# O-species and tensor algebras

### Nadiya Gubareni

Journal of Applied Mathematics and Computational Mechanics |
Download Full Text |

*O*-SPECIES and
TENSOR ALGEBRAS

Nadiya Gubareni

Institute of Mathematics, Czestochowa
University of Technology

Częstochowa, Poland

nadiya.gubareni@yahoo.com

**Abstract.** In
this paper we consider *O*-species and their representations. These *O*-species
are a type of a generalization of a species introduced by Gabriel. We also
consider the
tensor algebras of such *O*-species. It is proved that the category of all
representations
of an *O*-species and the category of all right modules over the
corresponding tensor algebra are naturally equivalent.

*Keywords: **species, O-species,
representations of O-species, tensor algebra, O-species of bounded
representation type, diagram of O-species*

1. Introduction

In this paper we consider *O*-species,
which generalize the notion of species
introduced by Gabriel in [1]. Recall this definition:

**Definition 1.1.** (Gabriel [1]). Let *I* be a finite index set. A **species ***L* = (*F _{i}*,

*)*

_{i}M_{j}

_{i,j }_{Î}

*is a finite family (*

_{ I}*F*)

_{i}

_{i }_{Î }

*of division rings together with a family (*

_{I}*)*

_{i}M_{j}

_{i,j }_{Î}

*of (*

_{ I}*F*,

_{i}*F*)-bimodules.

_{j}We say that (*F _{i}*,

*)*

_{i}M_{j}

_{i,j }_{Î}

*is a*

_{ I}**if all**

*K*-species*F*are finite dimensional and central over the common commutative subfield

_{i}*K*which acts centrally on

_{i}*M*, i.e.

_{j}*lm*=

*m*

*l*for all

*l*Î

*K*and all

*m*Î

_{i}*M*. We also assume that each bimodule

_{j}

_{i}*M*is a finite dimensional vector space over

_{j}*K*.

*K*-species is a

**if**

*K*-quiver*F*=

_{i}*K*for each

*i*.

**Definition 1.2.** A
**representation **(*V _{i}*,

_{j}*j*) of a species

_{i}*L*= (

*F*,

_{i}*)*

_{i}M_{j}

_{i,j }_{Î}

*(or an*

_{ I}**) is a family of right**

*L*-representation*F*-modules

_{i}*V*and

_{i}*F*-linear mappings:

_{j}(1.3) |

for each *i*, *j* Î *I*.
Such a representation is called **finite dimensional**, provided all
the spaces *V _{i}* are finite dimensional vector spaces.

Let *V *= (*V _{i}*,

_{j}*j*

*) and*

_{i}*W*= (

*W*,) be two

_{i}*L*-representations. An

*L*-morphism Y:

*V*®

*W*is a set of

*F*-linear maps a

_{i}*:*

_{i}*V*®

_{i}*W*such that

_{i}(1.4) |

Two representations (*V _{i}*,

_{j}*j*

*) and*

_{i}*W*= (

*W*,

_{i}

_{j}*y*

*) are called*

_{i}**equivalent**if there is a set of isomorphisms

*a*from the

_{i}*F*-module

_{i}*V*to the

_{i}*F*-module

_{i}*W*such that the (1.4) holds for all

_{i}*i,j*Î

*I*.

A representation (*V _{i}*,

_{j}*j*) is called

_{i}**indecomposable**, if there are no non-zero sets of subspaces (

*U*) and (

_{i}*W*) such that

_{i}*V*=

_{i}*U*Å

_{i}*W*and

_{i}

_{j}*j*=

_{i}

_{j}*y*Å

_{i}

_{j}*t*

_{i}_{ }, where

(1.5) |

(1.6) |

One defines the direct sum of two *L*-representations
in the obvious way.

Denote by Rep(*L*) the category of all *L*-representations,
and by rep(*L*) the category of finite dimensional *L*-representations,
whose objects are *L*-representations and whose morphisms are as defined
above.

