Let A be a finite dimensional algebra over an algebraically closed field k . Assume A is a basic connected and triangular algebra with n pairwise non-isomorphic simple modules. We consider the Coxeter transformation ϕ A T as the automorphism of the Grothendieck group K 0 A induced by the Auslander-Reiten translation τ in the derived category D b mod A of the module category mod A of finite dimensional left A -modules. In this paper we study the Mahler measure M χ A of the Coxeter polynomial χ A of certain algebras A . We consider in more detail two cases: (a) A is said to be cyclotomic if all eigenvalues of χ A are roots of unity; (b) A is said to be of Littlewood type if all coefficients of χ A are − 1 , 0 or 1 . We find criteria in order that A is of one of those types. In particular, we establish new records according to Mossingshoff’s list of Record Mahler measures of polynomials q with 1 < M q as small as possible, ordered by their number of roots outside the unit circle.
- finite dimensional algebra
- coxeter transformation
- derived category
- accessible algebra
- characteristic polynomial
- cyclotomic polynomial
- littlewod type
Assume throughout the paper that is an algebraically closed field. We assume that is a triangular finite dimensional basic -algebra, that is, of the form , where is an ideal of the path algebra for a quiver without oriented cycles. In particular, has finite global dimension. The Coxeter transformation is the automorphism of the Grothendieck group induced by the Auslander-Reiten translation in the derived category see . The characteristic polynomial of is called the Coxeter polynomial of , or simply see [15, 17]. It is a monic self-reciprocal polynomial, therefore it is , with for , and .
Consider the roots of , the so called spectrum of . There is a number of measures associated to the absolute values for in the spectrum of . For instance, the spectral radius of is defined as and the Mahler measure of defined as . Recently, some explorations on the relations of the Mahler measure and properties of the algebra have been initiated.
For a one-point extension , we show that . The strongest statements and examples will be given for the class of accessible algebras. We say that an algebra is accessible from if there is a sequence of algebras such that each is a one-point extension (resp. coextension) of for some exceptional -module . As a special case, a -algebra is called accessible if is accessible from the one vertex algebra .
We say that is of cyclotomic type if the eigenvalues of lie on the unit circle. Many important finite dimensional algebras are known to be of cyclotomic type: hereditary algebras of finite or tame representation type, canonical algebras, some extended canonical algebras and many others. On the other hand, there are well-known classes of algebras with a mixed behavior with respect to cyclotomicity. For instance, in Section 6 below we consider the class of Nakayama algebras. Let be the quotient obtained from the linear quiver with vertices
with relations . The Nakayama algebras are easily proven to be of cyclotomic type, while those of the form are of cyclotomic type as consequence of lengthly considerations in . The case is more representative: is of cyclotomic type for all except for and . Clearly, if is of cyclotomic type then , for . We show the following theorem.
Theorem 1: Let be an unimodular -matrix. The following are equivalent:
is of cyclotomic type;
for every positive integer , we have .
We also consider algebras of Littlewood type where has all its coefficients in the set . Among other structure results, we prove.
Proposition. The closure of the set of roots of Littlewood polynomials, equals the set of roots of Littlewood series.
Our results make use of well established techniques in the representation theory of algebras, as well as results from the theory of polynomials and transcendental number theory, where Mahler measure has its usual habitat. We stress here the natural context of these investigations on the largely unexplored overlapping area of these important subjects. Hence, rather than a comprehensive study we understand our work as a preliminary exploration where examples are most valuable.
2. Measures for polynomials
2.1 Self-reciprocal polynomials
A polynomial of degree is said to be self-reciprocal if . The following table displays the number of polynomials of degree (for small ) with non-zero, is the number of such polynomials which are additionally self-reciprocal, and is the number of those which are self-reciprocal and where is the square of an integer.
Indeed, there is an efficient algorithm to determine such polynomials of given degree , based on a quadratic bound for in terms of Euler totient function, .
Cyclotomic polynomials and their products are a natural source for self-reciprocal polynomials. Clearly, is not self-reciprocal, but all remaining (with ) are. Hence, exactly the polynomials with natural numbers and are self-reciprocal with spectral radio one and without eigenvalue zero.
