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Revista de la Unión Matemática Argentina
versão impressa ISSN 0041-6932
Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca jun. 2009
A description of hereditary skew group algebras of Dynkin and Euclidean type
Olga Funes
Abstract. In this work we study the skew group algebra Λ[G] when G is a finite group acting on Λ whose order is invertible in Λ. Here, we assume that Λ is a finite-dimensional algebra over an algebraically closed field k. The aim is to describe all possible actions of a finite abelian group on an hereditary algebra of finite or tame representation type, to give a description of the resulting skew group algebra for each action and finally to determinate their representation type.
In this work we assume that is a finite-dimensional algebra over an algebraically closed field
. Let
a finite group acting on
. The skew group algebra
is the free left
-module with basis all the elements in
and multiplication given by
for all
in
,
in
. We study the skew group algebra
when
is a finite group acting on
whose order is invertible in
.
There is an extensive literature about skew group algebras and crossed product algebras
, and their relationship with the ring
, given by elements in
that are fixed by
. It is of interest to study which properties of
are inherited by
,
or
. Some of these ideas are rooted in trying to develop a Galois Theory for non-commutative rings. See [1, 3, 7, 9, 10, 11, 14, 13] for more details.
It is of interest to find ways to describe in terms of
because the algebras
and
have many properties in common which are of interest in the representation theory of finite-dimensional algebras, like finite representation type, being hereditary, being an Auslander algebra, being Nakayama, see [2, 16] for more details. However, we must observe that there are properties which are not preserved by this construction, like being a connected algebra, so we are dealing with essentially different algebras.
It is well known [6] that a connected hereditary algebra is of finite representation type if and only if the underlying graph of its quiver is one of the Dynkin diagrams (
),
(
),
,
or
; some years later it was shown that a a connected hereditary algebra is of tame representation type if and only if the underlying graph of its quiver is one of the euclidean diagrams
(
),
(
),
,
or
, see [4, 12, 17].
The aim of this paper is to describe all possible actions of a finite abelian group on an hereditary algebra of finite or tame representation type, to give a description of the resulting skew group algebra for each action and finally to determinate their representation type.
Then, in order to classify the finite and tame representation type hereditary skew group algebras, it suffices to study the group actions on the Dynkin and the euclidean quivers. In order to do this description, we start by considering a short exact sequence of groups . We can express
in terms of the skew group algebra
or the crossed product algebra
. In this context, we describe when
is isomorphic to
, for
a finite group whose order is invertible in
. In section
we provide an introduction to the subject, that is, the definition of skew group algebra and crossed product algebra. Finally, in section
we consider hereditary algebras of finite representation type and in section
we consider hereditary algebras of tame type. In each one of these cases, that is, when the associated quiver is
(
),
(
),
,
,
,
(
),
(
),
,
or
, we get a connection between
and the crossed product algebra
with a complete description of all the possible groups
appearing in each case, where
is the subgroup of
consisting on all the elements acting trivially on a complete set of primitive orthogonal idempotents of the algebra
. As a consequence of all these results, we get that if
acts trivially on
then the crossed product algebras obtained in each description are skew group algebras. Finally, the case
is considered at the end of section 4.
This section consists of the preliminaries necessary for the proof of the main results.
Let be a finite-dimensional
-algebra and
a finite group acting on
. The skew group algebra
is the free left
-module with basis all the elements in
and multiplication given by
for all
in
,
in
. Clearly
is a finite-dimensional
-algebra. If we identify each
in
with
in
and each
in
with
in
, we have that
is the group of units of
and
is a
-subalgebra of
.
Let be a basic finite-dimensional algebra (associative with unity) over an algebraically closed field. A quiver
is a quadruple consisting of two sets
(whose elements are called points, or vertices ) and
(whose elements are called arrows), and two maps
which associate to each arrow
its source
and its target
. An arrow
of source
and target
is usually denoted by
. A quiver
is usually denoted briefly by
or even simply by
. Thus, a quiver is nothing but an oriented graph without any restriction as to the number of arrows between two points, to the existence of loops or oriented cycles.
