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FACTORIZATION STRUCTURES WITH A 2-DIMENSIONAL FACTOR
O´SCAR CORTADELLAS IZQUIERDO, JAVIER LO´PEZ PEN˜A, AND GABRIEL NAVARRO
Abstract. We introduce the notion of quantum duplicates of an (associative, unital)
algebra, motivated by the problem of constructing toy-models for quantizations of cer-
tain configuration spaces in quantum mechanics. The proposed (algebraic) model relies
on the classification of factorization structures with a two-dimensional factor. In the
present paper, main properties of this particular kind of structures are determined, and
we present a complete description of quantum duplicates of finite set algebras. As an ap-
plication, we obtain a classification (up to isomorphism) of all the algebras of dimension
4 (over an arbitrary field) that can be factorized as a product of two factors.
Introduction
Consider a manifold M representing some physical system. From a dual point of view,
this manifold can also be represented by some algebra of functions A (that could be taken,
for instance, to be A = C∞(M), the algebra of smooth functions on M) over some base
field k (usually k = R or k = C when dealing with actual physical systems). Now, if we
want to describe a physical system consisting on two “parallel” sheets M1 and M2, both
equal to M , the natural algebra to consider from a classical point of view is the direct
product A× A, which is isomorphic to the algebra A⊗ k2.
M1
M2
(a) Standard commutative duplicate
•
• •
•
•
•
• • •
M1
M2
(b) Noncommutative duplicate
However, assume that we keep studying this system while making the two sheets come
closer and closer. After we reach certain critical distance, one could expect that mea-
surements on M1 would start to interfere with measurements on M2, so that they do not
commute anymore. Under these premises, the algebra A⊗ k2 can no longer be assumed
(since it is commutative) to represent the measurements on the “parallel, but very close”
sheets M1 and M2. In the present work, our aim is to define and study, inside an algebraic
framework, a suitable replacement for the algebra A⊗ k2 in this quantized situation, i.e.
Keywords and phrases: twisting maps, factorization structures, twisted tensor product, quiver, path
algebra.
2000 Mathematics Subject Classification: 16S35, 16G20, 16W35.
Research supported by MTM2007-66666, FQM-1889 and FQM-266. J. Lo´pez was also supported by
Max-Planck Institut fu¨r Mathematik in Bonn.
1

2 O´SCAR CORTADELLAS IZQUIERDO, JAVIER LO´PEZ PEN˜A, AND GABRIEL NAVARRO
a (noncommutative) algebra that may be regarded as a deformation of A⊗k2 and retains
similar structural properties.
In particular, we would like to keep the linear dimension of the algebra A ⊗ k2, since
the fact that a system starts showing up quantum effects should not alter the number of
(linearly independent) quantities that we can meassure on it. Thus, we are led to finding
an algebra X, a “quantum duplicate” of A, which is isomorphic, as a vector space, to
A⊗ k2. In particular, if A is finite dimensional, the dimension of a quantum duplicate of
A should be twice the dimension of A.
The method we propose in order to build these so-called quantum duplicates is the use
of a factorization structure, or twisted tensor product involving the algebra A plus
a (necessarily commutative) 2–dimensional factor. The number of ways in which we can
choose this two dimensional factor depends on the field k. More concretely, if k admits
a degree 2 field extension k¯, (as happens, for instance, with the real numbers), then we
have three kinds of non-isomorphic algebras of dimension 2 (over k), namely:
(1) The trivial direct product k2,
(2) Quadratic field extensions of k,
(3) The ring of dual numbers k[ξ] ∼= k[x]/(x2).
On the other hand, if k does not admit nontrivial extensions (for instance, if k = C),
there are only two possible algebras: the direct product k2 and the dual numbers.
From a purely algebraic point of view, the notion of twisted tensor product comes
directly from the factorization problem :
Given some kind of (algebraic) object, is it possible to find to suitable subob-
jects, having minimal intersection and such that they generate our original
object?
The factorization problem has been intensively studied in the case of groups, coalgebras
and Hopf algebras, and algebras (cf. for instance [19], [4], [5], [1]). In the particular case
of algebras, a well known result (independently proven many times) establishes a one-
to-one correspondence between the set of factorization structures admitting two given
algebras A and B as factors and the set of so-called twisting maps, which are linear
maps τ : B ⊗ A → A ⊗ B satisfying certain compatibility conditions with respect with
the units and products of A and B.
Henceforth, the problem of constructing factorization structures with given factors boils
down to the problem of finding all the existing twisting maps for those factors. Under
suitable, very mild, conditions (for instance, whenever A and B are affine algebras), the
set T (A,B) of all the twisting maps τ : B ⊗ A → A⊗ B is an algebraic variety, and two
interesting problems arise:
Problem 0.1. Is it possible to describe explicitly the variety T (A,B)?
Problem 0.2. Once the variety T (A,B) is known, is it possible to determine which points
of the variety give rise to isomorphic algebras?
These two problems, even in the simplest cases, turn out to be very difficult. Though
there are many different methods that produce twisted tensor products of two given
algebras, not a single one that produces all the existing ones is known, let alone describing
the properties of the algebraic variety. Even harder is the problem of the determination

FACTORIZATION STRUCTURES WITH A 2-DIMENSIONAL FACTOR 3
of the isomorphism classes of algebras obtained from the same factors through different
tensor products, or finding any description of these isomorphism classes in terms of the
variety T (A,B).
Recently, Cibils showed in [7] that the set T (k2, A) of twisted tensor products between
any algebra A and the commutative, semisimple algebra k2 (also called the set of 2–
interlacings) is in one to one correspondence with couples of linear endomorphisms of
the algebra A satisfying certain conditions. If we take A = kn, these couples of linear
maps can be described by combinatorial means using certain families of colored quivers,
and this description gives a simple way to describe all the twisted tensor products kn⊗τ k2,
up to isomorphism (cf. [7, 15]). Some other partial steps in the classification problem for
factorization structures have been undertaken in [3] and the final sections of [11].
Beyond the physical motivation originating the notion of noncommutative duplicate
(defined by Cibils), the remaining choices of two-dimensional factors have their own source
of interest. In particular, the building of duplicates by means of dual numbers has yet
another physical interpretation. Being the algebra of dual numbers one of the simplest
examples of non-trivial superspaces, where the ξ–direction can be reinterpreted as the
fermionic direction, and the scalar component as the bosonic direction, the procedure of
duplicating a manifold using dual numbers can be regarded as a simple way of adding a
superstructure to the physical system described by the manifold. Moreover, quantizations
of the tensor product A⊗ k[ξ] admit an interpretation as infinitesimal deformations
(for a central formal parameter) in the framework of formal deformation theory (cf. [9]),
so twisted tensor products A⊗τ k[ξ] may be regarded as infinitesimal deformations with
respect to a non-central parameter, with the added advantage of the existence of such
a kind of deformations for algebras that are rigid from the formal point of view (like
separable algebras). Finally, the remaining case of quantum duplicates obtained using
a quadratic field extension have similar properties to complexifications of real algebras,
hinting the possibility of thinking about them as noncommutative scalar extensions.
In Section 1 we introduce the definition of quantum duplicates of an algebra A as
twisted tensor products A⊗τ B where B is any two-dimensional algebra, characterize the
set of twisting maps as a set of couples (f, δ) where f is an algebra endomorphism of A
and δ is an f -derivation, satisfying certain compatibility conditions. We also show how
to lift certain classes of endomorphisms and involutions to such kind of twisted tensor
products, with special attention to the case in which the algebra B is a quadratic field
extension of k, obtaining a simple criterion (similar to the one existing for telling whether
a vector space is a complexification of a real one) for determining whether or not certain
k-algebras factorize as quantum duplicates with a quadratic field extension.
Sections 2 and 3 deal with the classification problem of isomorphism classes of the
resulting twisted tensor products. More concretely, in Section 2 we classify (up to iso-
morphism) all quantum duplicates of the finite set algebras kn by means of combinatorial
techniques, obtaining results similar to the ones contained in [7].
The paper concludes with the complete classification, in Section 3, of all the algebras of
dimension 4 that can be obtained as a twisted tensor product, that should necessarily be
of two factors of dimension 2. For the case of an algebraically closed field, the resulting
algebras are displayed inside the diagram of all the four dimensional algebras, showing
that no apparent pattern relating those factorizable algebras appears.