It is not a coincidence that in the above tables we have for even and for n odd. Indeed, if is self-reciprocal of odd degree then , hence where is also self-reciprocal.
2.2 Mahler measure
Let be a finite dimensional -algebra with finite global dimension. The Grothendieck group of the category of finite dimensional (right) -modules, formed with respect to short exact sequences, is naturally isomorphic to the Grothendieck group of the derived category, formed with respect to exact triangles.
The Coxeter transformation is the automorphism of the Grothendieck group induced by the Auslander-Reiten translation . The characteristic polynomial of is called the Coxeter polynomial of , or simply . It is a monic self-reciprocal polynomial, therefore it is , with for , and .
Consider the roots of , the so called spectrum of . In , a measure for polynomials was introduced. Namely, the Mahler measure of is . By a celebrated result of Kronecker , see also [7, Prop. 1.2.1], a monic integral polynomial , with , has if and only if factorizes as product of cyclotomic polynomials. As observed in , is of cyclotomic type if and only if , that is, factorizes as product of cyclotomic polynomials.
2.3 Spectral radius one, periodicity
If the spectrum of lies in the unit disk, then all roots of lie on the unit circle, hence has spectral radius . Clearly, for fixed degree there are only finitely many monic integral polynomials with this property.
The following finite dimensional algebras are known to produce Coxeter polynomials of spectral radius one:
hereditary algebras of finite or tame representation type;
all canonical algebras;
(some) extended canonical algebras;
generalizing (2), (some) algebras which are derived equivalent to categories of coherent sheaves.
We put . Note that has degree . There are several reasons for this choice: first of all , second this normalization yields convincing formulas for the Coxeter polynomials of canonical algebras and hereditary stars, third representing a Coxeter polynomial — for spectral radius one — as a rational function in the ‘s relates to a Poincaré series, naturally attached to the setting.
|Dynkin type||Star symbol||-factorization||Cyclotomic factorization||Coxeter number|
In the column ‘-factorization’, we have added some extra terms in the nominator and denominator which obviously cancel.
Inspection of the table shows the following result:
Proposition. Let be an algebraically closed field and be a connected, hereditary -algebra which is representation-finite. Then the Coxeter polynomial determines up to derived equivalence.□
2.4 Triangular algebras
Nearly all algebras considered in this survey are triangular. By definition, a finite dimensional algebra is called triangular if it has triangular matrix shape
where the diagonal entries are skew-fields and the off-diagonal entries , , are -bimodules. Each triangular algebra has finite global dimension.
Proposition. Let be a triangular algebra over an algebraically closed field . Then is the square of an integer.
Proof. Let be the Cartan matrix of with respect to the basis of indecomposable projectives. Since is triangular and is algebraically closed, we get , yielding
Hence is the determinant of the skew-symmetric matrix . Using the skew-normal form of , see [16, Theorem IV.1], we obtain for some , where is a block-diagonal matrix whose first block is the zero matrix of a certain size and where the remaining blocks have the shape with integers . The claim follows. □
Which self-reciprocal polynomials of spectral radius one are Coxeter polynomials? The answer is not known. If arbitrary base fields are allowed, we conjecture that all self-reciprocal polynomials are realizable as Coxeter polynomials of triangular algebras. Restricting to algebraically closed fields, already the request that is a square discards many self-reciprocal polynomials, for instance the cyclotomic polynomials , , , . Moreover, the polynomial , which is the Coxeter polynomial of the non simply-laced Dynkin diagram , does not appear as the Coxeter polynomial of a triangular algebra over an algebraically closed field, despite of the fact that is a square. Indeed, the Cartan matrix
yields the Coxeter polynomial , where . The equation of Hurwitz-Markov type does not have an integral solution. (Use that reduction modulo only yields the trivial solution in .)
2.5 Relationship with graph theory
Given a (non-oriented) graph , its characteristic polynomial is defined as the characteristic polynomial of the adjacency matrix of . Observe that, since is symmetric, all its eigenvalues are real numbers. For general results on graph theory and spectra of graphs see .