We write a path in
as a composition of consecutive arrows
where
for all
, and we set
,
. The path algebra
is the
-vector space with basis all the paths in
, including trivial paths
of length zero, one for each vertex
. The multiplication of two basis elements is the composition of paths if they are composable, and zero otherwise. A relation from
to
is a linear combination
such that, for each
,
is a non-zero scalar and
a path of length at least two from
to
. A set of relations on
generates an ideal
, said to be admissible, in the path algebra
of
.
It is well-known that if is basic there exists a quiver
and a surjective algebra morphism
whose kernel
is admissible, where
, the ordinary quiver of
, is defined as follows:
-
If
is a complete set of primitive orthogonal idempotents of
, the vertices of
are the numbers
which are taken to be in bijective correspondence with the idempotents
;
-
Given two points
the arrows
are in bijective correspondence with the vectors in a basis of the
-vector space
.
Thus we have . We refer to [2] for more details.
If is acting on a basic algebra
, we can view
as
in such a way that the action of
on
is induced by an action of
on
which leaves
stable and preserves the natural grading on
by the length of paths. Then
is isomorphic to
, see [16, Proposition 2.1]. Moreover, if
contains no multiple arrows, the action of
on
is simple: each
permutes the vertices in
and maps each arrow
onto a multiple scalar of the unique arrow from
to
. From now on we assume
with the action of
as described above,
without double arrows.
Proposition 2.1. Let be a finite group acting on
, let
be the associated quiver of
,
without double arrows, and
. We consider the action of
on
induced by an automorphism of algebras which preserves the length of paths of
. Then
-
If
,
, then
for some
;
-
If
and
, then
for some arrow
,
. In particular, if
fixes the starting and ending point of
then
, with
;
-
If
is a source (sink) then
is a source (sink);
-
The cardinal of the set of arrows that start (end) in
is equal to the cardinal of the set of arrows that start (end) in
.
Proof.
-
Let
be a primitive idempotent in
. Since the action of
preserves the vector space generated by arrows,
. Moreover,
, then we have that
, and hence
, that is,
. On the other hand, suppose
. Then
But this is a contradiction because
es primitive. Then
for some
.
-
If
,
with
. Moreover if
then
, for some arrow
, because
has no double arrows. Then
.
-
Let
be an idempotent of
. Suppose that
is a source and
is not. Then, there exists an arrow
such that
, that is, there exists an index
such that
. Since the action of
on
is induced by an automorphism of algebras which preserves the length of paths, there exists an arrow
such that
. Then we have
because
is a source. This contradiction arises from the assumption that
is not a source. Similarly we prove that if
is a sink then
is a sink.
-
Let
and
be the
-vector space with basis
. If
, by ii) we know that the automorphism
induces an isomorphism
. Then the cardinal of
and
are equal.
2.1. Crossed product algebra . The purpose of this section is to present the crossed product algebras in order to study when
is isomorphic to
where
is a short exact sequence of groups. We start with the definition of crossed product algebras and prove, for completeness, Theorem 2.2 that connects the skew group algebras with the crossed product algebras. See [16] for more details.
Let be a ring,
a finite group acting on
,
the units of
and
, a map satisfying
-
for
,
,
;
-
for
,
the identity element of
;
-
for
,
.
Then the corresponding crossed product algebra has elements
;
. Addition is componentwise, and multiplication is given by
and
.
Let be a group and
be a short exact sequence of groups. Let
be a disjoint union of lateral classes. Then
where
,
.
Theorem 2.2. If is a short exact sequence of groups, then
where the action of on
is induced by the action of
, the action of
on
is defined by
and is defined by
, with
.
