni
, n0 = deg l̃, ni = deg q̃i, i = 1, 2.
We have by comparison of (8.6) and (8.7) that ϕ (λ) lj = (ϕ (λ))j lj for
2 ≤ j ≤ n0 and ψ (λ) qi,j = ψ (λ) (ϕ (λ))j qi,j for 1 ≤ j ≤ ni, i = 1, 2. We
denote n = max {n0, n1, n2}. We take µ = ϕ (λ) ∈ Imϕ \ {0} such that
ϕ (λ)j 6= 1 for every j = 1, . . . , n. ψ (λ) 6= 0, so lj = 0, qi,j = 0, hence
I . Shestakov, A. Tsurkov 121
l̃ = 0, q̃1 = q̃2 = 0 and
m1 ⊥ m2 = r (m1) + r (m2) + p (m1) + p (m2) = m1 +m2. (8.8)
We denote by WΨ−1
the system of words which fulfills conditions
Op1) and Op2) and corresponds to the automorphism Ψ−1. By wΨ−1
λ (m)
(λ ∈ k) we denote the word from WΨ−1
which corresponds to the operation
of the multiplication by the scalar λ. We denote by λ ∗
Ψ−1
the operation
defined by word wΨ−1
λ (m). By (8.2), (8.3), (8.4) and (8.5) wΨ−1
λ (m) =
ρ (λ) r (m) +σ (λ) p (m), where ρ, σ are monomorphisms of the field k. By
(8.8) for w+ we have only one possibility for every system of words which
fulfills conditions Op1) and Op2): w+ (m1,m2) = m1 +m2.
By
{
sΨ−1
F
}
we denote the systems of bijections corresponding to
automorphism Ψ−1. Ψ−1Ψ = ΨΨ−1 = I, where I is the identical au-
tomorphism. By consideration of the formula (7.1) we can conclude
that to the automorphism Ψ−1Ψ corresponds the systems of bijections{
sΨ−1
F sΨ
F | F ∈ ObΘ0
}
. On the other side to the automorphism I corre-
sponds the systems of bijections
{
idF | F ∈ ObΘ0
}
. So, we have
sΨ−1
F (m)s
Ψ
F (m) (λm) = sIF (m) (λm) = λm = λr (m) + λp (m) .
On the other side, by using of the formula (7.4),
sΨ−1
F (m)s
Ψ
F (m) (λm) =
= sΨ−1
F (m) (ϕ (λ) r (m) + ψ (λ) p (m)) = ϕ (λ) ∗
Ψ−1
r (m) +ψ (λ) ∗
Ψ−1
p (m) =
= (ρϕ (λ) rr (m) + σϕ (λ) pr (m)) + (ρψ (λ) rp (m) + σψ (λ) pp (m)) =
= ρϕ (λ) r (m) + σψ (λ) p (m) .
Therefore ρϕ = σψ = idk. Analogously ϕρ = ψσ = idk. Therefore
ϕ,ψ ∈ Autk.
Now we consider the case when Autk = {idk}. We denote by × the
operation defined by the word w[,] ∈ F (m1,m2).
m1 ×m2 = u (r (m1) , r (m2)) +
+ t1 (r (m1) , r (m2)) p (m1) + t2 (r (m1) , r (m2)) p (m2) ,
where u ∈ L (x1, x2), t1, t2 ∈ A (x1, x2). (λm1) ×m2 = λ (m1 ×m2) must
fulfill for every λ ∈ k.
122 Automorphic equivalence of the representations
λ (m1 ×m2) = λu (r (m1) , r (m2)) + λt1 (r (m1) , r (m2)) p (m1) +
+ λt2 (r (m1) , r (m2)) p (m2) . (8.9)
(λm1) ×m2 = u (λr (m1) , r (m2)) + t1 (λr (m1) , r (m2))λp (m1) +
+ t2 (λr (m1) , r (m2)) p (m2) . (8.10)
We decompose u = u0 + u1 + . . . + us0 , ti = ti,0 + ti,1 + . . . + ti,si
,
i = 1, 2 by homogeneous components according the degree of x1. By
comparison of (8.9) and (8.10) we have that λuj = λjuj for 0 ≤ j ≤ s0,
λt1,j = λj+1t1,j , for 0 ≤ j ≤ s1, λt2,j = λjt2,j , for 0 ≤ j ≤ s2. We denote
s = max {s0, s1, s2}. We take λ such that λj 6= λ for j = 0, 2, . . . , s + 1
and conclude that u = u1, t1 = t1,0, t2 = t2,1.
