Low Dimensional Representations Over Quantum Complete Intersections

A quantum complete intersection is an algebraA(q,n₁,...,n_t)=k x₁,...,x_t/x_iⁿⁱ,x_ix_j+q_{i j}x_jx_i,1?i,j?t.This notion originated from the quantum planes work by Yu.I.Manin^[1],studied by Luchezar L.Avramov,Vesselin N.Gasharov and Irena V.Peeva^[2],and named in^[3]and^[4]by quantum complete intersection(algebra).Along with the change of q and each n_i,the quantum complete intersection comes out a large number of(non-commutative)local Frobenius algebras with different nature.These algebras were used widely in fields of mathematics such as representation theory,non-commutative algebraic geometry and quantum algebra.The representation type of a quantum complete intersection is wild,except the case of t=2=n₁=n₂.It means that to classify all the indecomposable finite-dimensional modules is hopeless.Accordingly,although there were a lot of works on quantum complete intersec-tions,including the homology and cohomology,the representation dimension,and the type of stable Auslander-Reiten quivers,there still no study of the finite-dimensional indecomposable modules over quantum complete intersections.Precisely because the behavior of representation theory of quantum complete intersections is strange,these algebras have been used to construct counterexamples for many significant conjectures.In this thesis,we systematically study the regular representation,the reduction lemma,and the low dimensional indecomposable modules of quantum complete intersection.Al-though it is still impossible to classify all the finite dimensional indecomposable representations quantum over quantum complete intersection completely,the classification of low dimensional modules provides us with rich information to understand quantum complete intersections,and also shows the complexity of its indecomposable representations,which is of great help to un-derstand these algebras and construct counterexamples.In many cases,it has already provided the information we need.The structure and main results of this thesis are as follows.In the first chapter,we summarized the background and significance of this thesis,the current situation of quantum complete intersections research,and stated the main results.In the second chapter,we described the left regular module of a quantum com-plete intersection by the Loewy length:the unique finite-dimensional indecomposable left A(q,n₁,...,n_t)-module of Loewy lengthn_i-t+1 is the regular module.1?i?tIn the third chapter,we classified the indecomposable modules of dimension?5 over k x,y/x²,y²,x y+qy x by giving their diagram presentations.Another important observation was given in the fourth chapter,that is,the reduction lemma holds for quantum complete intersection,which makes it possible to classify the representations of quantum complete intersections in low dimensions.To be spe-cific,the d-dimensional indecomposable modules over the quantum complete intersection A(q,n₁,...,n_t)with d?n_ican be reduced to theindecomposable modules over A(q,d,...,d).The fifth chapter classified the indecomposable modules of dimension?3 over quan-tum complete intersection k x,y/x³,y³,x y+qy x.Then we got the classification of indecomposable modules of dimension?3 over k x,y/x^m,yⁿ,x y+qy x by reduction lemma.For example,when m,n?3 and q-1,we got 7 classes of non-isomorphic 3-dimensionial indecomposable modules,4 classes with no parameter and 3 classes determined by one parameter.In the last chapter,we classified the indecomposable modules of dimension?4 over k x,y/x⁴,y⁴,x y+qy x.Then by using the reduction lemma,we classified the indecompos-able modules over k x,y/x^m,yⁿ,x y+qy x of dimension?4 when m,n?2.For example,when m,n?4 and q{1,-1},the non-isomorphic 4-dimensionial inde-composable modules can be classified into 21 classes,9 classes with no parameter,10 classes determined by one parameter and 2 classes determined by two parameters.As another example,when m<4 or n<4,then the classification of indecomposable A(q,m,n)-modules of dimen-sion 4 can also seen from the classification of indecomposable A(q,4,4)-modules of dimension4:they are exactly those modules M satisfied x^mM=0=yⁿM.