There are important interactions between the theory of graph spectra and the representation theory of algebras, due to the fact that if is the Cartan matrix of , then is determined by the symmetrization of , since . We shall see that information on the spectra of provides fundamental insights into the spectral analysis of the Coxeter matrix and the structure of the algebra .
A fundamental fact for a hereditary algebra , when is a bipartite quiver, that is, every vertex is a sink or source, is that . This was shown as a consequence of the following important identity.
Proposition.  Let be a hereditary algebra with a bipartite quiver without oriented cycles. Then , where is the number of vertices of and is the characteristic polynomial of the underlying graph of .
Proof. Since is bipartite, we may assume that the first vertices are sources and the last vertices are sinks. Then the adjacency matrix of and the Cartan matrix of , in the basis of simple modules, take the form: , , where
for certain -matrix . Since , then . Therefore
The above result is important since it makes the spectral analysis of bipartite quivers and their underlying graphs almost equivalent. Note, however, that the representation theoretic context is much richer, given the categorical context behind the spectral analysis of quivers. The representation theory of bipartite quivers may thus be seen as a categorification of the class of graphs, allowing a bipartite structure.
Constructions in graph theory. Several simple constructions in graph theory provide tools to obtain in practice the characteristic polynomial of a graph. We recall two of them (see  for related results):
Assume that is a vertex in the graph with a unique neighbor and (resp. ) is the full subgraph of with vertices (resp. ), then
Let be the graph obtained by deleting the vertex in . Then the first derivative of is given by
The above formulas can be used inductively to calculate the characteristic polynomial of trees and other graphs. They immediately imply the following result that will be used often to calculate Coxeter polynomials of algebras.
Proposition. Let be a bipartite hereditary algebra. The following holds:
Let be a vertex in the graph with a unique neighbor . Consider the algebras and obtained as quotients of modulo the ideal generated by the vertices and , respectively. Then
The first derivative of the Coxeter polynomial satisfies:
where is an algebra obtained from by ‘killing’ a vertex .
Proof. Use the corresponding results for graphs and A’Campo’s formula for the algebras and its quotients . □
3. Important classes of algebras
In this section we give the definitions and main properties of such classes of finite dimensional algebras where information on their spectral properties is available.
3.1 Hereditary algebras
Let be a finite dimensional -algebra. For simplicity we assume for a quiver without oriented cycles and an ideal of the path algebra. The following facts about the Coxeter transformation of are fundamental:
Let be a complete system of pairwise non-isomorphic simple -modules, the corresponding projective covers and the injective envelopes. Then is the automorphism of defined by , where denotes the class of a module in .
For a hereditary algebra , the spectral radius determines the representation type of in the following manner:
is representation-finite if is not a root of the Coxeter polynomial .
is tame if .
is wild if . Moreover, if is wild connected, Ringel  shows that the spectral radius is a simple root of . Then Perron-Frobenius theory yields a vector with positive coordinates such that . Since is self reciprocal, there is a vector with positive coordinates such that . The vectors play an important role in the representation theory of , see [5, 17].
Explicit formulas, special values. We are discussing various instances where an explicit formula for the Coxeter polynomial is known.
star quivers. Let be the path algebra of a hereditary star with respect to the standard orientation, see
In particular, we have an explicit formula for the sum of coefficients of as follows:
This special value of has a specific mathematical meaning: up to the factor this is just the orbifold-Euler characteristic of a weighted projective line of weight type . Moreover,
if and only if the star is of Dynkin type, correspondingly the algebra is representation-finite.
if and only if the star is of extended Dynkin type, correspondingly the algebra is of tame (domestic) type.
if and only if is not Dynkin or extended Dynkin, correspondingly the algebra is of wild representation type.
The above deals with all the Dynkin types and with the extended Dynkin diagrams of type , , and , . To complete the picture, we also consider the extended Dynkin quivers of type () restricting, of course, to quivers without oriented cycles. Here, the Coxeter polynomial depends on the orientation: If (resp. ) denotes the number of arrows in clockwise (resp. anticlockwise) orientation (), that is, the quiver has type , the Coxeter polynomial is given by
Hence , fitting into the above picture.