Proof. We consider the action of on
induced by the action of
and the action of
on
given by
. We claim that the action is well defined since
is a normal subgroup of
. If
,
then
for some
, that is
. Let
be defined by
A direct computation shows that is a crossed product. Now let us see that the map
given by
is an isomorphism of -algebras, where
. Clearly
is a morphism of
-vector spaces. If
, then
because is a normal subgroup of
. On the other hand we have
and agrees with
. Furthermore it is clear that
is bijective, hence we get
.
Corollary 2.3. If is a short exact sequence of groups that splits on the right, then
Proof. We only have to prove that the map defined in the theorem above is such that
for any
,
where
, with
for some
. If the sequence
splits on the right, there exists a map
such that
. Let
. Since
, then
and hence we assume that
. Since
is a morphism of groups,
and therefore
. Now it is clear that
.
It is clear that the map which defines a crossed product is by definition a cocycle with respect to group cohomology, see [8] for more details. We shall prove that if the cocycle
is a coboundary, then we have
.
Proposition 2.5. [16, Lemma 5.6] Let be a map,
, and let
be given by
Then .
Proof. A direct computation shows that defines a crossed product. Let
be defined by
. Then
is a morphism of algebras because
Therefore is an isomorphism and hence
.
We say that acts trivially on an element
if
for all
. If
is a finite abelian group of order
acting trivially on
with
invertible in
, then
. In fact, by Maschke's theorem we have that
, see [14]. Now the map
given by
is an isomorphism of
-algebras.
Proposition 2.6. [16, Proposition 5.8] Let be a finite cyclic group acting on a commutative local algebra
with the order of
invertible in
. Then
.
We may infer from the previous proposition that if is a finite cyclic group with the order of
is invertible in
, and
(the center of
) is a local algebra,
because any cocycle is a coboundary. In particular, if
is a basic connected algebra without oriented cycles,
.
Corollary 2.7. Let be a finite abelian group acting on a basic connected algebra
where the associated quiver has no oriented cycles, with the order of
invertible in
. Let
be a subgroup of
which acts trivially on
, with
cyclic. Then
.
Proof. It follows from Theorem 2.2 that , and
by Maschke's theorem. Since
is abelian and acts trivially on
,
takes values in the set of invertible elements of the center of
. But
and
is cyclic, so Proposition 2.6 implies that
is a coboundary. From Proposition 2.5 we may deduce that
.
It is known that if is a finite group of order
acting trivially on the idempotents of
and
is invertible in
, then
is an abelian group, see [15, Proposition 2.7]. In fact, given
,
with
a
-root of unity. Hence
, and this equality holds for any arrow
. Moreover, for every
. So
.
Finally, we state a result that will be used in the proof of the main theorem in this work.
Theorem 2.8. [5, Theorem 8]. Let be a finite abelian group of order
acting trivially on a complete set of primitive orthogonal idempotents of a simply connected algebra
,
without double arrows and
invertible in
. Then
.
3. with
an abelian group and
an hereditary algebra of finite representation type
The aim of this section is to describe all possible actions of a finite abelian group on an hereditary algebra of finite representation type and to give a description of the skew group algebra for each action.
Gabriel has shown in [6] that a connected hereditary algebra is representation-finite if and only if the underlying graph of its quiver is one of the Dynkin diagrams ,
(
),
,
or
, that appear also in Lie theory, where the index in the Dynkin graph always refers to the number of points in the graph. Then, in order to classify the representation-finite hereditary skew group algebras, it suffices to study the group actions on the Dynkin quivers.
Before we present the results, we need some definitions.
Definition 3.1. We say that an quiver of type has symmetric orientation if it is symmetric with respect to the middle point
.
Definition 3.2. We say that an quiver of type ,
, has symmetric orientation if
or
.
Definition 3.3. We say that an quiver of type has
-
symmetric orientation of kind
if
or
; and,
-
symmetric orientation of kind
if
or
.
Definition 3.4. We say that the quiver of type
has symmetric orientation if it is symmetric with respect to the side
, that is,
-
and
, or
and
;
and,
-
, or
.