4 O´SCAR CORTADELLAS IZQUIERDO, JAVIER LO´PEZ PEN˜A, AND GABRIEL NAVARRO
Along this paper, k will denote a field. All the algebras will be unital, associative
k-algebras. The tensor product will be taken over k and all maps will be k-linear. An
algebra X is a factorization structure of the algebras A and B if there exist two
injective algebra maps iA : A →֒ X and iB : B →֒ X and the map ϕ : A⊗B → X defined
by ϕ(a⊗ b) = iA(a) · iB(b) is a linear isomorphism.
A k-linear map τ : B ⊗A → A⊗B is said to be a (unital) twisting map if
(0.1) τ ◦ (B ⊗ µA) = (µA ⊗B) ◦ (A⊗ τ) ◦ (τ ⊗ A)
(0.2) τ ◦ (µB ⊗ A) = (A⊗ µB) ◦ (τ ⊗ B) ◦ (B ⊗ τ),
(0.3) τ(1⊗ a) = a⊗ 1, τ(b ⊗ 1) = 1⊗ b for all a ∈ A, b ∈ B.
where µA and µB stand for the product of A and B respectively and A and B stand for
the identity maps on each algebra.
The map µτ := (µA ⊗ µB) ◦ (A⊗ τ ⊗ B) defines an associative product over A⊗ B if,
and only if, τ is a twisting map. So we can endow the vectorial space A ⊗ B with this
product and get a new algebra (A⊗B, µτ ), which will be denoted by A⊗τ B.
Proposition 0.3 ([6], [16]). Let (C, iA, iB) a factorization structure of C with factors A
and B. Then there exists a unique twisting map τ : B ⊗ A → A ⊗ B such that C is
isomorphic to A⊗τ B as a twisted tensor product.
When working with path algebras kQ of quivers Q (cf. for instance [2]) we shall assume
that arrows are multiplied as if they were maps, for instance, in the quiver
/.-,()*+1 α // .-,()*+2
β // .-,()*+3
the length-two path from /.-,()*+1 to /.-,()*+3 will be written as βα. All along the work, the ideal (Q≥2)
of KQ generated by all paths in Q of length greater than one will play a fundamental roˆle.
The quotient kQ/(Q≥2) will be denoted by kQ<2. We shall denote by Qop the opposite
quiver of Q, that is, the quiver that has the same set of vertices while arrows are reversed.
1. Generalities about quantum duplicates
Let A and B be two (unitary) k-algebras, with dimkB = 2, so that we may consider
it given as a quotient B = k[x]/(p(x)), where p(x) is a polynomial of degree two. All
along this work we write it as p(x) = x2 − αx + β where α, β ∈ k. We also denote by q
the polynomial q(x) = x2 + αx+ β, that is, the polynomial whose roots (in the algebraic
closure of k) are the opposite to the ones of p.
1.1. Basic definitions and properties. Our purpose is to describe the twisting maps
between A and B, that is, the k-linear maps
(1.4) τ : k[x]/(p(x))⊗ A −→ A⊗ k[x]/(p(x))
verifying the twisting conditions (0.1) and (0.2). Following the method developed in [7],
it is worth noting that A⊗ k[x]/(p(x)) ∼= A[x]/(p(x)) and then a twisting map as in (1.4)

FACTORIZATION STRUCTURES WITH A 2-DIMENSIONAL FACTOR 5
is determined by the values τ(x⊗ a) corresponding to the product xa in A[x]/(p(x)). For
any a ∈ A, we put τ(x⊗ a) = xa = δ(a) + f(a)x. Then,
x2a = (αx− β)a = α(δ(a) + f(a)x)− βa = αδ(a)− βa + αf(a)x(1.5)
x(xa) = x(δ(a) + f(a)x) = δ2(a)− βf 2(a) + fδ(a)x+ δf(a)x + αf 2(a)x(1.6)
for any a ∈ A. Thus the associativity x2a = x(xa) produces the equalities
p(δ) = δ2 − αδ + βidA = βf 2(1.7)
fδ + δf = α(f − f 2)(1.8)
As in the proof of [7, Proposition 2.10], the condition x(ab) = (xa)b for any a, b ∈ A,
produces that f : A → A is a morphism of algebras and δ : A → A is a left f -derivation.
It is also clear that if we have such an f and δ verifying (1.7) and (1.8), the linear map
defined by τ(x⊗ a) = δ(a)⊗ 1 + f(a)⊗ x is a twisting map between A and B. In other
words, we have proven the following result.
Lemma 1.1. The set of twisting maps τ : k[x]/(p(x))⊗A −→ A⊗ k[x]/(p(x)) is in one-
to-one correspondence with the set of pairs (f, δ), where f : A → A is an endomorphism
of algebras and δ : A → A is a left f -derivation, verifying the conditions
p(δ) = βf 2 and fδ + δf = α(f − f 2).
A twisting tensor product of the form A⊗τ B, where B is an algebra of dimension two,
will be referred by a quantum duplicate of A.
Example 1.2. If α = 1 and β = 0, then B ∼= k2 and we recover the approach given by
Cibils [7], since, in that case, the twisting maps are in one-to-one correspondence with the
pairs (f, δ) verifying that p(δ) = δ2 − δ = 0 and δf + fδ = f − f 2.
Remark 1.3. The previous lemma admits a refinement when the characteristic of k is
different from two. Taking the linear transformation φ(x) = (α/2)x + 1, we obtain that
p(φ(x)) = x2+γ and k[x]/(p(x)) ∼= k[x]/(x2+γ). Thus equations (1.7), (1.8) are rewritten
as follows:
δ2 = γ(f 2 − idA)(1.9)
fδ + δf = 0(1.10)
1.2. Characterization of certain quantum duplicates. When we have a real vector
space V , we can construct the complexification of V , called V C, as the tensor product
V ⊗R C . The original vector space V remains a real vector subspace of V C, and can
be recovered if we take advantage of the canonical conjugation map χ : V C → V C given
by χ(v ⊗ z) := v ⊗ z¯. When the two dimensional algebra l = k[x]/(p(x)) = k(η) is
a quadratic (Galois) field extension, twisted tensor products A ⊗τ l can be regarded as
”noncommutative scalar extensions” to l. Our goal in this subsection is to stablish a
result that guarantees us that a given k–algebra B can be factorized as A ⊗τ l for some
twisted tensor product τ . In the forecoming, l will be assumed to be some fixed (Galois)
quadratic field extension of k, with k a field of characteristic different from 2.

6 O´SCAR CORTADELLAS IZQUIERDO, JAVIER LO´PEZ PEN˜A, AND GABRIEL NAVARRO
Lemma 1.4. Let B a k–algebra endowed with a right l–module structure B ⊗ l → B.
Then the map i : l → B given by z 7→ 1B · z is an injective algebra map.
Proof. It is a straightforward calculation. 
Assume now that B is endowed with a conjugation map σ : B → B satisfying the
following conditions:
(1) σ2 = IdB,
(2) σ(ab) = σ(a)σ(b) for all a, b ∈ B.
(3) σ(aλ) = σ(a)λ¯ for all λ ∈ l, a ∈ B
and let
A := Bσ = {a ∈ B| σ(a) = a}.
We have an obvious algebra map (the inclusion map) iA : A → B.
Lemma 1.5. The mapping
ϕ : A⊗ l −→ B
a⊗ z 7−→ a · z
is a linear isomorphism.
Proof. Let b ∈ B, consider the elements
a1 :=
b + σ(b)
2
and a2 :=
b− σ(b)
2η
then, obviously, b = a1 + a2 · η. Now,
σ(a1) =
σ(b) + σ(σ(b))
2
=
σ(b) + b
2
= a1 and σ(a2) =
σ(b)− b
−2η = a2
and therefore a1, a2 ∈ A. Thus b = ϕ(a1 ⊗ 1 + a2 ⊗ η) and ϕ is surjective.
Let α ∈ B ⊗ C, we may write α = a ⊗ +b ⊗ η with a, b ∈ A, and then ϕ(α) =
ϕ(a ⊗ 1) + ϕ(b ⊗ η) = a + b · i. Assume ϕ(α) = 0, that is, a + b · η = 0. Applying σ,
a− b · η = 0. As a consequence, a = b = 0 and thus ϕ is injective.