Also m1 × (λm2) = λ (m1 ×m2) must fulfill for every λ ∈ k.
m1 × (λm2) = u1 (r (m1) , λr (m2)) + t1,0 (r (m1) , λr (m2)) p (m1) +
+ t2,1 (r (m1) , λr (m2))λp (m2) . (8.11)
Now we decompose u1 = u1,0 + u1,1 + . . . + u1,s3 , t1,0 = t1,0,0 + t1,0,1 +
. . .+ t1,0,s4 , t2,1 = t2,1,0 + t2,1,1 + . . .+ t2,1,s5 , by homogeneous components
according the degree of x2. And by comparison of (8.9) and (8.11) as
above we conclude that u = u1 = u1,1, t1 = t1,0 = t1,0,1, t2 = t2,1 = t2,1,0.
Therefore by (4.2)
m1 ×m2 = α [r (m1) , r (m2)] + β [r (m2) , p (m1)] + γ [r (m1) , p (m2)] ,
where α, β, γ ∈ k.
m1 ×m2 = −m2 ×m1 must fulfill.
m2 ×m1 = α [r (m2) , r (m1)] + β [r (m1) , p (m2)] + γ [r (m2) , p (m1)] =
−α [r (m1) , r (m2)] + γ [r (m2) , p (m1)] + β [r (m1) , p (m2)] .
Therefore γ = −β and
m1 ×m2 = α [r (m1) , r (m2)] + β [r (m2) , p (m1)] − β [r (m1) , p (m2)] .
In the case 1 we assume that β 6= 0.
The Jacobi identity
J (m1,m2,m3) = (m1 ×m2)×m3+(m2 ×m3)×m1+(m3 ×m1)×m2 = 0
(8.12)
must fulfill in F (m1,m2,m3).
I . Shestakov, A. Tsurkov 123
(m1 ×m2) ×m3 =
= (α [r (m1) , r (m2)] + β [r (m2) , p (m1)] − β [r (m1) , p (m2)]) ×m3 =
= α [α [r (m1) , r (m2)] , r (m3)] +
+ β [r (m3) , β [r (m2) , p (m1)] − β [r (m1) , p (m2)]] −
− β [α [r (m1) , r (m2)] , p (m3)] =
= α2 [[r (m1) , r (m2)] , r (m3)] + β2 [r (m3) , [r (m2) , p (m1)]] −
− β2 [r (m3) , [r (m1) , p (m2)]] − βα [[r (m1) , r (m2)] , p (m3)] .
F (m1,m2,m3) = L (r (m1) , r (m2) , r (m3)) ⊕
⊕
(
3⊕
i=1
A (r (m1) , r (m2) , r (m3)) p (mi)
)
,
so J (m1,m2,m3) =
3∑
i=0
Ji, where
J0 ∈ L (r (m1) , r (m2) , r (m3)) ,
Ji ∈ A (r (m1) , r (m2) , r (m3)) p (mi) ,
i = 1, 2, 3 and must fulfill Ji = 0, i = 0, . . . , 3.
J1 = β2 [r (m3) , [r (m2) , p (m1)]] − β2 [r (m2) , [r (m3) , p (m1)]] −
− βα [[r (m2) , r (m3)] , p (m1)] =
= β2 [r (m3) , [r (m2) , p (m1)]] − β2 [r (m2) , [r (m3) , p (m1)]] −
− βα [r (m2) , [r (m3) , p (m1)]] + βα [r (m3) , [r (m2) , p (m1)]]
by (4.2) and definition of representation of Lie algebra. So β2 + βα = 0
must fulfill and we have that β = −α, α 6= 0. It is easy to check that
β = −α enough for (8.12). Therefore
m1 ×m2 = α([r(m1), r(m2)] + [r(m1), p(m2)] −
− [r(m2), p(m1)]) = α [m1,m2] (8.13)
by (4.1) and (4.2).
In the case 2, if β = 0 we have that
m1 ×m2 = α [r (m1) , r (m2)] . (8.14)
124 Automorphic equivalence of the representations
If α = 0, then F (m1,m2) × F (m1,m2) = {0}, but
[F (m1,m2) , F (m1,m2)] 6= {0}. By condition Op2) F (m1,m2) ∼=
(F (m1,m2))∗
W . From this contradiction we conclude that hear also
α 6= 0.
