On monomial algebras with representation-finite enveloping algebras
Jianguo Zhou Yu-Zhe Liu Chao Zhang



On monomial algebras with repr-finite enveloping algebras]On monomial algebras with representation-finite enveloping algebras
^∗Corresponding author.
MSC2020:
16G10, 
16G60, 
16G70. 

Key words: opposite algebras, representation type, string algebras, tensor algebras.


School of Mathematics and statistics, Guizhou University, Guiyang 550025, P. R. China
14785841291@163.com


School of Mathematics and statistics, Guizhou University, Guiyang 550025, P. R. China
liuyz@gzu.edu.cn / yzliu3@163.com


School of Mathematics and statistics, Guizhou University, Guiyang 550025, P. R. China
zhangc@amss.ac.cn





























The present paper mainly considers the representation type of the enveloping algebra of monomial algebra.
Let A be a monomial algebra and A^e= A⊗_k A^op its enveloping algebra.
It is shown that A^e is representation-finite if and only if A ≅A_n/rad^2 A_n,
where A_n is the path algebra k𝒬 with 𝒬 = 1 ⟶ 2 ⟶⋯⟶ n.
Moreover, we show that the number of all isoclasses of indecomposable  (A_n/ rad^2A_n)^e-modules is 4/3n^3 + n^2-7/3n+1,
and classify all indecomposable modules over (A_n/ rad^2A_n)^e.
Finally, the Clebsch-Gordon problem over (A_n/ rad^2A_n)^e is studied.
 

[
    Chao Zhang^∗
    April 29, 2024
==================












§ INTRODUCTION


Throughout this paper, unless stated otherwise,  is an algebraically closed field, all
algebras are connected basic finite-dimensional associative
-algebras with identity, and all modules are finite-dimensional right modules. The concept of the tensor product of rings goes back to the research of group representations, which is introduced by Frobenius and also plays an important role in the representation theory of algebras.

Tensor algebra is closely related to the Clebsch-Gordon problem (see <cit.>) :


for any two indecomposable representations of a group or ring, how to decompose their tensor products into a direct sum of indecomposable representations?
To be more precise, for two finite-dimensional algebras A and B over a field ,
it is well-known that A⊗_ B is a finite-dimensional algebra and the tensor M⊗_ N of right A-module M_A and right B-module N_B can be seen as a right A⊗_B-module by the following formula

    M⊗_ N  × A⊗_ B → M⊗_ N, (m⊗ n)·(a⊗ b) := ma ⊗ nb.

Clebsch-Gordon problem concerns the decomposition of M⊗_ N over A⊗_B

    M⊗_ N = ⊕_i∈ I X_i

such that each X_i is an indecomposable right A⊗_B-module.
To some extent, Clebsch-Gordon problem concerns the description of the indecomposable A⊗_B-modules,
which is closely related to the representation-type of algebra. The representation-type of an algebra characterizes the complexity of this algebra,
and its research has greatly promoted the development of representation theory
<cit.>.
In <cit.>, Clebsch-Gordon problem over [x] was extensively studied and all tensor products coming from bialgebra structures were classified.
Subsequently, this problem on Kronecker algberas, algebras of type 𝔸_n, and polynomial algebras was studied in <cit.>.
It was proved in <cit.> that the bound quiver (_A⊗_B, _A⊗_B) of A⊗_B is the tensor of quivers _A and _B through Gabriel's works in <cit.>.


Generally, the homological properties of the tensor product of two rings are discussed in terms of these two rings.
For instance, for any Noetherian rings A and B, the Gorenstein global dimension Ggldim(A⊗_ B) can be described by that of A and B, see <cit.>.
Moreover, for an algebra A, we can choose a special algebra B whose properties have been thoroughly studied,
and use A⊗_ B to depict the properties of A, see <cit.>.

So it is natural to consider the representation type of A⊗ B in terms of the representation type of A and B. To be more precise, we propose a natural question as follows.

 
	Let A and B be two finite-dimensional -algebras.
	Which kind of algebras A and B can ensure that A⊗_ B is representation-finite?


As a special kind of tensor product of algebras, the enveloping algebra A^e of A is defined by A ⊗_ A^, where A^ is the opposite algebra of A (see <cit.> for more details).
Any (A,A)-bimodule M = _AM_A is natural right A^e-module M = M_A^e by

    M_A^e× A^e → M_A^e, m·(a⊗ a') := a'ma.

Many properties of A can be described by its enveloping algebra A^e,

such as <cit.>.

Moreover, the Hochschild cohomology and homology, defined over the enveloping algebra A^e,  play an important role in the
research of the properties of A including the simply connectedness, separability, deformation theory, the global dimension and so on (see <cit.>).


































In this paper, we consider the enveloping algebra of monomial algebras, and show the following result.

  Let A be a monomial algebra.
The enveloping algebra of A is representation-finite if and only if A≅_n/^2_n := _n/^2(_n), where _n is defined in Notation <ref>.


This theorem answers the Question <ref> for the case that B=A^ and A is a monomial algebra. Furthermore, we provide
a classification (up to isomorphism) and a counting formula for indecomposable modules over _n/^2_n as follows.

  If A=_n/ ^2_n, then we have the following statement.


  
  (1) Any indecomposable module over A^e is either a string module or an indecomposable projective-injective module.
  
  (2) The number of indecomposable modules over A^e is
  
    ♯( A^e) = 4/3n^3 + n^2-7/3n+1.




The above formula precisely classifies all indecomposable right modules over (_n/ ^2_n)^e (see Theorem <ref>).

Let  be a path on the underlying graph of the bound quiver (, ) of a monomial algebra.
A V-sequence ω is a path on  such that any subpath of ω in  is a walk without relations, see Definition <ref> for details.
Considering the Clebsch-Gordon problem for (_n/^2_n)^e, we obtain the following corollary.

  If A=_n/ ^2_n, then all indecomposable right A^e-bimodule can be described by two classes as follows.
  
    
  * The tensor of two indecomposable modules over A;
    
  * The string A^e-module corresponding to V-sequence of length ≥ 2.
  



This paper is organized as follows. In Section <ref>,
some preliminaries on the monomial algebras, string algebras and their enveloping algebras are recalled.
In Section <ref>, we prove some lemmas and calculate some representation-types of monomial algebras and their enveloping algebras.
In Section <ref>, we prove our main results.



§ MONOMIAL ALGEBRAS AND THEIR ENVELOPING ALGEBRAS
 

In this section, we recall some notations and concepts for monomial, biserial and string algebras provided by <cit.>.



 §.§ Monomial algebras and string algebras


Let  be an algebraically closed field. Throughout this paper, a quiver is a quadruple = (_0, _1, s, t), where _0 is the set of vertices, _1 is the set of arrows, and s and t are functions _1 →_0 sending any arrow a∈_1 to its source and target, respectively.
If a and b are arrows with t(a)=s(b),  we denote by ab the composition of a and b.
All algebras in our paper are finitely dimensional -algebras, and all modules are finite generated right A-modules.