As a consequence of the previous lemmas, by using [6, Proposition 2.7] (see also [20] or
[17]), we obtain the following corollary:
Corollary 1.6. An algebra B factorizes as a twisted tensor product A⊗τ l for some twist-
ing map τ if, and only if, B is endowed with a right l-module structure and a conjugation
map satisfying (1), (2) and (3).
1.3. Lifting of endomorphisms and involutions. Let A be an algebra, ϕ : A → A
an algebra map, and A ⊗τ B a quantum duplicate of A, with B = k[x]/(p(x)), induced
by the couple (f, δ). The map ϕ admits a natural lifting ϕ˜ : A ⊗τ B → A ⊗τ B defined
by ϕ˜(a⊗ b) := ϕ(a)⊗ b.
Assume that the lifting ϕ˜ is an algebra map. Then on the one hand we have
ϕ˜(xa) = ϕ˜(f(a)x + δ(a)) = ϕ(f(a))x+ ϕ(δ(a)),

FACTORIZATION STRUCTURES WITH A 2-DIMENSIONAL FACTOR 7
while, in the other one,
ϕ˜(xa) = ϕ˜(x)ϕ˜(a) = xϕ(a) = f(ϕ(a))x+ δ(ϕ(a)).
Thus, we have stablished the following result:
Theorem 1.7. An endomorphism ϕ of A can be lifted to an endomorphism of the twisted
tensor product A⊗(f,δ) B if, and only if
fϕ = ϕf, and δϕ = ϕδ.
In other words, the set of endomorphisms of A admitting a lifting to A⊗(f,δ) B coincides
with the set of morphisms that simultaneously commute with f and δ.
Assume now that A is an algebra endowed with an involution a 7→ a∗, and that l = k(η)
is a quadratic field extension of k, with char(k) 6= 2, so that the Galois automorphism
η 7→ −η gives us an involutive k–automorphism of l. Assume also that we are given a
twisting map τ induced from a couple (f, δ). Our purpose is to determine conditions in
f and δ in such a way that we can ensure the existence of an involution j in A⊗(f,δ) l.
First, recall that for any linear endomorphism ϕ of A we can define its conjugate ϕ by
ϕ(a) := ϕ(a∗)∗. An easy computation shows that if f is an algebra map, then so is f , and
if δ is a left f -derivation, then the conjugate δ is a right f -derivation.
Any involution j defined on a twisted tensor product A⊗τ B which is compatible with
the ones existing in A and B must satisfy
j(a⊗ b) = j((a⊗ 1)(1⊗ b)) = j(1⊗ b)j(a⊗ 1) =
= (1⊗ jB(b))(jA(a)⊗ 1) = τ(jB(b)⊗ jA(a)),
i.e., we necessarily have j = τ ◦ (jB ⊗ jA) ◦ τBA, where τBA is the flip map. Henceforth,
the involutions can be combined in a unique compatible way, provided that the twisting
map τ satisfies the involutive condition (τ ◦ (jB ⊗ jA) ◦ τBA)2 = IdA⊗B (cf. [21]). In our
particular situation, where the involution in B is given by η 7→ −η and the twisting map
is giving in terms of f and δ, the lifting of the involution of A becomes
(aη)∗ = ηa∗ = −ηa∗ = −f(a∗)η − δ(a∗)
so the involutive condition is
aη = ((aη)∗)∗ = (−f(a∗)η − δ(a∗))∗ =
= ηf(a∗)∗ + δ(a∗)∗ =
= f(f(a∗)∗)η + δ(f(a∗)∗)− δ(a∗)∗,
for all a ∈ A. We have proved the following result:
Proposition 1.8. If (A, ∗) is an algebra with involution, l = k(η) is a quadratic field
extension (with Galois automorphism η 7→ −η), and we have a twisting map induced by a
couple (f, δ), then the involution of A lifts in a compatible way to an involution of A⊗(f,δ) l
if, and only if, the following conditions are satisfied:
f ◦ f = IdA,
δ ◦ f = δ.

8 O´SCAR CORTADELLAS IZQUIERDO, JAVIER LO´PEZ PEN˜A, AND GABRIEL NAVARRO
2. Quantum duplicates of kn
In this section we describe and classify all quantum duplicates of kn for some natural
number n ≥ 2. Denote by {e1, . . . , en} the canonical basis of kn. Following Cibils’
procedure [7], the set of algebra morphisms f : kn → kn is in one-to-one correspondence
with the set of set maps ϕ : {e1, . . . , en} → {e1, . . . , en}, where, to each set map ϕ, we
associate the algebra morphism f defined as
(2.11) f(ei) =
∑
{ej |ϕ(ej)=ei}
ej , for any i = 1, . . . , n.
This is also in one-to-one correspondence with the set of quivers with n vertices verifying
that from each vertex starts precisely one arrow. From now on we denote by Qf the quiver
associated to the endomorphism f (by meanings of ϕ). We recall that Qf has {e1, . . . , en}
as set of vertices, and there is an arrow ei → ej in Qf if, and only if, ϕ(ei) = ej . For the
convenience of the reader we introduce the following notation:
Definition 2.1. For any positive integer s, we say that a connected component of Qf is
an s-cycle if the component has the following shape:
 ◦
~~
~~
~
◦oo
// ◦
@
@@
@@
◦
__@@@@@
oo
◦ ◦
??~~~~~
OO OO
where the central cycle has s vertices and, by a diagram
◦ oo
we mean a tree with ascendent orientation. Observe that, for any s-cycle, if we remove
the arrows of the central cycle, we obtain s disjoint trees with ascendent orientation. For

FACTORIZATION STRUCTURES WITH A 2-DIMENSIONAL FACTOR 9
instance, in the figure below we have a 3-cycle.
◦
◦ ◦
i1
i3
i2
◦ ◦
◦ ◦ ◦
◦
◦ ◦
◦
a1

a2
ii
a3
MM
)
))
)




)
))
)





%
%%
%



⇒ ◦
◦ ◦
◦
i1
i3
i2
◦
◦ ◦ ◦
◦
◦ ◦
◦
)
))
)




)
))
)





%
%%
%



We say that a connected component of Qf is an strict s-cycle if such component is
the quiver A˜s, i.e., the quiver
◦
zzuuu
uu
◦oo ◦oo
◦ ◦
ddIIIII
◦ // ◦ // ◦
::uuuuu
with s vertices. Observe that, according to this nomenclature, an strict 1-cycle component
is nothing more than a single vertex with a loop whilst an strict 2-cycle is the round-trip
quiver.
Therefore we have the following result, see [12] or [10]:
Lemma 2.2. For any algebra map f : kn → kn, each connected component of the quiver
Qf is a s-cycle for certain integer s.
With respect to the f -derivations, since kn is separable, all of them are inner, so each
δ ∈ Der(kn, f(kn)) is determined by certain element a = (a1, . . . , an) ∈ kn such that
(2.12) δ(ei) = (f(ei)− ei)a =
∑
ϕ(ej)=ei
ejaj − eiai, for any i = 1, . . . , n.
Observe that if ei is a loop vertex, then δ(ei) does not depend of the value of ai. For
that reason, we normalize the element a by taking ai = 0 for any loop vertex ei. Then
each inner derivation is given by a unique normalized element of kn (that, following the
nomenclature of [7], we shall call a coloration of Qf )
In order to characterize the derivations (=colorations) verifying (1.7) and (1.8), we will
need the following formulae:
f 2(ei) =
∑
ϕ2(ej)=ei
ej(2.13)
δ2(ei) =
∑
ϕ2(ek)=ei
akaϕ(ek)ek −
∑
ϕ(ej)=ei
aj(aj + ai)ej + a2i ei(2.14)
for any vertex ei of Qf . Therefore, we rewrite (1.7) and (1.8) as follows:

10 O´SCAR CORTADELLAS IZQUIERDO, JAVIER LO´PEZ PEN˜A, AND GABRIEL NAVARRO
∑
ϕ2(ek)=ei
(akaϕ(ek) − β)ek −
∑
ϕ(ej)=ei
(aj(aj + ai) + αaj)ej + (a2i + αai + β)ei = 0(2.15)
∑
ϕ2(ek)=ei
(ak + aϕ(ek) + α)ek −
∑
ϕ(ej)=ei
(aj + ai + α)ej = 0(2.16)
for any vertex ei of Qf . It is easy to see that the study of the possible colorations of the
quiver Qf can be reduced to the study of each connected component separately. Let us
start by showing up the case of an strict cycle.
Proposition 2.3. Let p(x) = x2 − αx + β ∈ k[x] be a polynomial of degree two. The set
of twisting maps (f, δ) : kn ⊗ k[x]/(p(x)) → k[x]/(p(x))⊗ kn, where Qf is a strict s-cycle
for some s > 1, is the following:
(1) If Qf is a strict 2-cycle, it is a one-parameter family of twisting maps, indexed in
the field, given by the colorations
'&%$ !"#a **/.-,()*+bjj where a + b = −α.
(2) If Qf is a strict s-cycle with s > 2, each twisting map is given by coloring each
vertex of the cycle with a root of q satisfying that, if a vertex is colored by a root r1,
the immediate predecessor and the immediate successor of such vertex are colored
by r2 = −α − r1.
/.-,()*+r1
yysss
sss
/.-,()*+r2oo /.-,()*+r1oo
/.-,()*+r2 /.-,()*+r2
eeKKKKKK
/.-,()*+r1 // /.-,()*+r2 // /.-,()*+r1
99ssssss
As a consequence, if s is even, there are as many colorations as different roots of
p; and, if s is odd, there exists a coloration (which would then be unique) if, and
only if, p has a unique (double) root. In particular, there are at most two twisting
maps.
Proof. Let us suppose that we treat with a strict 2-cycle, i.e., a connected component
with shape
◦ ** ◦jj .
We also denote by ei and ej the vertices of this component. Then (2.15) and (2.16) applied
to the vertex ei reduces to
(aiaj − β)ei − aj(aj + ai + α)ej + (a2i + αai + β)ei = 0(2.17)
(ai + aj + α)ei − (ai + aj + α)ej = 0(2.18)
Clearly, these equations hold if and only if ai + aj = −α.
Let us now consider an strict s-cycle with s > 2. Then, for any vertex ei of Qf ,
the component i of (2.15) provide us the equation q(ai) = 0. Therefore (2.15) holds if,
and only if, each vertex is colored with a root of q. Now, denote by ej the immediate
predecessor of ei. Then the component i of (2.16) provides us the equation ai+aj+α = 0.
Similarly, we may prove that the immediate successor of ei must be colored by −α−ai. 

FACTORIZATION STRUCTURES WITH A 2-DIMENSIONAL FACTOR 11
Theorem 2.4. Let p(x) = x2 − αx + β ∈ k[x] be a polynomial of degree two. The set
of twisting maps τ : kn ⊗ k[x]/(p(x)) → k[x]/(p(x))⊗ kn is in one-to-one correspondence
with the set of colored quivers Qf , where each connected component Qi of Qf is colored
according to the following rules:
(1) If Qi is a 1-cycle, any immediate predecessor of the loop vertex must be colored
by a root of q. Given an immediate predecessor of the loop vertex e colored by
r1, other vertex d in the same tree as e must be colored by r1 if t is even, and by
r2 = −α− r1 if t is odd, where t is the length of the path from d to e.
/.-,()*+r1
%%KK
KKK
K /.-,()*+r2
yysss
sss
/.-,()*+r1oo
/.-,()*+r2 // .-,()*+0
 /.-,()*+r1oo
/.-,()*+r1
99ssssss /.-,()*+r2
eeKKKKKK
As a consequence, each connected component of this kind increases the number
of twisting maps by at most a factor of 2c, where c is the number of immediate
predecessors of the loop vertex. Of course, if q has no roots in k, the only connected
components of this kind that may show up are strict 1-cycles.
(2) If Qi is a strict 2-cycle, any valid coloration must satisfy
'&%$ !"#a **/.-,()*+bjj where a + b = −α.
Connected components of this kind give rise to a one-parameter family of twisting
maps, the parameter ranging in the field,
(3) If Qi is either a non-strict 2-cycle or a s-cycle with s > 2, any valid coloration is
determined by choosing a coloration of the central cycle. Each vertex of the central
cycle must be colored by roots of q in alternating order, meaning that, if a vertex
is colored by a root r1, any immediate predecessor and the immediate successor of
such vertex are colored by r2 = −α − r1. Moreover, for any vertex e in the cycle,
the tree attached to e is also colored by alternating roots of q; i.e. if e is colored
by r1, any other vertex d in the same tree as e must be colored by r1 if t is even,
and by r2 = −α − r1 if t is odd, where t is the length of the path from d to e.
/.-,()*+r1
yysss
sss
/.-,()*+r2oo /.-,()*+r1oo /.-,()*+r1
yysss
sss
/.-,()*+r2oo
/.-,()*+r2 /.-,()*+r2
eeKKKKKK /.-,()*+r1oo /.-,()*+r2oo
/.-,()*+r1 // /.-,()*+r2 // /.-,()*+r1
99ssssss /.-,()*+r1
eeKKKKKK
As a consequence, if s is even, there are as many colorations as different roots of
p; and, if s is odd, there exists a (unique) coloration if, and only if, p has a unique
root. In particular, this kind of components increase the number of twisting maps
by at most a factor of 2.
Proof. The second case is proven in Proposition 2.3. Let us suppose that Qf is a 1-cycle.
The loop vertex is colored by 0 by the arguments given above. Since any of the other
vertices ei is not inside a cycle, by applying the same arguments as in Proposition 2.3, we
obtain that q(ai) = 0 and thus all the vertices remaining are colored by a root of q. Now,
let us consider an arrow /.-,()*+ei // 76540123ej inside one of the trees engaged to the loop vertex.
Then (2.16) in the component i provides us the equation ai + aj +α = 0. That is, for one

12 O´SCAR CORTADELLAS IZQUIERDO, JAVIER LO´PEZ PEN˜A, AND GABRIEL NAVARRO
of these trees, if one level is colored by a root r1, the next level is colored by −α− r1, see
Figure 1.
/.-,()*+r1
/.-,()*+r2
=={{{{{{ /.-,()*+r2
aaCCCCCC
/.-,()*+r1
??~~~~~~ /.-,()*+r1
__@@@@@@
/.-,()*+r1
__@@@@@@
where r2 = −α− r1
/.-,()*+r2
OO
/.-,()*+r2
OO
/.-,()*+r1
=={{{{{{ /.-,()*+r1
aaCCCCCC
Figure 1. Coloration of a tree engaged to the cycle
Finally, suppose that Qf is as stated in (3). Following the same proof as in Proposition
2.3, we deduce that the central cycle is colored by choosing to each vertex a root of q and,
if a vertex is colored by a root r1, the immediate predecessor and the immediate successor
of such vertex are colored by −α − r1. Now, applying the above reasoning to any of the
trees engaged to a vertex of the cycle, we get the statement of (3). It is clear that these
conditions are sufficient to have a coloration, thus the proof is completed. 
Corollary 2.5. Let p ∈ k[x] be an irreducible polynomial of degree two. The set of twisting
maps τ : kn⊗k[x]/(p(x)) → k[x]/(p(x))⊗kn is in one-to-one correspondence with the set
of colored quivers Qf such that each connected component is either a single loop vertex,
or a round-trip quiver '&%$ !"#a
**/.-,()*+bjj where a + b = −α.
Proof. By Theorem 2.4, if a connected component of Qf is neither a strict 1-cycle nor a
strict 2-cycle, the vertices are labeled by a root of q in k. Therefore there is no derivations
for an algebra map f producing such components. Thus all components of f must be
either a single loop vertex or a round-trip quiver. Now, apply Theorem 2.4 for such
cases. 
Corollary 2.6. Let p ∈ k[x] be a polynomial of degree two such that it has a unique root
s in k. The set of twisting maps τ : kn ⊗ k[x]/(p(x)) → k[x]/(p(x))⊗ kn is in one-to-one
correspondence with the set of colored quivers Qf such that:
(1) Qf is the quiver associated to a set map {1, . . . , n} → {1, . . . , n}.
(2) The coloration of a connected component different of a round-trip quiver is given
by putting 0 in the loop vertex and −s in the others.
(3) A round-trip connected component is colored by '&%$ !"#a
**/.-,()*+bjj , where a + b = −2s.
Now we are going to describe the isomorphism classes of the algebras that we have
obtained. There is no loss of generality on assuming that the quiver Qf is connected;
otherwise we reason over every connected component, and the resulting algebra will be
the direct product of the algebras associated to the distinct components (cf. [7]). As the
reader should note, the case of a round-trip quiver becomes very special. For that reason
we study this case separately.