The next table displays the -factorization of extended Dynkin quivers.
|Extended Dynkin type||Star symbol||Weight symbol||Coxeter polynomial|
Remark: As is shown by the above table, proposition 2.3 extends to the tame hereditary case. That is, the Coxeter polynomial of a connected, tame hereditary -algebra (remember, is algebraically closed) determines the algebra up to derived equivalence. This is no longer true for wild hereditary algebras, not even for trees.
3.2 Canonical algebras
Canonical algebras were introduced by Ringel . They form a key class to study important features of representation theory. In the form of tubular canonical algebras they provide the standard examples of tame algebras of linear growth. Up to tilting canonical algebras are characterized as the connected -algebras with a separating exact subcategory or a separating tubular one-parameter family (see ). That is, the module category accepts a separating tubular family , where is a homogeneous tube for all with the exception of tubes with of rank ().
Proposition. Let be a canonical algebra with weight and parameter data (,). Then the Coxeter polynomial of is given by □
The Coxeter polynomial therefore only depends on the weight sequence . Conversely, the Coxeter polynomial determines the weight sequence — up to ordering.
3.3 Incidence algebras of posets
Let be a finite partially ordered set (poset). The incidence algebra is the -algebra spanned by elements for the pairs in , with multiplication defined by . Finite dimensional right modules over can be identified with commutative diagrams of finite dimensional -vector spaces over the Hasse diagram of , which is the directed graph whose vertices are the points of , with an arrow from to if and there is no with .
We recollect the basic facts on the Euler form of posets and refer the reader to  for details. The algebra is of finite global dimension, hence its Euler form is well-defined and non-degenerate. Denote by , the matrices of the bilinear form and the corresponding Coxeter transformation with respect to the basis of the simple -modules.
The incidence matrix of , denoted , is the matrix defined by if and otherwise . By extending the partial order on to a linear order, we can always arrange the elements of such that the incidence matrix is uni-triangular. In particular, is invertible over . Recall that the Möbius function is defined by .
Lemma. a. .
b. Let . Then .
Proposition. If and are posets, then and .
4. Cyclotomic polynomials and polynomials of Littlewood type
4.1 Cyclotomic polynomials
We recall some facts about cyclotomic polynomials.
The -cyclotomic polynomial is inductively defined by the formula
The Möbius function is defined as follows:
A more explicit expression for the cyclotomic polynomials is given by
for , where .
4.2 Hereditary stars
A path algebra is said to be of Dynkin type if the underlying graph of is one of the ADE-series, that is, of type, , for some or , for .
There are various instances where an explicit formula for the Coxeter polynomial is known.
Let be the path algebra of a hereditary star with respect to the standard orientation, see .
Since the Coxeter polynomial does not depend on the orientation of we will denote it by . It follows that
In particular, we have an explicit formula for the sum of coefficients of as follows:
4.3 Wild algebras
Let be the real root of the polynomial , approximately . As observed in , a wild hereditary algebra associated to a graph without multiple arrows has spectral radius unless is one of the following graphs:
In these cases, for
where is the real root of the Coxeter polynomial
associated to any hereditary algebra whose underlying graph is . Observe that in these cases, the Mahler measure of the algebra equals the spectral radius.
4.4 Lehmer polynomial
In 1933, D. H. Lehmer found that the polynomial
has Mahler measure , and he asked if there exist any smaller values exceeding 1. In fact, the polynomial above is the Coxeter polynomial of the hereditary algebra whose underlying graph is depicted below.
We say that a matrix is of Mahler type (resp. strictly Mahler type) if either or (resp. ). Earlier this year, Jean-Louis Verger-Gaugry announced a proof of Lehmer’s conjecture, see
4.5 Happel’s trace formula
In , Happel shows that the trace of the Coxeter matrix can be expressed as follows:
where denotes the -th Hochschild cohomology group. In particular, if the Hochschild cohomology ring is trivial, that is, for and , then .