**Definition 1.7.** [2]
A species *L* = (*F _{i}*,

*)*

_{i}M_{j}

_{i,j}_{Î}

*is said to be of*

_{I}**finite type**, if the number of indecomposable non-isomorphic finite dimensional representations is finite.

A species *L* = (*F _{i}*,

*)*

_{i}M_{j}

_{i,j}_{Î}

*is said to be of*

_{I}**strongly unbounded type**if it possesses the following three properties:

1. *L* has indecomposable objects of
arbitrary large finite dimension.

2. If *L* contains a finite
dimensional object with an infinite endomorphism ring, then there is an
infinite number of (finite) dimensions *d* such that, for each *d*,
the species *L* has infinitely many (non-isomorphic) indecomposable
objects
of dimension *d*.

3. *L* has indecomposable objects of
infinite dimension.

Dlab and Ringel proved in [2, Theorem E]
that any *K*-species is either of finite or of strongly unbounded type.

With any species *L* = (*F _{i}*,

*)*

_{i}M_{j}

_{i,j}_{Î}

*one can define the tensor algebra in the following way. Let , and let . Then*

_{I}*B*is a ring and

*M*naturally becomes a (

*B*,

*B*)-bimodule. The

**tensor algebra**of the (

*B*,

*B*)-bimodule

*M*is the graded ring

(1.8) |

with component-wise addition and the multiplication induced by taking tensor products.

If *L* is a *K*-species, then *T*(*L*)
is a finite dimensional *K*-algebra.

**Theorem 1.9.** (Dlab, Ringel [2, Proposition 10.1]). Let *L*
be a *K*-species. Then the category Rep(*L*) of all representations
of *L* and the category Mod* _{r}*(

*T*(

*L*)) of all right

*T*(

*L*)-modules are equivalent.

2. *O*-species and their
representations

In this section we consider the notion of *O*-species,
which generalizes the notion of species considered in [1].

Let {*O _{i}*} be a family of
discrete valuation rings (not necessarily commutative)

*O*with radicals

_{i}*R*and skew fields of fractions

_{i}*D*, for

_{i}*i*= 1, 2, ...,

*k*, and let {

*D*}, for

_{j}*j*=

*k*+ 1, ...,

*n*, be a family of skew fields. Let (

*n*

_{1},

*n*

_{2}, ...,

*n*) be a set of natural numbers. Write

_{k} , |

which is a subring in the matrix ring . It is easy to
see that each
is a Noetherian serial prime hereditary ring. Write *F _{i}* = for

*i*= 1, 2, ...,

*k*, and

*F*=

_{j}*D*for

_{j}*j*=

*k*+ 1, ...,

*n*. Then, by the Goldie theorem, there exists a classical ring of fractions for

*i*=1,2, ...,

*n*.

Consider the following generalization of a species.

**Definition 2.1.** An ** O-species** is a set W = (

*F*,

_{i}*)*

_{i}M_{j}

_{i,j }_{Î I}, where

*F*=

_{i}_{ }for

*i*= 1, 2, ...,

*k*, and

*F*=

_{j}*D*for

_{j}*j*=

*k*+ 1, ...,

*n*, and moreover

*is an -bimodule, which is finite dimensional as a right*

_{i}M_{j}*D*-vector space and as a left

_{j}*D*-vector space.

_{i }An *O*-species W is called a (** D, O**)-

**species**if all

*O*have a common skew field of fractions

_{i }*D*, i.e. all

*D*are equal to a fixed skew field

_{i}*D*and

(2.2) |

for some natural number *n _{ij}* (

*i*= 1, 2, ...,

*n*).

An *O*-species W is called a (** K, O**)

**-species**, if all

*D*(

_{i}*i*= 1, 2, ...,

*n*) contain a common central subfield

*K*of finite index in such a way that

*lm*=

*m*

*l*for all

*l*Î

*K*and all

*m*Î

*(moreover, each bimodule*

_{i}M_{j}*is a finite dimensional vector space over*

_{i}M_{j}*K*). It is a (

*K*,

*O*)-quiver if moreover

*D*=

_{i}*D*for each

*i*.