We denote by p the operation defined by the word wp ∈ F (m).
p (m) = δr (m) + qp (r (m)) p (m) ,
where δ ∈ k, qp ∈ k [x]. p (λ ∗m) = λ ∗ p (m) must fulfill for every λ ∈ k.
λ ∗ p (m) = λδr (m) + λqp (r (m)) p (m) . (8.15)
p (λ ∗m) = δr (λ ∗m) + qp (r (λ ∗m)) p (λ ∗m) =
= δλr (m) + qp (λr (m))λp (m) . (8.16)
As above we decompose qp by homogeneous components according the
degree of x and conclude as above by comparison of (8.15) and (8.16)
that deg qp = 0 and
p (m) = δr (m) + εp (m)
where ε ∈ k.
p (p (m)) = p (m) must fulfill in F (m).
p (p (m)) = δr (δr (m) + εp (m))+εp (δr (m) + εp (m)) = δ2r (m)+ε2p (m) .
Therefore δ2 = δ, ε2 = ε.
If δ = ε = 1 then p (m) = r (m) + p (m) = m and p ((F (m))∗
W ) =
(F (m))∗
W but p (F (m)) 6= (F (m)) contrary to F (m) ∼= (F (m))∗
W . So it
is impossible that δ = ε = 1.
If δ = ε = 0, then p (m) = 0 and p ((F (m))∗
W ) = 0 but p (F (m)) 6= 0
contrary to F (m) ∼= (F (m))∗
W . As above we conclude that δ = ε = 0 is
impossible.
If δ = 1, ε = 0. Then p (m) = r (m), i.e. p = r. p must be a derivation
of (F (m))∗
W . In the case 2, by (8.14), we have that
p (m1 ×m2) = r (α [r (m1) , r (m2)]) = α [r (m1) , r (m2)] ,
p (m1) ×m2 +m1 × p (m2) = r (m1) ×m2 +m1 × r (m2) =
= α [rr (m1) , r (m2)] + α [r (m1) , rr (m2)] = 2α [r (m1) , r (m2)] .
I . Shestakov, A. Tsurkov 125
char (k) = 0, so p is not a derivation. In the case 1, by (8.13), we have
that
p(m1 ×m2) = r(α([r(m1), r(m2)] + [r(m1), p(m2)]−
− [r(m2), p(m1)])) = α[r(m1), r(m2)],
p(m1) ×m2 +m1 × p(m2) = r(m1) ×m2 +m1 × r(m2) =
= α([r(r(m1)), r(m2)] + [r(r(m1)), p(m2)] − [r(m2), p(r(m1))])+
+ α([r(m1), r(r(m2))] + [r(m1), p(r(m2))] − [r(r(m2)), p(m1)]) =
= α([r(m1), r(m2)]+[r(m1), p(m2)])+α([r(m1), r(m2)]−[r(m2), p(m1)]) =
= α(2[r(m1), r(m2)] + [r(m1), p(m2)] − [r(m2), p(m1)]).
In this case p also is not a derivation.
Therefore we have only one possibility: δ = 0, ε = 1. It means
p (m) = p (m), i.e., p = p.
And in the case 2, by (8.14), we have that
p (F (m1,m2) × F (m1,m2)) = 0
but
p [r (m1) , p (m2)] = [r (m1) , p (m2)] 6= 0,
so
p [F (m1,m2) , F (m1,m2)] 6= 0
contrary to F (m1,m2) ∼= (F (m1,m2))∗
W . Therefore the case 2 is impos-
sible.
Hence
m1 ×m2 = α ([r (m1) , r (m2)] + [r (m1) , p (m2)] −
− [r (m2) , p (m1)]) = α [m1,m2] ,
where α 6= 0.
From this fact, as in [4, end of the subsection 2.5], we conclude that
Ψ ∈ Y. So S = S ∩ Y and A/Y = {1}.
9. The main theorem
Theorem 9.1. If Autk = {idk} then automorphic equivalence of repre-
sentations of Lie algebras coincides with the geometric equivalence.
126 Automorphic equivalence of the representations
Proof. We assume that H1 = (L1, V1) , H2 = (L2, V2) ∈ Ξ are automorphi-
cally equivalent. By Theorem 6.1 we have that N1 = F (H1) , N2 = F (H2)
are automorphically equivalent. By [1, Proposition 9] and Theorem 8.1
we can conclude from this fact that N1, N2 are geometrically equivalent.
It means that ClN1 (F ) = ClN2 (F ) for every F ∈ ObΘ0.