-
The quiver
is symmetric with respect to the middle point
if that point is center of symmetry of the quiver.
-
The quiver
is symmetric with respect to the side
if the line obtained with the points
is a symmetry axis of the quiver.
As we have already mentioned, if is acting trivially on
, we have
. Hence, from now on, we will assume that
is acting non trivially on
. Let
. Clearly
is a normal subgroup of
. Let
, then
is a short exact sequence of groups.
Theorem 3.6. Let be an hereditary algebra, with
of type
(
),
(
),
,
or
, and
a finite abelian group of order
acting non trivially on
, with
invertible in
. Let
.
-
If
then
;
-
If
and
with
of type
then
is of type
, with symmetric orientation, the order of
is even and
;
-
If
and
with
of type
,
, then
has symmetric orientation, the order of
is even and
;
-
If
and
with
of type
then
-
has symmetric orientation of kind
, the order of
is even and
,
or
-
has symmetric orientation of kind
, the order of
is divisible by
or
, and
or
-
-
If
and
with
of type
then
has symmetric orientation,
is a group of even order and
;
-
If
with
of type
or
of type
then
and
.
Proof. In order to prove the theorem, we need a precise description of all the possible actions of on
, for each type and orientation of
. We use Proposition 2.1 to describe all possible actions of
on
with
of type
,
,
,
or
.
-
See Theorem 2.8.
-
Let
with
of type
and let
,
. If
then
. This implies
and
with
-roots of unity. Repeating this procedure we have that
implies
with
an
-root of unity, and this for all
. Hence the action of
is trivial on the idempotents
of
. So
, a contradiction. So
. In this case
and
,
will have to be sinks or sources, see Proposition 2.1. This determines the orientation of
and
. Moreover
. So
and
. Inductively,
and
, and this for all
, with
,
. If
is an even number we have that
,
and if
is the arrow
, then
, contradiction. We also get a contradiction if
. Then if the number of vertices is an even number, the unique possible action on the set of idempotents is the trivial one.
Let
with
acting non trivially on the set
of idempotents of
. Then
and
, hence the quiver
has symmetric orientation. Moreover,
for all
, so
. Then
has even order
and
, for all
. Let
,
. Since
,
does not act trivially on the set
of idempotents of
. By the previous reasoning, the unique non trivial action is given by
. Then
. As a consequence
, that is
. Then
because
, and hence
. Hence, if the group
does not act trivially on the set
of idempotents of
, in accordance with the previous analysis we have that
is an even number and
is of type
with symmetric orientation. In this case we have
. Hence
, see Theorem 2.2 and Theorem 2.8.
-
Let
with
of type
,
. Assume that the group
is not acting trivially on the set
of idempotents of
. We observe that all
must satisfy
, see Proposition 2.1. If
then
,
and
for all
. This determines the orientation of the arrows, that is,
has symmetric orientation, and
,
,
for all
, with
-roots of unity,
non zero. Then
for all
, that is,
. So
has even order
and
. Let
,
. By the previous reasoning,
and
act in the unique possible non trivial way on the complete set of idempotents of
. Then
,
and
for all
. Hence
, that is
, then
because
. Hence
and
is an even number.
It follows from the previous analysis that
is an even number and the quiver
has symmetric orientation. Hence we have
.
-
Let
with
of type
.
Let
,
. Necessarily
, by Proposition 2.1, and all possible cases are:
-
,
,
;
-
,
,
;
-
,
,
;
-
,
,
;
-
,
,
.
In fact
,
and
, so
in
. On the other hand, since
for all
with
and
and
is abelian, we have that
cannot contain simultaneously elements acting as
for all
with
and
. Consequently
or
. The cases i), ii) and iii) determine the orientation of the arrows
and
, that is,
has symmetric orientation of kind (a) or (b), and the cases iv) and v) determine the orientation of all the arrows, that is,
has symmetric orientation of kind (b).