Everywhere in this paper we will consider *O*-species
without oriented cycles and loops, i.e. we will assume that * _{i}M_{i}*
= 0, and if

*≠ 0, then*

_{i}M_{j}*= 0. A vertex*

_{j}M_{i}*i*is said to be

**marked**if

*F*=

_{i}_{ }.

We will also assume that all marked vertices
are minimal, i.e. * _{j}M_{i}* = 0 if

*F*=

_{i}_{ }, and that

*=*

_{i}M_{j }*= 0 if*

_{ j}M_{i}*i*,

*j*are marked vertices.

**Definition 2.3.** The **diagram **of an *O*-species W =_{ }{*F _{i}*,

*}*

_{i}M_{j}

_{i,j}_{ÎI}is defined in the following way:

1. The
set of vertices is a finite set *I* = {1, 2, ..., *n*}.

2. The
finite subset *I*_{0} = {1, 2, ..., *k*} of *I* is a set
of marked points.

3. The
vertex *i* connects with the vertex *j* by *t _{ij}*
arrows, where

moreover, we assume
that *n _{i}* = 1 if

*F*=

_{i}*D*.

_{i}Similar to species we can define
representations of *O*-species in the following way.

**Definition 2.4.** A **representation** (*M _{i}*,

*V*,

_{r}

_{j}*j*,

_{i}

_{j}*y*) of an

_{r}*O*-species W = {

*F*,

_{i}*}*

_{i}M_{j}

_{i,j}_{ÎI}is a family of right

*F*-modules

_{i}*M*(

_{i}*i*= 1, 2, ...,

*k*), a set of right vector spaces

*V*over

_{r}*D*(

_{r}*r*=

*k*+1,

*k*+1, ...,

*n*) and

*D*-linear maps:

_{j}for each *i* = 1, 2, ..., *k*; *j*
= *k* +1, *k* +2, ..., *n*; and

for each *r*, *j* = *k* +1, *k*
+2, ..., *n*.

**Definition 2.5.**
Two representations *M* = (*M _{i}*,

*V*,

_{r}

_{j}*j*,

_{i}

_{j}*y*) and are called

_{r}**equivalent**if there is a set of isomorphisms

*a*of

_{i}*F*-modules from

_{i}*M*to and a set of isomorphisms

_{i}*b*of

_{r}*D*-vector spaces from

_{r}*V*to

_{r}_{ }

_{ }such that for each

*i*= 1, 2, ...,

*k*;

*r*,

*j*=

*k*+ 1,

*k*+ 2, ...,

*n*the following equalities hold:

(2.6) |

(2.7) |

In a natural way one can define the notions of a direct sum of representations and of an indecomposable representation.

The set of all representations of an *O*-species
W =
(*F _{i}*,

*)*

_{i}M_{j}

_{i,j}_{ÎI}can be turned into a category

*R*(W), whose objects are representations

*M*= (

*M*,

_{i}*V*,

_{r}

_{j}*j*,

_{i}

_{j}*y*), and a morphism from object

_{r}*M*= (

*M*,

_{i}*V*,

_{r}

_{j}*j*,

_{i}

_{j}*y*) to object is a set of homomorphisms

_{r}*a*of

_{i}_{ }- modules

*M*to , and a set of homo- morphisms

_{i}*b*of

_{r}*D*- vector spaces from

_{r}*V*to such that for each

_{r}*i*= 1,2, ...,

*k*;

*r*,

*j*=

*k*+ 1,

*k*+ 2, ...,

*n*the equalities (2.6) and (2.7) hold.