For any arrow α∈_1, we denote by α^-1 its formal inverse with s(α^-1)=t(α) and t(α^-1)=s(α), and _1^-1:= {α^-1|α∈ Q_1 }.
For a path p=α_1α_2⋯α_r, its formal inverse is p^-1=α_r^-1α_r-1^-1⋯α_1^-1. A sequence w=w_1w_2⋯ w_n is a walk if
each w_i is of form α or α^-1 with α∈_1, and
s(w_i+1)=t(w_i) for i=1, 2, ⋯, n-1. For a walk w=w_1w_2⋯ w_n, we say its length ℓ(w)=n,
and naturally define that s(w)=s(w_1) and t(w)=t(w_n).

A monomial algebra is a finite-dimensional -algebra which is Morita equivalent to / such that  is generated by some paths of length ≥ 2.
A biserial algebra A=/ with admissible ideal  is a finite-dimensional algebra such that the radical of any non-uniserial indecomposable projective module is a sum of two uniserial modules whose intersection is either simple or zero.
Thus, it is easy to see that the bound quiver (, ) of A satisfies the following condition.

	
  * Each vertex v∈_0 is the source of at most two arrow.

A biserial algebra is said to be special biserial if it moreover satisfies the following condition.

	
  * Each vertex v∈_0 is the target of at most two arrows.
	
  * For each arrow β∈_1, there is at most one arrow α∈_1 and at most one arrow γ∈_1
	such that αβ and βγ are not belong to .

Furthermore, an algebra is a string algebra if it is both a monomial algebra and a special beserial algebra.


Let A=/ be an algebra. A bound quiver (', ') is called to be a bound full subquiver of (, ) determined by ', if ' is a subquiver of Q
with '_1={a∈_1| s(a),t(a)∈_0'}
and



    '  =    ⟨{a_1a_2⋯ a_n∈| a_1, a_2,⋯, a_n ∈'_1}∪{∑_i∈ I_'℘_i |∑_i∈ I℘_i ∈}⟩
        (= |_' for short),

where every ℘_i (i∈ I) is a path of length ≥ 2 and I_' is the subset of I such that I\ I_' is the set of all paths crossing at least one vertex belong in _0\_0'.
It is clear that if A=/ is a monomial (resp. biserial, special biserial, string) algebra, so is A'='/' for any bound full subquiver (', ').

 
Assume A=/ with =

    @R=0.3cm          9 [rd]^b_3'            
           6 [r]_b_2[ru]^b_2'    7 [r]_b_3    8 [rd]^b_4         
    
    1 [r]^a_1    2 [rr]^a_2[ru]^b_1[rd]_c_1       3 [rr]^a_3       4 [r]^a_4    5 
           10 [rr]_c_2       11 [ru]_c_3

and
=⟨ a_2a_3+b_1b_2b_3b_4+c_1c_2c_3, b_2b_3-b_2'b_3', b_4a_4 ⟩.

Take '_0=_0\{9} and _1'=_1\{b_2', b_3'}. Then '=('_0,'_1,s,t) is a full subquiver of .
For the relation b_2b_3-b_2'b_3' ∈, let ℘_=b_2b_3 and ℘_=b_2'b_3', then we have
∑_i∈_'℘_i = ℘_=b_2b_3.
Therefore, the ideal of the full bound subquiver (', ') is ' = ⟨ a_2a_3+b_1b_2b_3b_4+c_1c_2c_3, b_2b_3, b_4a_4 ⟩.
Since b_2'b_3'=0 in the quotient A/⟨ e_9 + ⟩ of A=/, we obtain

    b_2b_3+ = b_2'b_3'+ =  = 0 
     and     a_2a_3+b_1b_2b_3b_4+c_1c_2c_3+ = a_2a_3+ c_1c_2c_3+

in A/⟨ e_9 + ⟩. Furthermore, we have the natural epimorphism
π: A ⟶ A'='/'≅ A/⟨ e_9 + ⟩.


The bound quiver of string algebra contains two important classes of sequences which are called strings and bands respectively. 
The classification of indecomposable modules over any biserial algebra is provided by “V-sequences” and “primitive V-sequences” <cit.>. Butler and Ringel then classified all indecomposable modules over string algebra by strings and bands in <cit.>.

Next, we recall some notations about V-sequences, primitive V-sequences, strings and bands.
Note that the definitions of V-sequences and primitive V-sequences make sense for any algebra and the definitions of strings and bands make sense for any monomial algebra.


A V-sequence <cit.> is a walk ω = ω_1 ω_2  ⋯ω_n with ω_i ∈_1∪_1^-1 such that s(ω_i+1)=t(ω_i) and the following conditions hold.

    
  (1) ω has no subsequence which is of the form aa^-1 (a ∈_1∪_1^-1).
    
  (2) Any path on ω does not belong to , that is,
      for any subwalk ω=ω_tω_t+1⋯ω_t+ℓ with ω_t, ω_t+1, …, ω_t+ℓ∈_1 (resp., ∈_1^-1), we have ω∉ (resp., ω^-1∉).

Two V-sequences ω and ω' are called equivalent if  ω = ω' or ω = ω'^-1.
Furthermore, if A is a monomial algebra, then V-sequences is called a string.

A primitive V-sequence<cit.> is a V-sequence β such that the following conditions hold.

    
  (1) s(β) and t(β) coincide.
    
  (2) β^m is a V-sequence for any positive integer m.
    
  (3) β is not a non-trivial power of any V-sequence, that is,
    β≠α^s for any V-sequence α and positive integer s≥ 2.

Two primitive V-sequences β=β_1⋯β_m and β'=β_1'⋯β_n' are called equivalent if β = β'[t] or  β = (β'[t])^-1 for some integer t, where β'[t]=β_t'⋯β_n'β'_1⋯β'_t-1.
Furthermore, if A is a monomial algebra, then primitive V-sequences is called a band.


We denote by [ω] (resp. [β]) the equivalent class of V-sequence ω (resp. primitive V-sequence β), 𝒥 the set of all Jordan blocks with non-zero eigenvalue.
For any algebra A=/, we define

    (A)     = {[ω] |ω is a V-sequence on the bound quiver of  A }, 
    (A)     = {[β] |β is a primitive V-sequence on the bound quiver of  A }.

All indecomposable modules over special biserial algebra can be described by the following theorem.

   <cit.>
Let A be a biserial algebra. Then there is a surjection

    : (A) ∪ ((A)×𝒥) →( A)

such that Im(|_(A))∩Im(|_(A)×𝒥) = ∅, where 𝒥 is the set of all Jordan blocks with nonzero eigenvalue.
Moreover, if all projective-injective A-module are uniserial, then the above corresponding  is a bijection.


  
  
  (1) Note that the concept of V-sequence (resp. primitive V-sequence) is a generalization of the concept of string (resp. band). For the convenience, we call the indecomposable modules in Im(|_(A)) string modules and the indecomposable modules in Im(|_(A)×𝒥) band modules.