For an algebra and a left -module we call
the one-point extension of by . This construction provides an order of vertices to deal with triangular algebras, that is, algebras , where is an ideal of the path algebra for a quiver without oriented cycles.
4.6 One-point extensions
Let be an algebra and a -module. Consider the one-point extension . In  it is shown the Coxeter transformations of and are related by
where is the Cartan matrix of which satisfies and is the class of in the Grothendieck group . In case with an exceptional module, it follows that
We recall that the Euler quadratic form is defined as . Assume that for an algebra and an indecomposable module . In many cases, we get that , for the dimension vector of (for instance, if is preprojective, or if coincides with the Tits form of …)
Proposition. Let be an accessible algebra, such that for the dimension vector of , where for certain algebra and an indecomposable module . Then the following happens:
b. if and , then .
Proof. Assume that for an algebra and an indecomposable module such that for the dimension vector of . Then is also accessible. By induction hypothesis, . Then
This shows (a).
For (b) assume that and , then
4.7 Strongly accessible algebras
Theorem: A finite dimensional accessible algebra then it is strongly accessible if and only if .
Proof. Assume is strongly accessible from . Since , for a one-point extension of the subcategory of by the exceptional module (since then ). By the Proposition above
Conversely, assume that and write as a one-point extension of the subcategory of by the module . We shall prove that is exceptional.
Equality holds and , since is indecomposable, it follows that the extension ring of is trivial. □
4.8 Stable matrices
The following statement is Theorem 1 for stable matrices.
Proposition. Suppose is a stable unimodular -matrix. Let be its characteristic polynomial.
Suppose that for and certain integers and .
Then for all integers .
In particular, is of cyclotomic type.
Proof. Consider the coefficients of . Since is stable then and the signs alternate until we meet a with . Cayley-Hamilton theorem states that . Then
Let be the common value of the trace of this matrix.
Write for or . Consider the matrices
so that we get two expressions of as positive linear combinations of powers of .
Suppose that . By hypothesis we have . Moreover, since then
The claim follows by induction.
Otherwise, . The claim follows similarly.□
4.9 Theorem 1
Proof of Theorem 1. Observe that is a real unimodular matrix. One implication of the Theorem was shown before. Suppose that or equivalently, for . The Proposition above yields that is cyclotomic.□
4.10 Polynomials of Littlewood type
An integral self-reciprocal polynomial is of Littlewood type if every coefficient non-zero has modulus . A polynomial of Littlewood type with all , for , is said to be Littlewood.
Lemma. If is a root of a polynomial of Littlewood type, then
Proof. Suppose is a root of a polynomial of Littlewood type. Then
for some .
If then so . Since is the root of a polynomial of Littlewood type if and only if is, then .
Moreover, if , then and . Hence .□
4.11 Littlewood series
Definition. A Littlewood series is a power series all of whose coefficients are or .
Let is the root of some Littlewood polynomial .
a. Littlewood series converge for .
b. A point with lies in if and only if some Littlewood series vanishes at this point.
c. A Littlewood polynomial is not a Littlewood series. But any Littlewood polynomial, say yields a Littlewood series having the same roots with : indeed, consider the series
Thus , where is the set of roots of Littlewood series. We shall show the Proposition at the Introduction.
Proof. Let be the set of Littlewood series. Then , so with the product topology it is homeomorphic to the Cantor set. Choose . Let be the space of finite multisets of points with , modulo the equivalence relation generated by when .
Claim. Any Littlewood series has finitely many roots in the disc . The map sending a Littlewood series to its multiset of roots in this disc is continuous.
Since is compact, the image of is closed. From this we can show that , the set of roots of Littlewood series, is closed. Since Littlewood polynomials are densely included in and is continuous, we get that , the set of roots of Littlewood polynomials, is dense in . It follows that , as we wanted to show.□
5. An example
For a natural number and let . Let be an algebra formed by commutative squares. Consider the one-point extension with the unique indecomposable projective -module of -dimension . Observe that (resp. ) is given by the following quiver with vertices and commutative relations (resp. vertices and relations):
, for all . As consequence, the algebras and are of Littlewood type;
the number of eigenvalues of not lying in the unit disk is at least ;
Proof. (a): Consider , and the algebra such that and the perpendicular category in is derived equivalent to where is a proper quotient of an algebra derived equivalent to . Therefore
We shall calculate . Observe that is tilting equivalent to the one-point extension . Hence
As consequence of formula (a) we observe the following:
(b) By induction, we shall construct polynomials representing .