In accordance with Definition 3.3 and with the previous analysis for
of type
, we have that the quiver
has symmetric orientation of kind (a) and
, or has symmetric orientation of kind (b) and
or
. From Theorem 2.2 and Theorem 2.8 we have that, in the first case,
and
. In the second case, the order of
is divisible by
or
,
or
, and
or
-
-
We need again a precise description of all the possible actions of
on
with
of type
. Let
,
. By Proposition 2.1 ,
, and this implies that
. On the other hand
or
. If
, then
and
. This is a contradiction, because
. Then
, and this implies that
and
. This determines the orientation of the arrows, and we have
,
,
,
and
with
non zero and
an
-root of unity. Since
for all
, then
. So
has even order
and
,
. Let
be such that
. Hence,
for all
. Therefore
,
and
that is,
, and then
. Hence, if the group
does not act trivially on the set
of idempotents of
, in accordance with the previous analysis, we have that
is an even number,
has symmetric orientation and
. Hence
, see Theorem 2.2 and Theorem 2.8.
-
If we consider the cases
of type
or
, the unique possible action on the set of idempotents is the trivial one. Hence
and
and the result follows from i).
Corollary 3.7. Let be an hereditary algebra, with
of type
(
),
(
),
,
or
, and
an abelian group of order
acting on
, with
invertible in
. Suppose that
does not act trivially on the set
of idempotents of
and
acts trivially on
.
-
If
, with
of type
with symmetric orientation, then
;
-
If
, with
of type
,
, with symmetric orientation, then
;
-
If
, with
of type
with symmetric orientation of kind (a), then
;
-
If
, with
of type
with symmetric orientation of kind (b), then
or
.
-
If
, with
of type
with symmetric orientation, then
.
Proof. It follows from Theorem 3.6 and Corollary 2.7.
The following Corollary follows easily from [16, (2.3), (2.4)].
Corollary 3.8. Let be an hereditary algebra, with
of type
(
),
(
),
,
or
, and
a cyclic group of order
acting on
, with
invertible in
.
-
Let
be a field such that
. If
is an hereditary algebra with
of type
and
is acting non trivially on the set
of idempotents of
, then the skew group algebra
is Morita equivalent to an algebra
with
of type
. If
is acting non trivially on the set
of idempotents of
, then the skew group algebra
is Morita equivalent to an algebra
with
of type
.
-
Let
be a field such that
. If
is an hereditary algebra, with
of type
and
is acting non trivially on the set
of idempotents of
, then the skew group algebra
is Morita equivalent to an algebra
with
of type
if
and of type
if
.
-
Let
be a field such that
. If
is an hereditary algebra, with
of type
,
, and
is acting non trivially on the set
of idempotents of
, then the skew group algebra
is Morita equivalent to an algebra
with
of type
.
-
Let
be a field such that
. If
is an hereditary algebra, with
of type
, and
is acting non trivially on the set
of idempotents of
, then the skew group algebra
is Morita equivalent to an algebra
with
of type
.
For an example of Corollary 3.8 see [16, (2.3), (2.4)].
4. with
an abelian group and
an hereditary algebra of tame representation type
The aim of this section is to describe all possible actions of a finite abelian group on an hereditary algebra of tame representation type, to give a description of the skew group algebra for each action and finally to determinate their representation type.
It is well known that a connected hereditary algebra is of tame representation type if and only if the underlying graph of its quiver is one of the euclidean diagrams (
),
(
),
,
or
where an euclidean diagram
has
points. Then, in order to classify the tame representation type hereditary skew group algebras, it suffices to study the group actions on the euclidean quivers. It is necessary to clarify that the case
will be considered separately later on.
Before we present the results, we need some definitions.
Definition 4.1. We say that an quiver of type (
) has
-
symmetric orientation if
is odd and the quiver is symmetric with respect to an axis
,
-
cyclic orientation of order
if the full subquivers with vertices
are all equal, and
is minimal with respect to this property (
).