  
  (2) We call that a primitive V-sequence β is a primitive V-sequence without relation,
    if any path on β is not a component of some generator of .
    Assume β =
    
    v_1 @-@/^1pc/[rrr]^β_n@-[r]_β_1    v_2 @-[r]_β_2   ⋯@-[r]     v_n∈(A)

    
    (β_1, ⋯, β_n ∈_1∪_1^-1)

    is a primitive V-sequence without relation.
    If β is on a quiver of an algebra A, then (β, J_t(λ)) is an indecomposable representation
    
    ^t @-@/^1pc/[rrr]^J_n(λ)@-[r]_𝕀   ^t @-[r]_𝕀   ⋯@-[r]_𝕀   ^t

    of A with dimension _(β, J_t(λ)) = nt,
    where J_t(λ) ∈𝒥 is a t× t Jordan block
    (it is similar for general V-sequence without relation, see <cit.>).
    In this case, |_{β}×𝒥 is injective, and
    
    Im(|_{β}×𝒥) = {(β, J_t(λ)) |λ∈, t∈ℕ^+}

    is an infinite set. Therefore, A is representation-infinite.
    In particular, since any band is a primitive V-sequence without relation, if the bound quiver of a monomial algebra A contains a band, then A is representation-infinite.




The opposite algebra A^ of an algebra A=/ is the algebra which is isomorphic to ^/^,
where ^=(_0^, _1^, s^, t^), where 

  
  * _0^ = {v^| v∈_0};

  
  * _1^ = {a^| a∈_1};

  
  * s^(a^)=t(a) and t^(a^)=s(a) (∀ a∈_1);

  
  * ^ is naturally induced by , that is,
    
      
  * a_1a_2⋯ a_n ∈ if and only if a^_n⋯ a^_2a^_1∈^,
      
  * and ∑_i∈ I℘_i ∈ if and only if ∑_i∈ I℘_i^∈^
      (every ℘_i is a path of length ≥ 2).
    



 By the definition of opposite algebra, it is easy to see that the opposite algebra of any monomial (resp. string) algebra is also a monomial (resp. string) algebra.


Let A=/ be a finite-dimensional -algebra (, ) given by the quiver

    1 [r]_a @/^1pc/[rrr]^d     2[r]_b      3 [r]_c     4

and the admissible ideal =⟨ abc ⟩. This is a string algebra.
The opposite algebra A^ of A is isomorphic to ^/^, where ^=

    1^@<-[r]_a^@<-@/^1pc/[rrr]^d^    2^@<-[r]_b^    3^@<-[r]_c^    4^

and ^=⟨ c^b^a^⟩. In particular, A^ is also a string algebra.


  In this paper, we denote by
_n:=_n and _n:=_n, where

    _n = 
    1 [r]     2 [r]     3 [r]    ⋯[r]     n-1 [r]     n

and

    _n = 
    1 [r] @/_2pc/@<-[rrrrr]     2 [r]     3 [r]    ⋯[r]     n-1 [r]     n.

Moreover, for any algebra A=, we denote by ^2A the ideal ⟨αβ:α,β∈_1 ⟩ in our paper.




 §.§ The enveloping algebras

Let A = _A/_A and B=_B/_B be two finite-dimensional algebras, the -tensor algebra of A and B is the tensor A⊗_ B of A and B as -vector spaces.
By <cit.>, the quiver _A⊗_B of A⊗_ B = _A⊗_B / _A⊗_B is the quadruple ((_A⊗_B)_0, (_A⊗_B)_1, s, t), where 

    
  * (_A⊗_B)_0 = (_A)_0×(_B)_0 = {(i,j): i ∈ (_A)_0, j ∈ (_B)_0},
    
  * (_A⊗_B)_1={e_i⊗ b : i ∈ (_A)_0, b∈ (_B)_1 } ∪  {a⊗ e_j :a∈ (_A)_1, j∈ (_B)_0},
    
  * s: (_A⊗_B)_1 → (_A⊗_B)_0 is the function respectively sending any arrows e_i⊗ b and a⊗ e_j to the elements (i,s(b)) and (s(a), j),
    
  * and t: (_A⊗_B)_1 → (_A⊗_B)_0  is the function respectively sending any arrows e_i⊗ b and a⊗ e_j to the elements (i,t(b)) and (t(a), j).

The ideal _A⊗_B is induced by _A, _B, which is generated by the following relations.

  
  * e_i ⊗℘_B (i∈ (_A)_0), where ℘_B is a generator of _B,
  
  * ℘_A ⊗ e_j (j∈ (_B)_0), where ℘_A is a generator of _A,
  
  * (e_i⊗β)(α⊗ e_j') - (α⊗ e_i')(e_j⊗β),
    where α: i→ j is an arrow in (_A)_1 and β: i'→ j' is an arrow in (_B)_1,
    that is, any subquiver
    
    (i,j') [r]^α⊗ e_j'    (j,j')  
    
        (i,i') [u]^e_i⊗β[r]_α⊗ e_i'    (j,i') [u]_e_j⊗β

    of _A⊗_B is commutative.


Let A=/ be an arbitrary algebra. The -tensor algebra A^e := A ⊗_ A^ is called the enveloping algebra of A.


Obviously, we have

    A^e≅^e/^e,

where (^e, ^e) is the bound quiver given by

    
  * ^e_0={(i,j^): i ∈_0, j^∈_0^},
    
  * ^e_1={e_x⊗y^ : x ∈_0, y∈_1^} ∪  {x⊗e_y^ :x∈_1, y∈_0^},
    
  * and ^e=_A⊗_ A^ induced by  as above.



  Let A=/ be a finite-dimensional -algebra (, ) given by the quiver

    1 [r]^a     2[r]^b      3

and the admissible ideal =⟨ ab ⟩.
Then the enveloping algebra of A^e, is isomorphic to ^e/^e, where ^e=

    @C=1.5cm@R=1.5cm
    (1,1^) [r]^a ⊗ e_1^    (2,1^) [r]^b ⊗_ e_1^    (3,1^) 
    
    (1,2^) [r]^a ⊗ e_2^[u]^e_1 ⊗ a^   
    (2,2^) [r]^b ⊗ e_2^[u]^e_2 ⊗ a^   
    (3,2^) [u]^e_3 ⊗ a^
    
    (1,3^) [r]^a ⊗ e_3^[u]^e_1 ⊗ b^   
    (2,3^) [r]^b ⊗ e_3^[u]^e_2 ⊗ b^   
    (3,3^) [u]^e_3 ⊗ b^

and

    ^e =     ⟨{ (e_x ⊗ b^)(e_x ⊗ a^): 1≤ x≤ 3 }
       ∪{ (a ⊗ e_x^)(b ⊗ e_x^): 1≤ x≤ 3 }
       ∪{ (e_1⊗ a^)(a⊗ e_1^) - (a⊗ e_2^)(e_2⊗ a^), 
        (e_2⊗ a^)(b⊗ e_1^) - (b⊗ e_2^)(e_3⊗ a^), 
        (e_1⊗ b^)(a⊗ e_2^) - (a⊗ e_3^)(e_2⊗ b^), 
        (e_2⊗ b^)(b⊗ e_2^) - (b⊗ e_3^)(e_3⊗ b^)}⟩.