For , we have , which is represented by the polynomial .
Observe that then is represented by .
For , we define . We verify by induction on that represents :
which has changes of sign in the sequence of coefficients. According to Descartes rule of signs, has at most positive real roots. Since represents , then has at most roots in the unit circle. That is, has at least roots with .
We shall prove, by induction, that has at most positive real roots. Indeed, write
for some polynomial of degree with signs of its coefficients so that . Then
an addition of three polynomials with signs of coefficients given as follows:
Hence where the polynomial of degree has signs of its coefficients so that . Hence .
By the Lemma below, has at most roots in the unit circle. Equivalently, has at least roots outside the unit circle. Hence has at least roots satisfying .
Lemma. Let be a polynomial representing the polynomial . Assume accepts at most positive real roots, then has at most roots in the unit circle.
Proof. Let be the positive real roots of . Let be a root of with . Consider a complex number with . Then where . Then for some and . Hence
can be selected in two different ways.□
(c) For we have . Then
for and .
We observe that is a product of cyclotomic polynomials. Indeed, since we can write
for and .
Recall and . Moreover, , if is a power of . Altogether this yields
confirming the claim.
We estimate the Mahler measure of . Write , where is the cyclotomic summand. Observe that and apply Lemma (3.4) with to get
With the help of computer programs we calculate more accurate values of the Mahler measure of some of the above examples:
|No. vertices||No. roots outside unit disk||Mahler measure|
Comparing with the list of Record Mahler measures by roots outside the unit circle in Mossinghoff’s web page we see:
for the entry the Mahler measure is the same in both tables;
the entries and have a smaller Mahler measure in our table, establishing new records;
the entry of our table seems to be new. Further entries could be calculated.
6. Coefficients of Coxeter polynomials
6.1 Derived tubular algebras
There are interesting invariants associated to the Coxeter polynomial of a triangular algebra . For instance, the evaluation of the Coxeter polynomial for some integer . Clearly, this number is a derived invariant. A simple argument yields that in case has an odd number of vertices. In , it was shown that for a representation-finite accessible algebra with gl.dim the invariant equals zero or one. The criterion was applied to show that a canonical algebra is derived equivalent to a representation-finite algebra if and only if it has weight type , where and . In particular, the tubular canonical algebra of type is not derived equivalent to a representation-finite algebra, while the tubular algebras of type or are.
6.2 Strong towers
Recall from  that a strong tower of access to satisfies that or for some exceptional module in such a way that, in case (resp. ), the perpendicular category (resp. ) of in is equivalent to for some accessible algebra , . In the extension situation the perpendicular category (resp. in the coextension situation) in is equivalent to and is derived equivalent to a one-point (co-)extension of . An algebra as above is called an i-th perpendicular restriction of the tower , observe that it is well-defined only up to derived equivalence. We denote by the number of connected components of the algebra ; in particular, .
There are many examples of strongly accessible algebras, that is, algebras derived equivalent to algebras with a strong tower of access. The following are some instances:
A canonical algebra of weight is strongly accessible if and only if , in that case, is derived-equivalent to a representation-finite algebra if and only if the weight type does not dominate .
The following sequence of poset algebras defines strong towers of access:
6.3 Towering numbers
Consider a strong tower of access to such that is an one-point (co)extension of by and the corresponding i-th perpendicular restriction of . Let have connected components, . Define the first towering number of as the sum .
Theorem. Let be a strongly accessible algebra with vertices, then the first towering number of is a derived invariant, that is, depends only on the derived class of . It is , where is the coefficient of the quadratic term in the Coxeter polynomial of .