Remark 4.2. Supose you have with a fixed oritentation. Choose
such that
, for any action
. This set, a non-empty set of the natural numbers, has a first element and this is the
of the definition.
Definition 4.3. We say that an quiver of type ,
, has
-
symmetric orientation of kind
if
-
, or
-
, or
-
, or
-
,
-
-
symmetric orientation of kind
if
is even and the quiver is symmetric with respect to the middle point
;
Definition 4.4. We say that an quiver of type has symmetric orientation of order
if the number of arrows starting at the vertex
is equal to
, for
.
Definition 4.5. We say that an quiver of type has
-
symmetric orientation of kind
if
,
,
or
,
and
is a source or a sink;
-
symmetric orientation of kind
if it is not symmetric of kind (a) and it is symmetric with respect to the side
.
Definition 4.6. We say that an quiver of type has symmetric orientation if it is symmetric with respect to the side
.
-
We say that the quiver
is symmetric with respect to the middle point
if that point is center of symmetry of the quiver
.
-
We say that the quiver
is symmetric with respect to an axis
, if the line obtained with the points
is symmetry axis of the quiver.
Let be a group and we will assume that
is acting trivially on
, we have
. Hence, from now on, we will assume that
is acting non trivially on
. Let
. Clearly
is a normal subgroup of
. Let
, then
is a short exact sequence of groups.
Theorem 4.8. Let be a tame hereditary algebra, with
of type
(
),
(
),
,
or
, and
a finite abelian group of order
acting non trivially on
, with
invertible in
.
-
If
then
;
-
If
and
with
of type
then
-
is symmetric not cyclic,
if it is symmetric with respect to one axis, or
if it is symmetric with respect to a pair of perpendicular axes, the order of
is divisible by
or
respectively, and
or
;
-
is cyclic of order
, not symmetric,
is the smallest natural number such that
is divisible by
, the order of
is divisible by
and
;
or
-
is symmetric and cyclic of order
,
, the order of
is divisible by
or
and
or
.
-
-
If
and
with
of type
,
then
-
has symmetric orientation of kind
, not (a), the order of
is even and
,
-
has symmetric orientation of kind
, not
, the order of
is divisible by
or
, and
or
,
or
-
has symmetric orientation of kind
and (b), the order of
is divisible by
or
and
,
or
.
-
-
If
and
with
of type
then
-
is symmetric of order
or
, the order of
is divisible by
or
and
or
;
-
is symmetric of order
, the order of
is divisible by
or
and
or
;
or
-
is symmetric of order
, the order of
is divisible by
,
or
and
,
,
or
;
-
-
If
and
with
of type
then
-
has symmetric orientation of kind (a), the order of
is divisible by
or
and
or
;
or
-
has symmetric orientation of kind (b), the order of
is divisible by
and
;
-
-
If
and
with
of type
then
has symmetric orientation,
is a group of even order and
;
-
If
with
of type
then
and
.
Proof. In order to prove the theorem, we need a precise description of all the possible actions of on
, for each type and orientation of
. We use Proposition 2.1 to describe all possible actions of
on
with
of type
,
,
,
or
. We observe that we identify the elements of
with the natural numbers
in the indexes of the idempotents
.
-
Theorem 2.8 cannot be applied because
is not simply connected. Using [16, (2.3), (2.4)] for this case we have
(where
,
,
,
under the conditions of [16, (2.3)]) .
-
Let
with
of type
and let
,
. Assume that
fixes at least one point, say
. Then
or
. In the first case, repeating this procedure we have that
for all
, and so
, a contradiction. In the second case, we get that
, for all
, and this determines the orientation of the arrows. If
, we have
and
, a contradiction since there is only one arrow joining
and
. So
and
has symmetric orientation. Moreover,
for all
, so
.