§ THE REPRESENTATION-TYPES OF ENVELOPING ALGEBRAS OF MONOMIAL ALGEBRAS


Homological dimensions of algebras can be described by their enveloping algebras <cit.>. We intend to describe the properties of quivers for monomial algebras by the representation-types of enveloping algebras in our paper.
To do this, we prove that the enveloping algebras of some monomial algebras are representation-infinite in this section.



 §.§ The representation types of algebras


We always assume that ≅/ is an algebra whose quiver  is connected.
We denote by () the set of all isoclasses of indecomposable objects lying in , and Γ( A) the Auslander-Reiten quiver of A.
An algebra  is called representation-finite (resp. representation-infinite) if () is a finite set (resp. an infinite set).
By Auslander-Reiten theory, the vertex set Γ( A)_0 of Γ( A) describes all elements in (), i.e., ♯Γ( A)_0=♯().
Thus, the representation-types of algebras can be judged directly by their Auslander-Reiten quiver.

 
Let _2 be the finite-dimensional -algebra given by Example <ref>.
Then the enveloping algebra _2^e of _2 is representation-finite.
Indeed, we can compute a connected component 𝒞 of Γ(_2^e) as follows

    @C=0.18cm@R=0.18cm
                [ 0 0 1; 0 0 1; 0 0 0 ][rd]       [ 0 0 0; 0 1 0; 0 1 0 ][rd]       [ 0 0 0; 0 0 0; 1 0 0 ][rd]       [ 0 0 0; 1 1 0; 0 0 0 ][rd]       [ 0 1 1; 0 0 0; 0 0 0 ][rd]             
    
    
       [ 0 0 0; 0 0 1; 0 0 1 ][rd]
          [ 0 0 0; 0 0 1; 0 0 0 ][ru] [rd]       [ 0 0 1; 0 1 1; 0 1 0 ][rd] [ru]       [ 0 0 0; 0 1 0; 1 1 0 ][ru] [rd] [r]   [ 0 0 0; 1 1 0; 1 1 0 ][r]    [ 0 0 0; 1 1 0; 1 0 0 ][ru] [rd]       [ 0 1 1; 1 1 0; 0 0 0 ][ru] [rd]       [ 0 1 0; 0 0 0; 0 0 0 ][rd]       [ 1 0 0; 1 0 0; 0 0 0 ][rd]    
    
    
    [ 0 0 0; 0 0 0; 0 0 1 ][ru] [rd]       [ 0 0 0; 0 0 1; 0 1 1 ][r] [ru] [rd]    [ 0 0 0; 0 1 1; 0 1 1 ][r]    [ 0 0 0; 0 1 1; 0 1 0 ][ru] [rd]       [ 0 0 1; 0 1 1; 1 1 0 ][ru] [rd]       [ 0 0 0; 0 1 0; 0 0 0 ][ru] [rd]        [ 0 1 1; 1 1 0; 1 0 0 ][ru] [rd]        [ 0 1 0; 1 1 0; 0 0 0 ][r] [rd] [ru]    [ 1 1 0; 1 1 0; 0 0 0 ][r]    [ 1 1 0; 1 0 0; 0 0 0 ][ru] [rd]       [ 1 0 0; 0 0 0; 0 0 0 ]
    
    
       [ 0 0 0; 0 0 0; 0 1 1 ][ru]
          [ 0 0 0; 0 0 0; 0 1 0 ][rd] [ru]       [ 0 0 0; 0 1 1; 1 1 0 ][ru] [rd]       [ 0 0 1; 0 1 1; 0 0 0 ][ru] [rd] [r]   [ 0 1 1; 0 1 1; 0 0 0 ][r]    [ 0 1 1; 0 1 0; 0 0 0 ][ru] [rd]       [ 0 1 0; 1 1 0; 1 0 0 ][ru] [rd]       [ 0 0 0; 1 0 0; 0 0 0 ][ru]       [ 1 1 0; 0 0 0; 0 0 0 ][ru]    
    
    
                [ 0 0 0; 0 0 0; 1 1 0 ][ru]       [ 0 0 0; 0 1 1; 0 0 0 ][ru]       [ 0 0 1; 0 0 0; 0 0 0 ][ru]       [ 0 1 0; 0 1 0; 0 0 0 ][ru]       [ 0 0 0; 1 0 0; 1 0 0 ][ru]

All indecomposable modules provided by the component 𝒞 are finite dimensional, i.e., they are finite length modules.
Thus, by <cit.>, we have 𝒞=Γ(_2^e).


  Assume that _3=/, where (, ) is given by

    = x@(ur,dr)^α and=⟨α^m⟩ (m ≥ 2).

Then _3^≅_3, and thus the enveloping algebra of _3 is

    _3^e ≅_3⊗__3 ≅⟨ x,y⟩/⟨ x^m, y^m, xy-yx⟩

which is a representation-infinite algebra (cf. <cit.>).


As to the representation type, we have the following well-known lemma.

  Let (, ) be a bound quiver of an algebra A.

  
  (1) Suppose that  (', ') is a bound full subquiver of (, ).
    If A'='/' is representation-infinite, then A is representation-infinite.

  
  (2) If A is representation-infinite, then so is A^e.




(1)  Assume

    ' = ⟨{a_1a_2⋯ a_n∈| a_1, a_2,⋯, a_n ∈'_1}∪{∑_i∈ I_'℘_i |∑_i∈ I℘_i ∈}⟩,

where every ℘_i (i∈ I) is a path of length ≥ 2 and I_' is the subset of I such that I\ I_' is the set of all paths crossing at least one vertex belong in _0\_0'.
Let e = ∑_t∈_0\_0'e_t. Then A/⟨ e⟩≅'/'=A' because for any ∑_i∈ I℘_i = ∑_i∈ I_'℘_i + ∑_i∈ I\ I_'℘_i in ,
we have ∑_i∈ I\ I_'℘_i equals to zero in A/⟨ e+⟩.
Thus, we have an epimorphism A→ A', and then, naturally, any indecomposable A'-module can be seen as an indecomposable A-module.
Therefore, if A'='/' is representation-infinite then so is A.

(2) By the presentation of A^e, if (, ) contains no loop, then the bound quiver (, ) of A=/ can be seen as a bound full subquiver of the bound quiver (^e, ^e) of A^e = ^e/^e.
Thus, the statement holds by (1) in this case.
If (, ) contains a loop, then (^e, ^e) contains a subquiver '=x@(ur,dr)^α@(ul,dl)_α^ with
α^t=(α^)^t=0 (t∈ℕ^+, t≥ 2) and αα^=α^α.
Therefore, A^e is representation-infinite because α^-1α^ is a band.
Indeed, in this situation, the representation-type of A^e is independent on the representation-type of A.