FACTORIZATION STRUCTURES WITH A 2-DIMENSIONAL FACTOR 13
Proposition 2.7. Let p(x) = x2 − αx + β ∈ k[x] be a polynomial of degree two. The
twisted tensor product k2 ⊗(f,δ) k[x]/(p(x)), where (f, δ) is the twisting map associated to
the quiver
/.-,()*+1
**/.-,()*+2jj colored by '&%$ !"#a
**/.-,()*+bjj , where a + b = −α,
is isomorphic to one of the following algebras:
(1) The matrix ring M2(k) if a and b are not roots of q.
(2) The quotient algebra kQ<2, otherwise, where Q is the round-trip quiver.
Proof. Let us assume that a and b are not roots of q. In order to show the isomorphism,
it is enough to give the image of e1 ⊗ 1, e2 ⊗ 1 and 1 ⊗ x which, for brevity, we denote
simply by e1, e2 and x, respectively. Since e1 + e2 = 1, we map
e1 7→
(
1 0
0 0
)
e2 7→
(
0 0
0 1
)
x 7→
(
−a 1
−q(a) −b
)
Since q(a) 6= 0, this provides us an isomorphism of algebras.
Now, suppose that a and b are roots of q. Let us denote the elements of Q as follows:
'&%$ !"#u
R
++ '&%$ !"#v
S
kk
Then we consider the isomorphism of k-algebras Φ : kQ<2 → k2 ⊗(f,δ) k[x]/(p(x)) given
by:
u 7→ e1 v 7→ e2 R 7→ e2(x+ b) = (x+ a)e1 and S 7→ e1(x+ a) = (x+ b)e2
This is clearly well-defined. For instance, observe that
Φ(R)Φ(S) = (e2x + be2)(e1x+ ae1) = −q(a)e2 = 0,
as a is a root of q. This completes the proof. 
For the convenience of the reader, in order to state the main result of this section, we
remind the following quiver-transformation introduced by Cibils [7]. Let us consider an
1-cycle Q colored by means of the procedure explained above. Then we construct the
forest Q̂ obtained as follows:
• Remove the loop vertex and all the arrows ending at it. This produces a finite
number of trees.
• Insert two isolated vertices, numbered by 1 and 2.
• For each tree of the ones obtained above, insert an arrow from the root node to 1
if that vertex is colored by r1, or to 2 if it is colored by r2.
Theorem 2.8. Let p(x) = x2 −αx+ β ∈ k[x] be a polynomial of degree two. The twisted
tensor product R = kn ⊗(f,δ) k[x]/(p(x)), where (f, δ) is the twisting map associated to a
connected colored quiver Q, is isomorphic to the following algebra:
(a) When Q is an strict 2-cycle,
(i) If Q is colored by roots of q, then R ∼= kQ<2.
(ii) Otherwise R ∼= M2(k).
(b) When Q is a 1-cycle,
(i) If p has no root, n = 1 and therefore R ∼= k[x]/(p(x)).

14 O´SCAR CORTADELLAS IZQUIERDO, JAVIER LO´PEZ PEN˜A, AND GABRIEL NAVARRO
/.-,()*+0

/.-,()*+r1
==|||||| /.-,()*+r2
OO
/.-,()*+r2
aaBBBBBB
/.-,()*+r2
=={{{{{{ /.-,()*+r2
OO
/.-,()*+r1
OO
/.-,()*+r1
aaCCCCCC
/.-,()*+r1
OO
/.-,()*+r2
aaCCCCCC
/.-,()*+r2
OO
/.-,()*+r2
=={{{{{{ /.-,()*+r2
OO
/.-,()*+r1
OO
/.-,()*+r1
aaCCCCCC
=⇒
/.-,()*+1 /.-,()*+2
/.-,()*+r1
OO
/.-,()*+r2
==|||||| /.-,()*+r2
OO
/.-,()*+r2
=={{{{{{ /.-,()*+r2
OO
/.-,()*+r1
OO
/.-,()*+r1
aaCCCCCC
/.-,()*+r1
OO
/.-,()*+r2
aaCCCCCC
/.-,()*+r2
OO
/.-,()*+r2
=={{{{{{ /.-,()*+r2
OO
/.-,()*+r1
OO
/.-,()*+r1
aaCCCCCC
Figure 2. Cibils’ transformation
(ii) If p has a single root, then R ∼= k(Qop)<2.
(iii) If p has two different roots, then R ∼= k((Q̂)op)<2.
(c) Otherwise, R ∼= k(Qop)<2.
Proof. (a) is Proposition 2.7. (b)− (i) is given by Corollary 2.5 since Q must be a single
loop vertex. Then the statement follows trivially.
Let us prove (b) − (ii) and (c). We remind that each vertex in Qop is the target of a
unique arrow. For any vertex ei in Qop, the twisting map given by (f, δ), say τ , verifies
that
(2.19) τ(x⊗ ei) = δ(ei)⊗ 1 + f(ei)⊗ x =
∑
ej
ǫj(ej ⊗ 1)− ǫi(ei ⊗ 1) +
∑
ej
ej ⊗ x,
where the ej ’s are the target of the arrows starting at ei in Qop and, ǫi and ǫj are the labels
of ei and ej , respectively. Then we define a morphism Φ : kQop → kn ⊗(f,δ) k[x]/(p(x)) as
follows:
• For any vertex ei in Qop, we set Φ(ei) = ei ⊗ 1.
• For any arrow αi in Qop, we set Φ(αi) = ei ⊗ x+ ǫi(ei ⊗ 1), where ei is the target
of αi in Qop, and ǫi is its color.
It is a straightforward calculation to prove that Φ is a surjective algebra map such that
(Qop≥2) = Ker Φ, and it is left to the reader. For instance, by using (2.19),
Φ(αi)Φ(ei) = (ei ⊗ x + ǫi(ei ⊗ 1))(ei ⊗ 1) = −ǫi(ei ⊗ 1) + ǫ(ei ⊗ 1) = 0
if ei is not a loop vertex, whilst
Φ(ei)Φ(αi) = (ei ⊗ 1)(ei ⊗ x + ǫi(ei ⊗ 1)) = (ei ⊗ x+ ǫi(ei ⊗ 1)) = Φ(αi).
In order to prove (b)− (iii) we use a reformulation of the morphism Φ. Then we define
Φ̂ : K(Q̂)op → kn ⊗(f,δ) k[x]/(p(x)) as follows:
• For any vertex ei 6= /.-,()*+1 , /.-,()*+2 ; we set Φ̂(ei) = ei ⊗ 1.
• Φ̂(/.-,()*+1 ) = e0 ⊗
(
x−r1
r2−r1
)
and Φ̂(/.-,()*+2 ) = e0 ⊗
(
x−r2
r1−r2
)
, where e0 is the loop vertex in Q.
• For any arrow αi that does not start neither at /.-,()*+1 nor /.-,()*+2 , we set Φ̂(αi) = ei ⊗ x+
ǫi(ei ⊗ 1), where ei is the target of αi and ǫi is its color.
• For any arrow αk : /.-,()*+1 → ek, we set Φ̂(αk) = α
(
ek ⊗ x−r1r2−r1
)

FACTORIZATION STRUCTURES WITH A 2-DIMENSIONAL FACTOR 15
• For any arrow αk : /.-,()*+2 → ek, we set Φ̂(αk) = α
(
ek ⊗ x−r2r1−r2
)
We leave to the reader the details of the proof, see [7, Theorem 4.2] 
Corollary 2.9. Let p ∈ k[x] be an irreducible polynomial of degree two. Any twisted
tensor product
kn ⊗(f,δ) k ∼= M2(k)t × (k)r,
where k = k[x]/(p) and t and r are the number of strict 2-cycles and strict 1-cycles in
Qf , respectively.
Theorem 2.10. Any quantum duplicate of kn, kn ⊗τ k[k]/(p(x)), where p ∈ k[x] is a
reducible polynomial, is isomorphic to an algebra of type
(M2(k))t × kQ<2,
where t is a natural number, and Q an appropriate quiver.
Corollary 2.11. There is a finite number of quantum duplicates of kn for any fixed n > 2.
Example 2.12. Let us describe all quantum duplicates of k3, R = k3 ⊗τ k[x]/(p(x)). If
τ is given by a pair (f, δ), the only possibilities for the quiver Qf are:
(Q1)    (Q2)  //   (Q3)  //  // 
(Q4) 
)) ii oo (Q5)  //  oo (Q6) 
?
??
?