Now let
,
,
for all
. Let
. If
, the previous reasoning says that there must exist a middle point between
and
that will be fixed by
, a contradiction. So
, and inductively we get that
. This determines the orientation of the arrows, and so
is cyclic of order
, where
is the first element in the set
. Let
be such that
. Let
, with
. Then
. If
, we get a contradiction to the minimality of
. So
and
and
in
.
We denote by
We have already proved that
if and only if
has symmetric orientation, and
if and only if
is cyclic of order
.
Assume first that
is cyclic of order
and is not symmetric. We have seen that
in
for any
. Moreover,
, so
if and only if
is divisible by
. Let
be the smallest natural number such that
is divisible by
. We conclude that
in this case.
Assume now that
is symmetric but not cyclic, and let
, that is,
and
for some
and
,
. We assume, without loss of generality, that
. If
, then
, so
is divisible by
. Since
, we have that
for
. If
then
and hence
, that is,
in
, and hence
in this case. If
then
and
is symmetric with respect to the axes
and
and in this case
.
Finally, assume that
is symmetric (
) and cyclic of order
, and let
and
, that is,
and
. If
, then
, so
is divisible by
. Since
, we have that
and in this case
and
.
Finally, from Theorem 2.2 and [16, (2.3)] we have the conclusions, that is
or
.
-
Let
with
of type
,
. Let
,
. We observe first that
, see Proposition 2.1.
Assume first that
and
. Then
for all
. Since
, we must have
or
. This implies that
has symmetric orientation of kind
and all possible actions are given by:
-
,
,
,
;
-
,
,
,
;
-
,
,
,
.
Since
and
, we conclude that
or
.
Assume now that
and
. Then, using the same argument as in the proof of Theorem 3.6 in the case of
, we conclude that
is symmetric of kind
. If
is not of kind
, the unique possible non trivial action on the complete set of idempotents is given by
,
,
,
and
for all
. In this case,
.
To finish with this case, we have to assume that
is symmetric of kind
and
. Then all the possible non trivial actions are given by
-
,
,
,
,
for all
;
-
,
,
,
,
for all
;
-
,
,
,
,
for all
;
-
,
,
,
,
for all
;
-
,
,
,
,
for all
;
-
,
,
,
,
for all
;
-
,
,
,
,
for all
.
Now
,
,
and
. Moreover, if
, then
implies that
is equal to
,
,
,
,
or
. Moreover
. Hence
,
or
.
-
-
Let
with
of type
. Let
,
. Necessarily, by Proposition 2.1,
. If
is symmetric of order
or
, the same reasoning made for
in the proof of Theorem 3.6 holds, and hence
or
.
If
is symmetric of order
, assume that
. Hence all the possible cases for
are:
-
,
,
,
;
-
,
,
,
;
-
,
,
,
.
In fact
,
and
for all
with
and for all
. Consequently
or
.
If
is symmetric of order
, all the possible cases for
are in one to one correspondence with the non trivial permutations of
. Hence
,
,
or
(all the possible abelian subgroups of
).
-
-
This case follows from an argument similar to what has been done in the proof of Theorem 3.6 for the case
(for any
,
and the action of
on
is uniquely determined by the action of
in
and
).
-
Let
with
of type
, and let
. By Proposition 2.1,
and then
. Now
or
. In the first case we get that
for all
, and so
, a contradiction. Then
and this determines completely the orientation of the arrows, that is,
has symmetric orientation, and the action of
on the complete set of idempotents of
. Since
, we can deduce that
is an even number. Let
,
. By the previous reasoning,
and
act in the unique possible way on the complete set of idempotents of
. Then
for all
, hence
, that is,
in
. So
and
, see Theorem 2.2 and Theorem 2.8.
-
If we consider the case
of type
, the unique possible action on the set of idempotents is the trivial one. Hence
,
and the result follows from i).