 An alternative proof of the first statement of the above lemma can be provided using the cleaving functor method originally introduced in <cit.> (see also <cit.>).
Let (_A, _A) and (_B, _B) be bound quivers of algebras A and B, respectively.
View bounded quivers as categories naturally, there is a  -linear functor F: B→ A given by the bound quiver homomorphism (_B, _B) → (_A, _A) deduces a retraction functor F_*:  A → B sending M∈ A to F_*M=M∘ F.
The functor F_* admits as a left adjoint functor F^*:  B→ A, say extension functor, which is uniquely determined up to isomorphism by the requirement that it commutes with direct limits  (i.e. with cokernels and direct sums) and that F^*_(-,x)≃_'(-,Fx), ∀ x∈ B.
The composition F_*F^* is connected to the identity 𝕀_ B by the canonical transformation Φ_F: 𝕀_ B→ F_*F^* which is uniquely determined by the requirement that Φ_F(_B(-, M)) coincides for each M∈ B with the map _B(-, M) →_A(F(-), FM) associated with F.
The functor F above is called cleaving if Φ_F is a retraction.

Take a bound full subquiver (',') of (_A, _A) and an idempotent e=∑_v∈(_A)_0\_0'e_v.
Assume B=eAe=_B/_B, then we obtain a bound quiver homomorphism F: (_B, _B) → (_A, _A) sending any path on (_B, _B) onto a path on (_A, _A) because eAe is a -linear subspace of A up to -linear isomorphism.
Then F induces a -functor F: eAe → A and obtain a retraction functor F_*: A → B sending any M∈ A to Me ∈ B.
By the adjoint isomorphism

    _A(-⊗_B eA, □) ≃_B(-,_A(□, eA)) ≃_B(-, F_*(□)),

we obtain F^*≃ -⊗_B eA. Then F_*F^* ≃ (-⊗_B eA) e = -⊗_B B ≃𝕀_ B.
Thus, F is a cleaving functor and deduces a functor F_*: ('/') → A which is an embedding. Therefore, if A'='/' is representation-infinite then so is A.




 §.§ The representation-types of enveloping algebras of monomial algebras

In this subsection, some results on representation-type of enveloping algebras  in this paper are deduced, which are useful in the proof of our main result in the next section.
We assume that A is a monomial algebra whose quiver is connected.

 
The enveloping algebra A^e of A = _n/^2_n (n∈^+) is representation-infinite.



The case of n=1 is given by Example <ref>.

Suppose that n≥ 2, and A ≅_A/_A with _A =

    1 [r]_a_1@/^1pc/@<-[rrr]^a_n    2 [r]_a_2   ⋯[r]_a_n-1    n,

= ⟨ a_1a_2,a_2a_3,...,a_n-1a_n, a_na_1 ⟩.
By the definition of enveloping algebra, we have A^e ≅_A^e/_A^e, where _A^e is given by Figure <ref> with ijrs is the arrow from the vertex ij to the vertex rs, and _A^e is induced by _A.



We consider the walk 𝔟 marked in Figure <ref>, where

    𝔟 =    1112·n212^-1·n2n3·n-13n3^-1·
       ⋯·3n-13n·2n3n^-1·2n21·1121^-1.

If n>2, we consider the subquiver (', ') of (, ) determined by the walk 𝔟, where

    @C=0.5cm@R=0.3cm      12@<-[r]    n2[r]    n3@<-[r]    ⋯[r]    n-ii+3   
    
    	   '=11[ru] [rd]                   n-i-1i+3.  [lu] [ld] 
    
    	      21@<-[r]    2n[r]    3n@<-[r]    ⋯[r]    n-i-1i+4

and '=|_'=0.
Note that (', ') is a bound full subquiver of (, ) if n>2, in this case,
A'='/' (=') is representation-infinite (see the classification of indecomposable module on biserial algebra provided by <cit.>, or refer to <cit.>),
then A^e is representation-infinite by Lemma <ref>.
For case for n=2, it is clear to check that A^e is representation-infinite by Remark <ref>.




Let A=/ be a string algebra with underlying graph

    𝔸_3 = ∙@-@/^1pc/[rr] @-[r]    ∙@-[r]    ∙.

If (, ) contains a string of length two, then A^e is representation-infinite. Indeed, if =0, then A is a representation-infinite hereditary algebra,
so is A^e (cf. <cit.>).

If 0, then  is one of

    @R=0.25cm1 [r]_a     2[r]_b     3 @/_1pc/[ll],       @R=0.25cm1 @<-[r]_a     2[r]_b     3 @<-@/_1pc/[ll],       and      @R=0.25cm1 [r]_a     2@<-[r]_b     3 @<-@/_1pc/[ll]. 
     Case 1  (ab0)         Case 2                   Case 3

Thus, the quiver of A^e is one of the forms given in <ref>.

The bound quiver of A^e has a bound full subquiver (', ') (see <ref>) containing a primitive V-sequence without relation,
then by Remark <ref>, '/' is representation-infinite and so is A^e by Lemma <ref>.


 
If the bound quiver (, ) of a monomial algebra A=/ contains a string of length l≥ 2, then the enveloping algebra A^e of A is representation-infinite.



Assume that (, ) contains a string ϖ with  l(ϖ)≥ 2,
then there exists a substring ω of length 2 which is of the form x @-[r]^α    y @-[r]^β    z
satisfying that if t(α)=s(β) (resp.  t(β)=s(α)), then αβ∉ (resp.  βα∉)).

Let _i--j be the set of all arrows with vertices i and j,
and ♯_i--j the number of elements in it.
If ♯_x--y (resp. ♯_y--z) ≥ 2, we have two situations:

  
  *  contains a band with vertices x and y (resp. y and z).
    In this case, A is representation-infinite, so is A^e (see Lemma <ref>).

  
  *  contains an oriented cycle with vertices x and y (resp. y and z) but contains no band.
    Then A^e is representation-infinite because, in this case, (, ) contains a primitive V-sequence (similar to the proof of Lemma <ref>) which provides an infinite number of indecomposable modules.

If ♯_x--y = ♯_y--z = ♯_z--x = 1, the representation-infiniteness of A^e can be given by Example <ref>.
Thus, ♯_x--y = ♯_y--z =1 and ♯_z--x = 0.


Now we divide this proof into three cases as follows.


  
  (1) x, y and z are pairwise distinct.
  In this case, the bound bound quiver (_A^e, _A^e) of A^e contains a bound full subquiver
  (', ') whose underlying graph is 𝔻_4 and ideal ' is zero by the construction of (_A^e, _A^e).
  Thus, ' is representation-infinite,
  and A^e is representation-infinite by Lemma <ref>.

  
  (2) x=y or y=z. We only prove the first case and the proof of the other one is similar.
  In this case, ω admits a bound subquiver (_α, _α) of the form
  
  
    _α = x@(ur,dr)^α and_α =⟨α^t⟩ (t ≥ 2).