?? oo
(T )  )) ii 
We should also consider the Cibil’s transformation of the quivers with a 1-cycle com-
ponent
(Q̂1) ◦ ◦ ◦
◦ ◦ ◦
(Q̂2) ◦ // ◦ ◦
◦ ◦
(Q̂3) ◦ // ◦ // ◦
◦
(Q̂5) ◦ // ◦ ◦oo
◦
(Q˜5) ◦ ◦oo
◦ // ◦
(T̂ ) ◦ ** ◦jj ◦
◦
where Q̂5 and Q˜5 are the two possibilities depending on the coloration of Q5.
By Corollary 2.9, if p is irreducible, Qf must be Q1 or T . Then R ∼= (k[x]/(p(x)))3
or R ∼= M2(k) × k[x]/(p(x)). Otherwise, if Qf 6= T , R is isomorphic to one of the
(truncated) path algebras of the opposite quivers of Q1, Q2, Q3, Q4, Q5 ,Q6, Q̂1, Q̂2, Q̂3,
Q̂5 or Q˜5. In case of Qf = T , R depends on the coloration of the round-trip component
of T . Then we obtain that R is the truncated path algebra of T or T̂ if it is colored by
roots of q; or R ∼= M2(k)× k[x]/(x2), or R ∼= M2(k)× k2 if not.

16 O´SCAR CORTADELLAS IZQUIERDO, JAVIER LO´PEZ PEN˜A, AND GABRIEL NAVARRO
3. Factorization structures of dimension 4
The simplest nontrivial algebras that can be factorized as twisted tensor product ought
to have factors of dimension at least 2, and thus the dimension of the product has to
be greater or equal than 4. In the present Section, our purpose is to classify, up to
isomorphism, all the algebras of dimension 4 that can be factorized. This problem turns
out to have a strong dependence on the base field k. Basically we face two different
situations:
(1) The field k does not admit a field extension of degree 2 (for instance, if k is alge-
braically closed). In this case, there are only two different algebras of dimension
2 that can appear as factors, the semisimple algebra k2, and the algebra of dual
numbers k[ξ]. In the realization of these algebras as quotients k[x]/(p(x)), the
algebra k2 corresponds to the cases in which p has two distinct roots in k, and k[ξ]
to those for which p has a double root.
(2) If k admits quadratic extensions, or equivalently, there are polynomials of degree
2 with no roots in k, we have to take into account all these extensions as possible
factors. The number of non-isomorphic quadratic extensions may range among
one (for instance, if k = R the field of real numbers) to an infinite family of them
(as happens for k = Q the field of rational numbers).
Some of the possible combinations of the previous kinds of algebras to form 4 dimen-
sional factorization structures have already been classified, concretely all the algebras of
the form k2 ⊗τ k2 (classified in [7] and [15]), and all the factors of the form k2 ⊗τ A, with
A = k[x]/(p(x)) 2-dimensional, are described as a particular case of Theorem 2.8. For the
sake of completeness, we relist the resulting algebras (without any proof).
3.1. Twisted tensor products of the form k2 ⊗τ k2. Every twisted tensor product of
the form k2 ⊗τ k2 is isomorphic to one of the following algebras:
(1) The commutative algebra k4.
(2) The algebra of matrices M2(k).
(3) The quotient kQ<2 of the path algebra kQ of the round-trip quiver
Q = ◦ ** ◦jj
(4) The path algebra kQ˜ of the quiver
Q˜ =
◦
◦ // ◦
3.2. Twisted tensor products of the form k2 ⊗τ k[ξ]. Every twisted tensor product
of the form k2 ⊗τ k[ξ] is isomorphic to one of the following algebras:
(1) The commutative algebra k[ξ]× k[ξ].
(2) The algebra of matrices M2(k).
(3) The quotient kQ<2 of the path algebra kQ of the round-trip quiver
Q = ◦ ** ◦jj

FACTORIZATION STRUCTURES WITH A 2-DIMENSIONAL FACTOR 17
(4) The quotient kQ′<2 of the path algebra kQ′ of the quiver
Q′ = ◦ // ◦

3.3. Twisted tensor products of the form k[ξ]⊗τ k[ξ]. Since the algebra k[ξ] of dual
numbers is not separable, the description of twisting maps by means of endomorphisms
and derivations is not as simple as it is when we use k2. However, for this particular
situation, the equations derived from the twisting conditions are simple enough to solve
by a direct approach. If we consider the two copies of k[ξ] respectively generated by x
and y with x2 = y2 = 0, and identify x and y with their images in the twisted tensor
product k[ξ]⊗τ k[ξ], the twisting map is given by yx = a + bx + cy + dxy, and imposing
the twisting conditions we obtain a system of equations that can be reduced to
(3.20)



a(1 + d) = 0
b = 0
c = 0
and thus the variety T (k[ξ], k[ξ]) of twisting maps has two lines as irreducible components,
giving rise to two one-parameter families of algebras, corresponding respectively to the
solutions a = 0 and d = −1 of the previous system:
Aq := k〈x, y| x2 = y2 = 0, yx = qxy〉(3.21)
Xt := k〈x, y| x2 = y2 = 0, yx+ xy = t〉(3.22)
both families intersect at the algebra A−1 = X0 (which is the exterior algebra
∧
k2 of k2).
Lemma 3.1. For any t 6= 0, the algebra Xt is isomorphic to the matrix ring M2(k).
Proof. Just take the isomorphism given by
x 7→
(
0 1
0 0
)
, y 7→
(
0 0
t 0
)
.

The other irreducible component of the twisting variety actually gives rise to a complete
family of non-isomorphic algebras. More concretely, we have the following result (whose
proof is a straightforward computation):
Lemma 3.2. The algebras Aq and Ah are isomorphic if, and only if, q = h or q = h−1.
Remark 3.3. Remarkable algebras contained in the family Aq are A−1 =
∧ k2, the exte-
rior algebra of k2, and A1 ∼= k[x, y]/(x2, y2), the only commutative algebra in the family,
corresponding to the classical tensor product k[ξ] ⊗ k[ξ]. It is also worth noticing that,
unlike in the previous situations, the twisting variety T (k[ξ], k[ξ]) is connected.
3.4. Twisted tensor products of the form k2 ⊗τ l. Twisted tensor products of the
form k2 ⊗τ l, with l = k[x]/(p(x)) a quadratic field extension of k, are again classified
using Theorem 2.8, or more precisely its Corollary 2.9. According to it, all the resulting
algebras are of the following form:
(1) The commutative algebra k2 ⊗ l ∼= l × l.
(2) The matrix algebra M2(k).