Proof. Assume and such that for an exceptional -module and let be the algebra such that is derived equivalent to the perpendicular category formed in . Then . Write and . By induction hypothesis we may assume that . Then . Moreover, since is a direct sum accessible algebras, then . Hence .□
Corollary. Let be a strong tower of access to . Let for with exceptional and a perpendicular restriction of via . Consider the Coxeter polynomials and , then , with equality if and only if is connected. In particular, .
Proof. First recall that for a connected accessible algebra the linear term of the Coxeter polynomial has coefficient . Let be the Coxeter polynomial of . If is the direct sum of connected accessible algebras , then . Therefore, . By induction hypothesis, we get .□
Let be the algebra given by the following quiver with relation :
which is derived equivalent to the quiver algebra with the zero relation as depicted in the second diagram. Clearly, , where is a quiver algebra of type and is an indecomposable module with the category of modules of the disconnected quiver , that is . Moreover and . On the other hand such that is not hereditary. A calculation yields and , obviously implying that .
Some properties of the invariant s:
Let and be accessible algebras and be accessible from , then . Equality holds exactly when .
Let be an accessible schurian algebra (that is for every couple of vertices , ), then for every convex subcategory we have .
6.4 Totally accessible algebras
An accessible algebra with vertices, and , is said to be totally accessible if there is a family of (not necessarily connected) algebras satisfying:
is derived equivalent to ;
for each , there is a strong tower of access to ;
is an -th perpendicular restriction of , that is, is a one-point (co)extension of by a module and is a perpendicular restriction of via .
The tower is said to be a j-th derivative of the tower .
Examples that we have encountered of totally accessible algebras are:
Hereditary tree algebras: since for any conneceted hereditary tree algebra with at least vertices, there is an arrow with a source (or dually a sink) and such that the perpendicular restriction of via is the algebra hereditary tree algebra obtained from by deleting the vertices .
Accessible representation-finite algebras with gl.dim , since then the perpendicular restrictions of any strong tower (which exists by ) satisfy the same set of conditions.
Certain canonical algebras: for instance the tame canonical algebra of weight type is an extension of a hereditary algebra of extended Dynkin type by a module in a tube of rank , then the perpendicular restriction of via is the hereditary algebra of extended Dynkin type , see for example [?](10.1). Since is totally accessible, so is. Moreover .
Let be an accessible algebra of the form for an algebra and an exceptional module and let the perpendicular restriction of via . If is totally accessible, then and are totally accessible.
The following results extend some of the features observed in the examples above.
Proposition. a. Assume that is a totally accessible algebra, then .
b. Assume that is an accessible but not totally accessible algebra with gl.dim , then one of the following conditions hold:
i. for every exceptional -module such that and any perpendicular restriction of via , then is not accessible;
ii. there exists a homological epimorphism such that .
Proof. (a): Consider the perpendicular restriction of via , such that . Therefore and moreover, is totally accessible. Then by induction hypothesis, for a totally accessible algebra with number of vertices or . Clearly, is either , or hereditary of type , which yields the desired result.
(b): Assume is an accessible algebra with gl.dim and such that for every homological epimorphism we have . Let for an accessible algebra and an exceptional -module such that is a perpendicular restriction of via . Since gl.dim then there is a homological epimorphism and gl.dim . Observe that for every homological epimorphism (resp. ) there is a homological epimorphism (resp. ), hence (resp. ) is or . By induction hypothesis, is totally accessible. Moreover if is accessible, then the induction hypothesis yields that is totally accessible and also is totally accessible, a contradiction. Therefore is not accessible.□
7. On the quadratic coefficient of the Coxeter polynomial of a totally accessible algebra
7.1 Derived algebras of linear type
Recall that an extended canonical algebra of weight type is a one-point extension of the canonical algebra of weight type by an indecomposable projective module. As in (1.3), the extended canonical algebras of type is strongly accessible. Moreover, the extended canonical algebra of type (with 12 points) has Coxeter polynomial which is also the Coxeter polynomial of a linear hereditary algebra with vertices. Clearly and are not derived equivalent.
The following generalizes a result of Happel who considers the case of Coxeter polynomials associated to hereditary algebras .