  By Lemma <ref>, (_α/⟨α^t ⟩)^e is representation-infinite.
  Notice that the bound quiver of (_α/⟨α^t ⟩)^e='/' is a bound full subquiver of (_A^e, _A^e) with _α = |_',
  thus A^e is representation-infinite by Lemma <ref>.

  
  (3) x=z y.
 The bound quiver (_ω, _ω) corresponding to ω is one of the following forms.
  
    @C=0.5cm@R=0.5cm
     ·@/^1pc/[rr] @/_1pc/[rr]       ·      and      
     x @/^1pc/[rr]^α@<-@/_1pc/[rr]_β       y 
       (i)                  (ii)

  
    
  (i) In this case, _ω=0 and
       A_ω = _ω/_ω is isomorphic to 2-Kronecker algebra.
       Thus A_ω^e is representation-infinite, see Lemma <ref>.
    
  (ii) In this case, _ω is an oriented cycle and _ω is generated by some paths on _ω. Note that _ω≤ℛ:=⟨αβ, βα⟩,
      there exists an epimorphism
      
    π: A_ω = _ω/_ω→_ω/ℛ

      sending any element ℘+_ω in A_ω to ℘+ℛ, which induces a natural epimorphism
       
    π^e: A_ω^e → (_ω/ℛ)^e.

      Then π^e induces a full embedding from the category of (_ω/ℛ)^e modules to the category of A_ω^e modules.
      By Lemma <ref>, we have (_ω/ℛ)^e is representation-infinite,
      then A_ω^e is representation-infinite.
      Furthermore, by Lemma <ref>, A^e is representation-infinite.
  
 



The following result is a direct corollary of Lemma <ref> and Proposition <ref>.


 
If the quiver  of a monomial algebra A=/ contains a cycle, then A^e is representation-infinite.



Suppose that (, ) contains a cycle C.
If =C=c_0c_1⋯ c_n-1 is an oriented cycle with c_ic_i+1∈, where i equals to i modulo n,
then the representation-infiniteness of A^e is given by Lemma <ref>.
Otherwise, by the definition of monomial algebra, there is a string ω on C with l(ω)≥ 2,
then the representation-infiniteness of A^e follows by Proposition <ref>.




§ MAIN RESULT


We prove our main results in this section.



 §.§ The monomial algebras with representation-finite enveloping algebras

The present subsection mainly provides an explicit characterization of monomial algebras with representation-finite enveloping algebras.
The following result classifies the monomial algebras by their bound quivers.


 
Let A=/ be a connected monomial algebra. Then one of the following holds.

  
  (1) A is either _n/ ^2_n or _n/ ^2_n, where  _1/^2_1 = _1 is a simple algebra.

  
  (2) The bound quiver (,) of A has a string of length ≥ 2.



 
Note that if ^2() then it is clear that A has a string of length ≥ 2.
It suffices to prove that, for arbitrary monomial algebra A=/ with = ^2(), A is either _n/ ^2_n or _n/ ^2_n.

If the quiver  of A has cycles, then any cycle of the quiver  is oriented, since any non-oriented cycle provides a string of the form ab^-1 and a^-1b with a, b∈_1.
Next, we assume that  contains an oriented cycle, say C = c_0c_1⋯ c_n-1 with s(c_i)=v_i,
then c_ic_i+1∈ hold for all i ∈^+, where i equals to i modulo n,
and we have two cases as follows.



  Case1. = C. Then A is _n/ ^2_n.


  Case2. There exists an arrow a c_i for any i such that s(a) = v_j (resp.  t(a) = v_j ) for some j.
      Then a^-1c_j (resp. c_j-1a^-1) is a string of length two.


If  contains no (oriented) cycle, i.e.,  is a tree. If = _n, then A=_n/^2(_n). Assume that _n,
then there is a vertex v∈_0 such that at least two arrows ending or starting at v.
Let α_1 and α_2 (resp.  β_1 and β_2) be two distinct arrows with t(α_1)=t(α_2)=v (resp.  s(β_1)=s(β_2)=v).
Then α_1^-1α_2 (resp. β_1β_2^-1) is a string of length two.


Now we show our main result in this part.

  
Let A=_A/_A be a monomial algebra. Then A^e is representation-finite if and only if A≅_n/ ^2_n.


 If A≅_n/ ^2_n, then A^e is a special biserial algebra.
Assume that A^e = /, then  is generated by some zero relations of length two and some commutative relations.
All commutative relations of the form of (a⊗ e_j'^)(e_j⊗ b^) - (e_i ⊗ b^)(a⊗ e_i'^) corresponding to a commutative block of the form (<ref>) provide all primitive V-sequences on (,).
For any primitive V-sequence β=(a⊗ e_j'^)(e_j⊗ b^) ((a⊗ e_i'^)(e_i ⊗ b^))^-1, the map  in Theorem <ref> sends the pair (β, J_r(λ)) to projective-injective module or zero. More precisely,

    (β, J_r(λ)) =
    
    P((i,i')),     if  r=1  and λ = 1; 
    
    0,     otherwise.

Moreover, the rest indecomposable modules are provided by V-sequences on (,), thus A^e is representation-finite.

Conversely, suppose that A^e is representation-finite algebra. By the classification in Lemma <ref>, one of the following statement holds.

    
  (1) A = _n/^2_n  for some n∈^+.
    
  (2) A  = _n/^2_n for some n≥ 2.
    
  (3) A is a monomial algebra whose bound quiver has a string of length ≥ 2.

By Lemma <ref> and Proposition <ref>, A = _n/^2_n  for some n∈^+.


In particular, the Auslander-Reiten quiver Γ((A^e)) of A^e (see <ref>) can be explicitly described by
<cit.> and <cit.>,
which is the unique connected component of Γ((A^e)).



 §.§ The isoclasses of indecomposable right (_n/^2_n)^e-modules

In this subsection, we calculate the number ♯(((_n/^2_n)^e)) of isoclasses of indecomposable right (_n/^2_n)^e-modules.
We denote by ♯ S the number of elements in the set S.








If A=_n/ ^2_n, then we have the following statement.

  
  (1) Any indecomposable module over A^e is either a string module or an projective-injective module.
  
  (2) The number of indecomposable modules over A^e
  
    ♯( A^e) = 4/3n^3 + n^2-7/3n+1.





(1) The Auslander-Reiten quiver Γ^(n) of A^e (see <ref>)
can be explicitly described by <cit.> and <cit.>.



(2) Any indecomposable right A^e-module X_i corresponds to one of the vertices of the Auslander-Reiten quiver Γ((A^e)) = (Γ^(n)_0, Γ^(n)_1, s, t), say Γ^(n) for short, of A^e.