18 O´SCAR CORTADELLAS IZQUIERDO, JAVIER LO´PEZ PEN˜A, AND GABRIEL NAVARRO
3.5. Twisted tensor products of the form k[ξ]⊗τ l. For l a quadratic field extension
of k, by Lemma 1.1, twisted tensor products of the form k[ξ] ⊗τ l are given by couples
(f, δ), being f an algebra endomorphism of l, and δ and f–derivation such that
δ2 = 0(3.23)
fδ + δf = 0(3.24)
Now, if l is a Galois extension of k (which is always the case if char k 6= 2), its k-
linear endomorphisms are in one-to-one correspondence with elements of the Galois group
Gal(l/k) ∼= Z2, which in this case correspond to the identity map and the nontrivial
morphism σ that exchanges the roots of p(x). Moreover, since every Galois extension
is separable, all derivations in l are inner. If f is the identity map on l, there are no
nontrivial inner derivations, so we only get one couple (Idl, 0), that trivially satisfies
(3.23) and (3.24). The twisting map associated to the couple (Idl, 0) is the classical
tensor product k[ξ]⊗ l ∼= l[ξ].
Let us describe the σ–derivations for the nontrivial morphism σ. Assume that l is
generated over k by η, a root of the polynomial p(x) = x2 − αx+ β. As any derivation is
inner, there exists θ ∈ l such that δ(x) = (σ(x)− x)θ, so in particular we get
δ(η) = (σ(η)− η)θ = (α− 2η)θ.
For the study of this example it is useful to take into account the characteristic of the
field. If char k 6= 2, as stated in Remark 1.3, the change of variables x 7→ (α/2)x+1 takes
the polynomial p into p′(x) = x2 + β, and p′ generates the same field extension l as p, so
we may just assume that α = 0 and obtain δ(η) = −2ηθ. Writing θ = a+ bη, and taking
into account that η2 = −β, we get
δ(η) = −2aη − 2bη2 = 2bβ − 2aη.
Plugin this expression into equations (3.23) and (3.24) leads to conditions
4aη = 0(3.25)
4a2η − 4abβ = 0(3.26)
that are satisfied if, and only if, a = 0. Thus, the only σ-derivations providing valid
twisting maps are of the form δq(η) = q for some q ∈ k, and we get a 1-parameter family
of twisting maps given by the couples (σ, δq), leading to the family of algebras
Bq := k〈x, y| x2 = 0, y2 = γ, xy + yx = q〉
where in order to simplify notation we are writing γ in the place of −β.
Lemma 3.4. For all q 6= 0, the algebra Bq is isomorphic to the matrix algebra M2(k).
Proof. Just take the isomorphism given by
x 7→
(
0 0
q/γ 0
)
, y 7→
(
0 γ
1 0
)
.

Lemma 3.5. The algebra B0 is isomorphic to the invariant ring (lQ<2)G, where Q =
◦ ** ◦jj is the round-trip quiver, and G denotes the group generated by the non-trivial

FACTORIZATION STRUCTURES WITH A 2-DIMENSIONAL FACTOR 19
automorphism that exchanges the vertices and the arrows of Q and conjugate the scalar
elements of l with respect to the nontrivial element of the Galois group Gal(l/k).
Proof. Let us denote by u, v the vertices of Q, and by R, S its arrows. First, realize that
the l–algebra B0 ⊗ l = l〈x, y| x2 = 0, y2 = γ, xy + yx = q〉 is isomorphic to the algebra
lQ<2 via the automorphism defined through
x 7−→ R + S, y 7−→ √γu−√γv.
The automorphism σ of the Galois group lifts in a trivial way to B0 ⊗ l, and obviously
B0 ∼= (B0 ⊗ l)σ; now, it is straightforward to check that, under the above mapping, the
image of B0 is invariant under the action of G, resulting in the desired isomorphism
B0 ∼= (lQ<2)G.

Let us study now the case for k a field with char k = 2. In this case, for the twisted
derivation we get
δ(η) = αa + αbη,
which leads to equations
α2b = 0(3.27)
α2ab + α2b2η = 0,(3.28)
that are satisfied if, and only if, b = 0 or α = 0. Assume that α 6= 0, that is, that the
polynomial p has two distinct roots on l; this is always the case if k is a perfect field,
otherwise the situation is very different, and will be treated separately. Then from the
above equations we obtain b = 0, yielding δ(η) = q ∈ k, which gives us exactly the same
family of algebras
Bq := k〈x, y| x2 = 0, y2 = γ, xy + yx = q〉
previously mentioned. Same proof as in Lemma 3.4 tells us that Bq ∼= M2(k) whenever
q 6= 0, but in this case for q = 0 the algebra B0 is commutative and thus B0 ∼= k[ξ]⊗l ∼= l[ξ].
The remaining case, namely when char k = 2 and p(x) = x2+β, is a special one. Realize
that in this case the irreducible polynomial of η over k has η as its unique (double) root.
Under these premises, l is not a Galois extension of k, since it fails to be separable, and the
former reasoning cannot be applied. In particular, as there is only one root in p, there are
no nontrivial automorphisms of l, so the only possible choice for f is the identity; however,
as the extension is not separable anymore, now we must take into account that there can
be derivations that are not inner. In the present situation, with f = Id, equations (3.23)
and (3.24) become simply
δ2 = 0,(3.29)
2δ = 0,(3.30)
of which the second one is trivially satisfied, Thus, we only need to classify derivations of
square 0.
For any derivation δ we have
δ(1) = δ(12) = 2δ(1) = 0,(3.31)
δ(η) = aη + b(3.32)

20 O´SCAR CORTADELLAS IZQUIERDO, JAVIER LO´PEZ PEN˜A, AND GABRIEL NAVARRO
for some a, b ∈ k. It is straightforward to check that any choice of a and b leads to a valid
derivation in k(η), and we have that δ2 = 0 if, and only if,
0 = δ2(η) = δ(aη + b) = a2η + ab,
and this equation is satisfied if, and only if, a = 0. Henceforth, we have a one-parameter
family of square 0 derivations that give us twisting maps, and these derivations are given
by δ(η) = b. Once again, the resulting twisted tensor products are the family of algebras
Bq := k〈x, y| x2 = 0, y2 = γ, xy + yx = q〉,
so we obtain the same twisted tensor products regardless of the characteristic of the field.
3.6. Twisted tensor products of the form l⊗τ l′. In order to classify twisted tensor
products of two field extensions, we shall initially assume that the characteristic of the
base field is different from 2. As it happened in the last case, for fields of characteristic 2
weird phenomena may show up, so we will study them separately.
So, take k such that char k 6= 2, and assume (without loss of generality) that the field
extensions l and l′ are given as splitting fields of the polynomials x2 − α and x2 − β,
respectively, and let us denote by η, ζ their respective generators over k, so that l =
k〈η| η2 = α〉, and l′ = k〈ζ | ζ2 = β〉. Using again Lemma 1.1, twisting maps τ : l′⊗l → l⊗l′
are in one-to-one correspondence with couples (f, δ) where f is a k–linear endomorphism
of l, satisfying the corresponding compatibility conditions. Since l is a Galois extension
of k of degree 2, there are only two k–linear endomorphisms of l, namely the identity and
the map σ given by σ(η) = −η. Since l is separable, all derivations are inner, and thus
there are no nontrivial derivations associated to the identity map. For the map σ, we
need to find all the σ–derivations satisfying
δ2 = 0(3.33)
δσ + σδ = 0,(3.34)
where we are using the fact that σ2 = Idl. Realize that the above conditions are trivially
satisfied for the identity morphism and the trivial derivation (yielding the usual tensor
product l ⊗ l′). If the derivation δ, which is inner, is induced by an element θ = a + bη,
this leads us again to the same equations that showed up in the former paragraph:
4aη = 0(3.35)
4a2η + 4abα = 0(3.36)
with solutions given by δ(η) = q ∈ k, and once again we obtain the 1–parameter family
of algebras leading to the family of algebras
Cq := k〈x, y| x2 = α, y2 = β, xy + yx = q〉.
However, the classification of these algebras is not as easy as in the former situations, and
depends strongly on the ground field k. As a first approximation for the classification of
the algebras Cq, we have the following result:
Lemma 3.6. The algebra Cq is isomorphic to the generalized quaternion algebra αkt, with
t = q2−4αβ4α2 . In particular, the algebras Cq form a family of linked quaternion algebras.