3. Tensor algebra of O-species

For any *O*-species W = (*F _{i}*,

*)*

_{i}M_{j}

_{i,j }_{Î I}one can construct a tensor algebra of bimodules

*T*(W). Let , . Then

*B*is an (

*A*,

*A*) - bimodule and we can define a tensor algebra

*T*(

_{A}*B*) of the bimodule

*B*over the ring

*A*in the following way:

T(_{A}B) = A Å B Å B^{2}
Å ... Å B Å ...^{n} | (3.1) |

is a graded ring,
where *B ^{n}* =

*B*Ä

_{A}*B*

^{n}^{‒1}for

*n*>

_{ }1, and multiplication in

*T*(

_{A}*B*) is given by the natural

*A*-bilinear map:

B
´ ^{n}B
® ^{m}B
Ä^{n}_{A}B=^{m } B^{n+m} | (3.2) |

Then *T*(W) = *T _{A}*(

*B*) is the tensor algebra corresponding to an

*O*-species W.

**Proposition 3.3.** Let
W be
an *O*-species. Then the category (W) of all
representations of W and the category Mod_{r}* T*(W) of all right *T*(W)-modules are
naturally equivalent.

*Proof.* Form two functors *R*: Mod_{r}* T*(W) ® (W) and *P*: (W) ® Mod_{r}* T*(W)
in the following way. Let *X _{T}*

_{(}

_{W)}be a right

*T*(W)-module. Since

*A*is a subring in

*T*(W),

*X*can be considered as a right

*A*-module. Then

(3.4) |

where *M _{i}*
is an -module, and

*V*is a

_{r}*D*-vector space; moreover, for

_{r}*i*≠

*j*, and

*V*= 0 for

_{r}D_{s}*r*≠

*s*. Since

*B*is an (

*A*,

*A*)-bimodule, one can define an

*A*-homomorphism

*j*:

*X*Ä

_{A}*B*®

*X*. Taking into account that

_{A}*M*Ä

_{i}

_{A}*= 0 for*

_{s}M_{j}*i*≠

*s*, the map

*j*is defined in the following way:

(3.5) |

Since *M _{i}* Ä

_{A}*is mapping into*

_{i}M_{j}*V*, and

_{j}*V*Ä

_{r }

_{A}*is mapping into*

_{r}M_{j }*V*,

_{j}*j*defines a set of

*D*-homomorphisms:

_{j}(3.6) |

(3.7) |

for *i* = 1, 2, ..., *k*; *r*, *j*
= *k* + 1, ..., *n*.

Now one can define *R*(*X _{T}*

_{(}

_{W)}) = (

*M*,

_{i}*V*,

_{r}

_{j}*j*,

_{i}

_{j}*y*). Let

_{r}*X*,

*Y*be two right

*T*(W)-modules, let a:

*X*®

*Y*be a homomorphism, and let

*R*(

*X*) = (

*M*,

_{i}*V*,

_{r}

_{j}*j*,

_{i}

_{j}*y*),

_{r}*R*(

*Y*) = . Let's define a morphism from

*R*(

*X*) to

*R*(

*Y*). Since

*a*is an

*A*-homomorphism,

*a*(

*M*) Í

_{i}*N*,

_{i}*a*(

*V*) Í

_{r}*W*, i.e.,

_{r}*a*defines a family of -homomorphisms

*a*:

_{i}*M*®

_{i}*N*and a family of

_{i}*D*-homomorphisms

_{r}*b*:

_{r}*V*®

_{r}*W*, which are the restrictions of a to

_{r}*M*and

_{i}*V*. Therefore one can set

_{r}*R*(

*a*) = {(

*a*), (

_{i}*b*)}. Since a is a

_{r}*T*(W)-homomorphism,

(3.8) |

and

(3.9) |

for *i* = 1, 2, ..., *k*; *r*,
*j* = *k *+ 1, ..., *n*. Therefore *R*(*a*) is a
morphism in the category *R*(W).