By the symmetry of Γ^(n), we only consider the subquiver ℒ(Γ^(n)) of Γ^(n)
containing all vertices lying on the left side of the y-axis.
For the convenience of counting, we mark the leftmost column of Γ^(n) as x_1-th column, all columns of Γ^(n) containing at least one vertex corresponding to the projective-injective module from left to right as x_2-th column, ⋯, x_n-th column respectively.
Then ℒ(Γ^(n)) is divided to n-1 parts 𝒫_1, 𝒫_2, ⋯, 𝒫_n-1,
where 𝒫_t (1≤ t≤ n-1) is the subquiver of Γ^(n) containing all vertices in x_t-th column for x_t≤x_t < x_t+1. Thus,

    ♯𝒫_t = t(t+1)+t^2+(2t-1) = 2t^2+3t-1.

Indeed, the number of all vertices in the inner of 𝒫_t equals to t(t+1)+t^2 and that of all vertices on x_t-th column equals to 2t-1. Therefore,

    ♯Γ^(n) =  2∑_t=1^n-1♯𝒫_t + (2n-1)
    = 4/3n^3 + n^2-7/3n+1.



Now we are ready to provide an answer for Clebsch-Gordon problem over the tensor algebra (_n/^2_n)^e.

 
Suppose that A=_n/^2_n. Then the following statements holds.

  
  (1)
    The tensor of indecomposable right A-module and right A^-module is indecomposable.
  
  (2)
    All indecomposable right A^e-module can be described by two classes as follows:
    
      
  * the tensor M⊗_ N of a right A-module M and a right A^-module N;
      
  * the string right A^e-module corresponding to string of length ≥ 2.
    




(1) Assume that the bound quiver of A=_n/^2_n is (, ) with =

    1 [r]^a_1    2 [r]^a_2   ⋯[r]^a_n-1    n  and  = ⟨ a_ia_i+1| 1≤ i < n-1 ⟩.

Note that the dimension of any indecomposable right A-module equals to either 1 or 2,
and the string corresponded by indecomposable right A-module is either a vertex or an arrow of the quiver of A, see <cit.>.



  (i)
For any indecomposable right A-module M and indecomposable right A^-module N of dimension two,
there are two arrows (viewed as strings) a_i and a_j^ corresponding to M and N, respectively.
The representation of M⊗_N can be computed by

    _((M⊗_ N) (e_r⊗ e_s^))
        = _(Me_r⊗ Ne_s^) 
        = _Me_r·_Ne_s^
        = 
        1,     if _Me_r = _Ne_s^ = 1; 
    
        0,     otherwise ,

where e_r and e_s^ are idempotents corresponding to the vertices r and s^ of the quivers of A and A^, respectively.
Thus it follows that

    _ ((M⊗_ N) (e_r⊗ e_s^))
    = 
      1,     if  (r,s)∈{(i,(j+1)^), (i,j^),
                                       ((i+1), j^), ((i+1), (j+1)^)},
    
      0,     otherwise.

Moreover, the linear maps corresponding to arrows e_i⊗ a_j^, a_i⊗ e_(j+1)^, a_i⊗ e_j^ and e_i+1⊗ a_j^ are identity up to isomorphism.
Then M⊗_N is an indecomposable projective-injective right A^e-module.


  (ii)
If _M=1 and _N=2, then similar discussion as above, M⊗_N is a string right A^e-module with dimension two determined by the arrow e_i ⊗α_j^ of the quiver _A^e of A^e,
where e_i is the string in (, ) of length zero corresponding to M and α is the string in (^, ^) of length one corresponding to N.


  (iii)
The case that _M=2 and _N=1 is dual to  (ii).


  (iv)
If _M=_N=1, then _ (M⊗_N)
= 1, and M⊗_N is a  simple A^e-module corresponding to e_i ⊗ e_j^.

Therefore, M⊗_N is indecomposable in A^e.

(2) Let X be an indecomposable A^e-module. If X is a projective-injective module,
then there is a set V={(i,j^), (i+1,j^), (i,(j+1)^),(i+1,(j+1)^)} of four vertices of A^e such that

    Xe_(x,y^) = X(e_x⊗ e_y^) = e_y^Xe_x ≅,     (x,y^)∈ V; 
    
     0,    otherwise.

Then the quiver representation of A^e corresponded by X is given by

    @C=1.5cm
    Xe_(i,(j+1)^)[r]^a_i+1⊗ e_j+1^    Xe_(i+1,(j+1)^)
    
    Xe_(i,j^)[r]^a_i+1⊗ e_j^[u]^e_i⊗ a_j^    Xe_(i+1,j^)[u]_e_i+1⊗ a_j^

Similar to the case (i) in the proof of statement (1), X ≅(a_i+1)_A⊗_(a_j^)_A^.

If X is not a projective-injective module, then there exists a V-sequence ω on (,) such that X≅(ω).
If ℓ(ω)=1, then, by the cases (ii) and (iii) in the proof of (1), X is isomorphic either


(a_i⊗ e_j^)≅(a_i)_A⊗_ S(j^)_A^
 or(e_⊗ a_^) ≅ S()_A⊗_(a_^)_A^.


If ℓ(ω)=0, then X is isomorphic to some simple module S((i,j^)) corresponded by the vertex (i,j^) of the quiver of A^e.
Thus, similar to (iv) in the proof of (1), X≅ S(i)_A⊗_ S(j^)_A^.


Assume A=_n/^2_n. All indecomposable A^e-modules can be divided into four classes up to isomorphism by Proposition <ref>:

  
  (1) simple modules;
  
  (2) indecomposable projective-injective modules;
  
  (3) indecomposable A^e-modules corresponded by string of length 1;
  
  (4) indecomposable A^e-modules corresponded by string of length ≥ 2.

Moreover, the numbers of indecomposable A^e-modules which belong to four classes are

    ♯(_A^e)_0 = n^2,  
        (n-1)^2 = n^2-2n+1,  
       ♯(_A^e)_1 = 2· n(n-1) = 2n^2-2n,  and 
        2 ∑_k=1^n-1(([ 2k+1;    2 ]) - 2k)
    = 4/3n^3 - 3n^2 + 5/3n.

Therefore,

    ♯Γ( A^e) = (<ref>) + (<ref>) + (<ref>) + (<ref>)
    = 4/3n^3 + n^2 - 7/3n + 1,

which coincides with the formula (<ref>) given in Theorem <ref> (2).
Notice that _n/^2_n is isomorphic to its opposite algebra, thus, for a monomial algebra Λ, we can show that, by similar method, if Λ⊗_Λ representation-finite then Λ≅_n/^2_n.


 In Example <ref>, _2^e=(_3/^2_3)^e ≅ (_3/^2_3) ⊗_ (_3/^2_3) is representation-finite. Then we obtain that the number of isoclasses of indecomposable right _2^e-modules is

    4/3n^3 + n^2 - 7/3n + 1 |_n=3 = 36+9-7+1 = 39

by Proposition <ref>.