FACTORIZATION STRUCTURES WITH A 2-DIMENSIONAL FACTOR 21
Proof. Take the isomorphism Cq → αkt given by
x 7→ i, y 7→ q
2αi+ ij,
where i and j are the generators of αkt = k〈i, j| i2 = α, j2 = t, ij + ji = 0〉. 
We can draw some immediate consequences from the previous lemma. Firstly, it is
a well known fact that the quaternion algebra αkt is a central simple algebra whenever
α, t 6= 0, and since in our present situation α 6= 0, and t = 0 if, and only if, q2 = 4αβ; which
might happen for at most two values of q. Henceforth, all the algebras Cq (except maybe
two of them) are central simple. In particular, since char k 6= 2, the field k must have at
least three elements, and we can assure that there always exists a twisted tensor product
l ⊗τ l′ which is simple, giving some supporting evidence to the following conjecture, due
to J. Go´mez-Torrecillas and F. van Oystaeyen:
Conjecture 3.7. For any algebra A, there exists a twisting map τ : A⊗A → A⊗A such
that A⊗τ A is simple.
The solution to the isomorphism problem for quaternion algebras over a generic field
is not explicitly known; however, we have the following result establishing necessary and
sufficient conditions for two linked quaternion algebras to be isomorphic:
Lemma 3.8. Two linked quaternion algebras akb and akc are isomorphic if, and only if,
b/c ∈ Nl/k(l×), being l = k(
√a), and Nl/k : l → k the norm map of the extension l/k. As
a consequence, akb is a matrix ring if, and only if, b ∈ Nl/k(l×).
This result, as well as some others dealing with the problem of classifying quaternion
algebras, can be found in [18, Section 1.7] (cf. also [13, Chapter III] for a more recent
revision). Applied to our concrete situation, and taking into account that for a field
extension l = k(√α) the norm map is given by Nl/k(x+ y
√α) = x2 − αy2, we obtain the
following result:
Theorem 3.9. Let q, h ∈ k such that 4αβ − q 6= 0, 4αβ − h 6= 0.
(1) The algebras Cq and Ch are isomorphic if, and only if, there exist x, y ∈ k such
that
(3.37) x2 − αy2 = q
2 − 4αβ
h2 − 4αβ
(2) Cq is isomorphic to the matrix ring M2(k) if, and only if, there exist x, y ∈ k such
that
(3.38) x2 − αy2 = q2 − 4αβ
In other words, the isomorphism classes of twisted tensor products of the form l ⊗τ l′
are given by:
(1) The orbits of the action of Nl/k(l×), seen as a multiplicative subgroup of k×, that
intersect the image of the map q 7→ q2 − 4αβ.
(2) The algebra Cq ∼= C−q, provided that q = 2
√
αβ belongs to k.
(3) The commutative algebra l ⊗ l′.

22 O´SCAR CORTADELLAS IZQUIERDO, JAVIER LO´PEZ PEN˜A, AND GABRIEL NAVARRO
The degenerate case (2) only shows up under very special conditions. Assume that
√
αβ ∈
k. Since obviously √α ∈ l = k(√α), this means that also
√
β ∈ l, and henceforth l = l′.
Conversely, if l = l′ = k(√α), we may take q = 2α and obtain the exceptional algebra.
Proposition 3.10. Let l = l′ = k(√α); the algebra
C2α := l ⊗ C2α = l〈x, y| x2 = y2 = α, xy + yx = 2α〉.
is isomorphic to the truncated path algebra lQ<2 of the round-trip quiver Q = ◦
** ◦jj .
As a consequence, we have an isomorphism between C2α and the algebra of invariants
(lQ<2)G, being G the group generated by the nontrivial automorphism of kQ that conju-
gates scalars while exchanging vertices and arrows of Q.
Proof. The proof of this result follows the same lines as the one of lemma 3.5. In this
case, we use the isomorphism C2α ⊗ l → lQ<2 given by
x 7−→
√
αu−
√
αv + R + S, y 7−→
√
αu−
√
αv.

For the family of central simple algebras (1), as previously mentioned, the number of
isomorphism classes (or orbits of the group action) depends strongly on the ground field
k. A nice recent survey on quaternion algebras over different ground fields can be found
in [14] (cf. also [18] and [13]). Recall that any quaternion algebra must be isomorphic
to either a division ring over k, or to the matrix ring M2(k) (this can easily be proven
by using Artin-Wedderburn structure theorem). For some familiar fields, a more concrete
description can be given:
(1) If k is the field R of real numbers, α, β < 0 then the isomorphism class of Cq ∼= αRb,
with b = (4αβ−q2)/4α, depends on the sign of b, which is the sign of q2−4αβ, More
concretely, if |q| > 2
√
αβ, then Cq ∼= M2(R). If, on the other hand, |q| < 2
√
αβ,
then Cq ∼= H, the usual quaternion algebra.
(2) If k = Fn is a finite field, a well-known theorem by Wedderburn states that any
division ring over k is commutative. Since the quaternion algebras are noncom-
mutative, they must all be isomorphic to the matrix ring, and thus Cq ∼= M2(k)
for all values of q.
(3) If k is an algebraic number field, i.e. a finite extension field of the rational numbers
Q, then there exist an infinite number of nonisomorphic quaternion algebras over
k. Though there is no easy way to list the isomorphism classes, given any concrete
couple of algebras of type Cq, it is possible to tell wether they are or not isomorphic
in a finite number of steps by studying the existence of rational points in the conic
given by equation 3.10. The existence (or not) of such solutions is obtained as a
consequence of the Hasse-Minkowski principle and Hensel’s lemma.
(4) If k = Qp is a field of p–adic numbers, there is only one isomorphism class of
quaternion algebras, which is never isomorphic to the matrix ring. This follows
from the theory of quadratic forms over the p–adic numbers, see [13, Chapter VI]
for details.
The remaining case, of field extensions of characteristic 2, cannot be described in such
a fancy way using quaternion algebras. However, doing some computations (left to the
reader) similar to the ones at the end of section 3.5, we obtain the following result:

FACTORIZATION STRUCTURES WITH A 2-DIMENSIONAL FACTOR 23
Theorem 3.11. Let k be a field with char k = 2, and let l and l′ be quadratic field
extensions of k generated by polynomials p(x) = x2 + αx + β and p′(x) = x2 + α′x + β ′.
Then, all twisted tensor products l ⊗τ l′ of l and l′ are described as
Dq = k〈x, y| x2 = αx+ β, y2 = α′y + β ′, xy + yx = q〉,
where the algebra D0 corresponds to the usual tensor product l ⊗ l′.
Let us finish the paper writing down the “indecomposable” (i.e., non-decomposable as a
non-trivial twisted tensor product) algebras of dimension four over an algebraically closed
field. A complete list of all algebras of dimension four can be found in [8]. We reproduce
the scheme given there, highlighting the “decomposable” algebras putting them into a
box.
k4

k2 × k[ξ]
%%LL
LLL
LLL
xxppp
ppp
M2(k)

kΓ<2
<
<<
<<
<<
<<
 




k × k[x]
(x3)
%%JJ
JJJ
JJJ

k[ξ]× k[ξ]
{{vvv
vv
vv
vv
v
kQ<2
(
((
((
((
((
((
((
((
((
((
((
((
((
((
k(∆1)<2
?
??
??
??
??
k(∆2)<2





k[x]
(x4)

G
++XXXX
XXXXX
XXXXX
XXXXX
XXXXX
XXXXX
XXXXX
XXXXX
XXXXX
XXXX

A0
**TTT
TTTT
TTTT
TTTT
TTTT
TTTT
TTTT
TTTT k ×
k[x, y]
(x, y)2
!!C
CC
CC
CC
CC
A1

Aq
 





k[x, y]
(x3, xy, y2)

∧k2
--[[[[[[[[
[[[[[[[[
[[[[[[[[
[[[[[[[[
[[[[[[[[
[[[[[[[[ kΣ<2
xxrrr
rr
k[x, y, z]
(x, y, z)2
where:
(1) The scalar q 6= 1,−1, 0.
(2) G = k〈x, y〉
(x2, y2 + xy, xy + yx)
(3) The quivers appearing above are the following:
∆1 : ◦

// ◦ ∆2 : ◦
 ◦oo Q : ◦ ** ◦jj
Σ : ◦ //// ◦ Γ : ◦
◦ // ◦

24 O´SCAR CORTADELLAS IZQUIERDO, JAVIER LO´PEZ PEN˜A, AND GABRIEL NAVARRO
(4) The arrow A → B means that the algebra B can be obtained by a degeneration
of the structure of A.
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FACTORIZATION STRUCTURES WITH A 2-DIMENSIONAL FACTOR 25
Department of Algebra, University of Granada, Avda. Fuentenueva s/n, E-18071,
Granada, Spain
E-mail address : ocortad@ugr.es
Mathematics Research Centre, Queen Mary University of London, Mile End Road,
London E1 4NS, United Kingdom
E-mail address : j.lopez@qmul.ac.uk
Department of Computer Sciences and AI, University of Granada, C/ El Greco s/n,
E-51002, Ceuta, Spain
E-mail address : gnavarro@ugr.es