We will consider the arbitrary W1 = (L (X1) , A (X1)Y1) ∈ ObΞ0.
There are X2 ⊂ X0, Y2 ⊂ Y 0 such that X1 ⊆ X2, Y1 ⊆ Y2 and W2 =
(L (X2) , A (X2)Y2) ∈ Ξ′. By Theorem 4.2 there exists F ∈ ObΘ0 such
that F = F (W2). By Proposition 6.1 we can conclude from ClN1 (F ) =
ClN2 (F ) that ClH1 (W2) = ClH2 (W2). And by Theorem 3.1 we can
conclude that ClH1 (W1) = ClH2 (W1). So H1 and H2 are geometrically
equivalent.
10. Acknowledgements
We acknowledge the support by FAPESP - Fundação de Amparo à
Pesquisa do Estado de São Paulo (Foundation for Support Research of
the State São Paulo), projects No. 2010/50948-2 and No. 2010/50347-9.
References
[1] B.Plotkin, Algebras with the same algebraic geometry, Proceedings of the Steklov
Institute of Mathematics, MIAN, 242, (2003), pp. 176–207.
[2] B. Plotkin, A. Tsurkov, Action type geometrical equivalence of representations of
groups. Algebra and Discrete Mathematics, 4 (2005), pp. 48 - 79.
[3] Plotkin B.I., Vovsi, S.M. Varieties of Groups Representations, Zinatne. Riga, 1983,
(Russian).
[4] B. Plotkin, G. Zhitomirski, On automorphisms of categories of free algebras of some
varieties, Journal of Algebra, 306:2, (2006), pp. 344 – 367.
[5] A. Tsurkov, Automorphic equivalence of algebras. International Journal of Algebra
and Computation. 17:5/6, (2007), pp. 1263–1271.
Contact information
I. Shestakov,
A. Tsurkov
Institute of Mathematics and Statistics.
University São Paulo,
Rua do Matão, 1010, Cidade Universitária, São
Paulo - SP - Brasil - CEP 05508-090
E-Mail: shestak@ime.usp.br,
arkady.tsurkov@gmail.com
Received by the editors: 15.12.2012
and in final form 15.12.2012.
|
| id | nasplib_isofts_kiev_ua-123456789-152257 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-11-30T11:09:43Z |
| publishDate | 2013 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Shestakov, I. Tsurkov, A. 2019-06-09T13:36:43Z 2019-06-09T13:36:43Z 2013 Automorphic equivalence of the representations of Lie algebras / I. Shestakov, A. Tsurkov // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 1. — С. 96–126. — Бібліогр.: 5 назв. — англ. 1726-3255 2010 MSC:17B10. https://nasplib.isofts.kiev.ua/handle/123456789/152257 In this paper we research the algebraic geometry of the representations of Lie algebras over fixed field k. We assume that this field is infinite and char (k) = 0. We consider the representations of Lie algebras as 2-sorted universal algebras. The representations of groups were considered by similar approach: as 2-sorted universal algebras - in [3] and [2]. The basic notions of the algebraic geometry of representations of Lie algebras we define similar to the basic notions of the algebraic geometry of representations of groups (see [2]). We prove that if a field k has not nontrivial automorphisms then automorphic equivalence of representations of Lie algebras coincide with geometric equivalence. This result is similar to the result of [4], which was achieved for representations of groups. But we achieve our result by another method: by consideration of 1-sorted objects. We suppose that our method can be more perspective in the further researches. We acknowledge the support by FAPESP - Fundação de Amparo à Pesquisa do Estado de São Paulo (Foundation for Support Research of the State São Paulo), projects No. 2010/50948-2 and No. 2010/50347-9. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Automorphic equivalence of the representations of Lie algebras Article published earlier |
| spellingShingle | Automorphic equivalence of the representations of Lie algebras Shestakov, I. Tsurkov, A. |
| title | Automorphic equivalence of the representations of Lie algebras |
| title_full | Automorphic equivalence of the representations of Lie algebras |
| title_fullStr | Automorphic equivalence of the representations of Lie algebras |
| title_full_unstemmed | Automorphic equivalence of the representations of Lie algebras |
| title_short | Automorphic equivalence of the representations of Lie algebras |
| title_sort | automorphic equivalence of the representations of lie algebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/152257 |
| work_keys_str_mv | AT shestakovi automorphicequivalenceoftherepresentationsofliealgebras AT tsurkova automorphicequivalenceoftherepresentationsofliealgebras |