§ ACKNOWLEDGEMENTS

Yu-Zhe Liu is supported by National Natural Science Foundation of China (Grant No. 12171207) and Scientific Research Foundation of Guizhou University (Grant Nos. [2022]53 and [2022]65).
Chao Zhang is supported by National Natural Science Foundation of China (Grant Nos. 11961007 and 12061060).
The authors would like to thank Yafeng Zhang and Yu Wang for their helpful discussions. We are also grateful to referees for many helpful comments.














'
99








ASS2006

Assem, I., Simson, D.,  Skowroński, A.:
https://doi.org/10.1017/CBO9780511614309
     Elements of the Representation Theory of Associative Algebras,
      Volume 1 Techniques of Representation Theory.
Cambridge University Press, Cambridge,
2006





A1974repr

Auslander, M.:
https://doi.org/10.1080/00927877409412807
     Representation theory of Artin algebras II.
Comm. algebra, 
1, 269–310 (1974)


A1976repr

Auslander, M.:
https://doi.org/10.1016/B978-0-12-339050-9.50006-7
     Large modules over artin algebras.
In Algebra, Topology, and Category Theory, pages 1–17, Elsevier,
1976


BGMS2005Hochschild

Buchweitz, R., O.,  Green, E., L., Madsen, D., Solberg, O.:
https://dx.doi.org/10.4310/MRL.2005.v12.n6.a2
     Finite Hochschild cohomology without finite global dimension.
Math. Res. Lett., 
12, 805–816 (2005)


BGRS1985repr

Bautista, R., Gabriel, P., Roiter, A.,  V., Salmerón, L.:
https://doi.org/10.1007/BF01389052
     Representation-finite algebras and multiplicative bases.
Invent. Math., 
81, 217–285 (1985)


BR1987

Butler, M.,C., R., Ringel, C., M.:
https://www.tandfonline.com/doi/abs/10.1080/00927878708823416
     Auslander-Reiten sequences with few middle terms and applications to string algebras.
Comm. Algebra, 
15, 145–179 (1987)


CE1956

Cartan, H., Eilenberg, S.:
Homological algebra.
Princeton University Press, Princeton, 1956

Gab1972

Gabriel, P.:
https://linkspringer.53yu.com/article/10.1007/BF01298413Unzerlegbare Darstellungen I.
Manuscripta Math., 
6, 71–103 (1972)


H2006Hochschild

Happel., D.:
Hochschild cohomology of finite-dimensional algebras.
In Séminaire d'Algébre Paul Dubreil et Marie-Paul Malliavin: Proceedings,
         
         
pages 108–126, Springer, 2006

Han06

Han, Y.:
Hochschild (co)homology dimension.
J. Lond. Math. Soc. 
73, 657–666 (2006)

Her2003

Herschend, M.:
https://www.diva-portal.org/smash/get/diva2:306215/FULLTEXT01.pdf
     Solution of the Clebsch-Gordan problem for Kronecker representations (PhD Thesis).
Uppsala: Uppsala University, 2003

Her2005JAA

Herschend, M.:
https://doi.org/10.1142/S0219498805001332
     Solution to the Clebsch-Gordan problem for representations of
      quivers of type 𝔸̃_n.
J. Algebra Appl., 
4, 481–488 (2005)


Her2007

Herschend, M.:
https://www.infona.pl/resource/bwmeta1.element.bwnjournal-article-doi-10_4064-cm109-2-3
     Galois coverings and the Clebsch-Gordon problem for quiver representations.
Colloq. Math., 
109, 193–215 (2007)



Her2008

Herschend, M.:
http://dx.doi.org/10.1016/j.jpaa.2007.06.004
     Tensor products on quiver representations.
J. Pure Appl. Algebra, 
1, 452–469 (2008)
















Hochs1945

Hochschild G.:
https://doi.org/10.2307/1969145
     On the cohomology groups of an associative algebra.
Ann. of Math., 
46, 58–67 (1945)


Les1994

Leszczyński, Z.:
http://eudml.org/doc/212020On the representation type of tensor product algebras.
Fund. Math., 
144, 143–161 (1994)

Mou2019

Mousavand K.:
https://link.springer.com/article/10.1007/s10468-022-10170-1
     τ-tilting finiteness of biserial algebras.
Algebr. Represent. Theory, 
26, 2485–2522 (2022)


MT2015

Mahdou, N., Tamekkante, M.:
https://doi.org/10.56947/gjom.v3i2.159
     On Gorenstein global dimension of tensor product of algebras over a field.
Gulf Journal of Mathematics,
3, 30–37 (2015)


MV2004

Martsinkovsky, A., Vlassov, A.:
Https://citeseerx.ist.psu.edu/doc_view/pid/f42add2c9ca3020a0cf7f9057a211a7b6c393005
     The representation ring of k[x].
Preprint, 2004.

R1980repr

Ringel., C., M.:
https://doi.org/10.1007/BFb0089780
     Report on the Brauer-Thrall conjectures: Rojter's theorem and
      the theorem of Nazarova and Rojter (on algorithms for solving vectorspace problems. I).
In 
https://core.ac.uk/download/pdf/15976546.pdfRepresentation Theory I.
      Proceedings of the Workshop on the Present Trends in Representation Theory,
      
      

pages 104–136,

Springer-Verlag,
1980


RT1975QF3ring

Ringel, C., M., Tachikawa, H.:
https://doi.org/10.1515/crll.1975.272.49QF-3 rings.
J. Reine Angew. Math., 
272, 49-72 (1975)



Roi1968repr

Roiter, A., V.:
https://iopscience.iop.org/article/10.1070/IM1968v002n06ABEH000727
     Unboundedness of the dimension of the indecomposable representations
      of an algebra which has infinitely many indecomposable representations.

Mathematics of the USSR-Izvestiya,
2, 1223 (1968)


Sim1974repr

Simson, D.:
Functor categories in which every flat object is projective.
Bullet De L'academie Polonaise des Sciences Série des Sciences Mathematics, Astronomy, et Physics,
22, 375–380 (1974)

Sim2003repr

Simson, D.:
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/960/489
     On large indecomposable modules and right pure semisimple rings.
Algebra Discrete Math., 
2, 93–117 (2003)

Vos2001

Vossieck D.:
https://core.ac.uk/reader/82184880The algebras with discrete derived category.
J. Algebra, 
243, 168–176 (2001)


WW1985

Wald, B., WaschbüSch, J.:
https://doi.org/10.1016/0021-8693(85)90119-XTame biserial algebras.
J. Algebra, 
95, 480–500 (1985)


Y1978repr

Yamagata, K.:
https://doi.org/10.1016/0021-8693(78)90155-2
     On artinian rings of finite representation type.
J. Algebra, 
50, 276–283 (1978)


ZH16
Zhang, C., Han, Y,:
Brauer-Thrall type theorems for derived module categories.
Algebr. Represent. Theory, 
19, 1369–1386 (2016)

Zhang2018

Zhang, C.:
https://doi.org/10.1080/00927872.2017.1399404
     Derived representation type and cleaving functors.
Comm. Algebra, 
46, 2696–2701 (2018)