Corollary 4.9. Let be an hereditary algebra, with
of type
(
),
(
),
,
or
, and
an abelian group of order
acting on
, with
invertible in
. Suppose that
does not act trivially on the set
of idempotents of
and
acts trivially on
.
-
If
with
of type
(
) and
-
is symmetric not cyclic,
then
or
;
-
is cyclic of order
, not symmetric, then
;
or
-
is symmetric and cyclic of order
,
, then
or
.
-
-
If
with
of type
,
, and
-
with symmetric orientation of kind
, not (a), then
,
-
with symmetric orientation of kind
, not
, then
or
,
or
-
with symmetric orientation of kind
and (b) then
,
or
.
-
-
If
with
of type
and
-
is symmetric of order
or
then
or
;
-
is symmetric of order
then
or
;
or
-
is symmetric of order
then
,
or
;
-
-
If
with
of type
and
-
with symmetric orientation of kind (a) then
or
;
or
-
with symmetric orientation of kind (b) then
;
-
-
If
with
of type
and
with symmetric orientation then
.
Proof. It follows from Theorem 4.8 and Corollary 2.7.
The following corollary follows easily from [16, (2.3), (2.4)].
Corollary 4.10. Let be an hereditary algebra, with
of type
(
),
(
),
,
or
, and
an cyclic group of order
acting on
, with
invertible in
.
-
If
with
of type
and
is acting non trivially on the set
of idempotents of
then the skew group algebra
is Morita equivalent to an algebra
with
of type
.
-
If
with
of type
,
and
is acting non trivially on the set
of idempotents of
then the skew group algebra
is Morita equivalent to an algebra
with
of type
.
-
If
with
of type
, and
is acting non trivially on the set
of idempotents of
then the skew group algebra
is Morita equivalent to an algebra
with
of type
. If
is acting non trivially on the set
of idempotents of
then the skew group algebra
is Morita equivalent to an algebra
with
of type
. If
is acting non trivially on the set
of idempotents of
then the skew group algebra
is Morita equivalent to an algebra
with
of type
.
-
If
with
of type
,
and
is acting non trivially on the set
of idempotents of
and the action of
on
is induced by a reflection in the quiver, then the skew group algebra
is Morita equivalent to an algebra
with
of type
.
-
If
with
of type
,
and
is acting non trivially on the set
of idempotents of
and the action of
on
is induced by a reflection with respect the middle point
in the quiver,
-
if
is of type
then the skew group algebra
is Morita equivalent to an algebra
with
of type
;
-
if
is of type
,
then the skew group algebra
is Morita equivalent to an algebra
with
of type
.
-
-
If
with
of type
and
is acting non trivially on the set
of idempotents of
then the skew group algebra
is Morita equivalent to an algebra
with
of type
. If
is acting non trivially on the set
of idempotents of
then the skew group algebra
is Morita equivalent to an algebra
with
of type
.
-
If
with
of type
and
is acting non trivially on the set
of idempotents of
then the skew group algebra
is Morita equivalent to an algebra
with
of type
.
The case is not considered in Theorem 4.8 because the techniques we use do not hold in this case. In fact,
is the Kronecker algebra and Proposition 2.1 does not hold since this algebra has double arrows. Moreover, Theorem 2.8 cannot be applied because
is not simply connected. We will only consider the case of a cyclic group acting on the Kronecker algebra, then it is possible to apply directly [16, (2.3)].
If is a cyclic group acting on
, the Kronecker algebra,
with
invertible in
then all possible actions are given by:
-
,
,
,
and in this case the skew group algebra
. [16, (2.3)]
-
,
,
,
with
,
-
,
,
,
with
, and in this case the skew group algebra
is hereditary of type
. [16, (2.3)]
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Olga Funes
Universidad Nacional de la Patagonia San Juan Bosco, Comodoro Rivadavia, Argentina.
funes@ing.unp.edu.ar
Recibido: 20 de octubre de 2006
Aceptado: 10 de marzo de 2008