Conversely, let W = (*F _{i}*,

*)*

_{i}M_{j}

_{i,j}_{ÎI}and there is given a representation

*M*= (

*M*,

_{i}*V*,

_{r}

_{j}*j*,

_{i}

_{j}*y*). Then one can define

_{r}*P*(

*M*) in the following way:

(3.10) |

We define an action of

(3.11) |

on *M _{i}*
by means of the projection

*A*®

_{ }and an action of

*A*on

*V*by means of the projection

_{r}*A*®

*D*. We define an action of

_{r}*B*on

^{n}*X*by induction of

*j*

^{ (n)}:

*X*Ä

*®*

_{A }B^{n}*X*as follows:

If *a* = {{*a _{i}*}, {

*b*}} is a morphism of a representation

_{r}*M*= (

*M*,

_{i}*V*,

_{r}

_{j}*j*,

_{i}

_{j}*y*) to a representation ,

_{r}*X*=

*P*(

*M*),

*Y*=

*P*(), then

(3.12) |

is a *T*(W)-homomorphism and
therefore *P*(*a*) = *j*.

It is not difficult to show that *R*, *P*
are mutually inverse functors and they give an equivalence of categories Mod_{r}*
T*(W) and (W).

Recall that an Artinian ring *A* is of **finite
representation type** if *A* has only a finite number of
indecomposable finitely generated right *A*-modules up to isomorphism.

A ring *A* is of (right) **bounded
representation type** (see [3, 4]) if there is an upper bound on the
number of generators required for indecomposable finitely presented right *A*-modules.

Denote by
*m*(*M _{i}*)
the minimal number of generators of an -module

*M*, and denote by

_{i}*d*

_{r}_{ }=

_{ }the dimension of vector space

*V*over

_{r}*D*. The dimension of a representation

_{r}*M*= (

*M*,

_{i}*V*,

_{r}

_{j}*j*,

_{i}

_{j}*y*) is the number

_{r}(3.13) |

**Definition 3.14.** An
*O*-species W is said to be of **bounded representation type
**if the dimensions of its indecomposable finite dimensional representations
have
an upper bound.

**Corollary 3.15.**
An *O*-species W is of bounded representation type if and only
if the tensor algebra *T*(W) is of bounded representation type.

*Proof*. If W is an *O*-species
of bounded representation type, then there exists *N* > 0 such
that dim*M* < *N* for any indecomposable finite dimensional
representation *M*. Then for any finitely generated *T*(W)-module *X *we
have m(*X*) < *N*_{1}, where *N*_{1} is
some fixed number depending on *N*, i.e. *T*(W) is a ring of bounded
representation type. The converse also holds: if *T*(W) is a ring of
bounded representation type, then W is an *O*-species of bounded representation
type.

**Corollary 3.16.** Let W_{1} be a *D*-species, which is a
subspecies of a (*D*, *O*)-species W. If W is of bounded
representation type, then W_{1} is of finite type.

*Proof*. Since W is of bounded representation type, each of its subspecies is of bounded
representation type as well. So W_{1 }is of
bounded representation type, and, by corollary 3.15, its tensor algebra is of
bounded representation type, as well. Since W_{1} is a *D*-species, its tensor algebra is an Artinian ring. So
it is of finite representation type, by [5]. Therefore, W_{1} is also of finite representation type.

3. Conclusion

In this paper we introduced *O*-species and
the tensor algebras corresponding
to them. These *O*-species are some generalizations of species first
introduced by Gabriel in [1]. We consider the notion of a representation of an *O*-species.
In this paper we prove that the category of all representations of *O*-species
W and
the category of all right modules over a tensor algebra T(W) are naturally
equivalent.

References

[1] Gabriel P., Indecomposable representations I, Manuscripta Math. 1972, 6, 71-103.

[2] Dlab V., Ringel C.M., On algebras of finite representation type, Journal of Algebra 1975, 33, 306-394.

[3] Warfield R.B., Serial rings and finitely presented modules, Journal of Algebra 1975, 37, 2, 187-222.

[4] Gubareni N., On right hereditary SPSD-rings of bounded representation type I, Scientific Research of the Institute of Mathematics and Computer Science 2012, 3(11), 57-70.

[5] Auslander M., Representation theory of Artin algebras II, Comm. Algebra 1974, 1, 269-310.