Theorem 1. Let be a totally accessible algebra with vertices and let be the Coxeter polynomial of . The following are equivalent:
let be a strong tower of access to and the -th perpendicular restriction of , for all , then the algebras are connected;
A is derived equivalent to a quiver algebra of type .
Proof. (i) (ii): Let be a strong tower of access to . In case each is connected, then , that is . If , then with each . (i) (iii): We know that an algebra derived equivalent to a quiver algebra of type has , in particular, . Assume that an accessible algebra has the quadratic coefficient of its Coxeter polynomial . Let for an accessible algebra and an exceptional module . Since is also totally accessible with a tower satisfying (ii), then the quadratic coefficient of the Coxeter polynomial of is and we may assume that is derived equivalent to a quiver algebra of type . In particular, is representation-finite with a preprojective component such that the orbit graph is of type (recall that the orbit graph has vertices the -orbits in the quiver with Auslander-Reiten translation and there is an edge between the orbit of and the orbit of if there is some numbers and an irreducible morphism ). Observe that for any in not in the orbit of , there is some translation belonging to , implying that in case has two neighbors in the orbit graph then is not connected, that is and , a contradiction. Therefore, has just one neighbor in , hence is derived of type .□
7.2 Theorem 2
Consider a tower of accessible algebras where is a one-point (co)extension of by the indecomposable and is such that is derived equivalent to . Assume that , for , are the connected components of the category . Consider the corresponding Coxeter polynomials:
where clearly, .
Lemma. () For every , we have .
() For every , we have and .
Proof. We shall check that () implies (), then we show that (a’) holds by induction on .
Indeed, assume that () holds and proceed to show () by induction on . If , then . Assume that and and . Then
Let . If we have and . Moreover . Assume () holds for , then.
Theorem 2. Let be a totally accessible algebra with Coxeter polynomial , then:
, for every ;
if for some , we have then is derived equivalent to a hereditary algebra of type .
Proof. Keep the notation as in (4.1). Then (a) is the case of the Lemma above.
We shall prove (b) by induction on the number of vertices of . Let and assume , then (3.1) implies that is derived equivalent to . Consider now and assume that , we get:
The last inequality due to (a), hence . Induction hypothesis yields that is derived equivalent to and its Auslander-Reiten quiver consists of a preprojective component . In particular, , which implies that , that is, for some exceptional module such that is derived equivalent to for a connected algebra , that is, and by (3.1), is derived equivalent to a hereditary algebra of type .□
If is a representation-finite accessible algebra with gl.dim , then is totally accessible. On the other hand the algebra with quiver:
and is representation-finite and accessible (but not gl.dim ). The Coxeter polynomial of is:
Then observe that the -th coefficient is but the algebra is not derived equivalent to Dynkin type .
8. On the traces of Coxeter matrices
Let be an algebra such that not all roots of are roots of unity. By the result of Kronecker , not all of the spectrum of lies in the unit disk. Equivalently, the spectral radius . Arrange the eigenvalues of so that have absolute values and multiplicities , respectively. Therefore and
We define the critical power as the minimal such that
Since is a simple eigenvalue of , then it follows that is well defined due to the existence of satisfying the following chain of inequalities:
The following is a reformulation of Theorem 2.
Theorem. Let be an algebra such that not all roots of are roots of unity. We have
Proof. Indeed, suppose that is not of cyclotomic type and , that is, for all . Observe that is a unimodular matrix and therefore, Theorem 2 implies that is of cyclotomic type, which yields a contradiction.□
Remark: We consider explicitly the case in the above Theorem. Obviously, the Cartan matrix of is of the form
for some . Then has the indicated shape. If is not cyclotomic, then and .
9. Stability of a real matrix
9.1 Stability of matrices and the Lyapunov criterion
Let be a real invertible -matrix with eigenvalues , for some numbers and . We will say that is stable (resp. semi-stable) if the real part of the argument of the eigenvalue is positive (resp. non-negative), for every . The following is well-known, we sketch a proof for the sake of completeness.
Proposition. Let be a stable (resp. semi-stable) -matrix. Then the characteristic polynomial has coefficients satisfying (resp. ), for ;
Proof. Observe that is the product of polynomials with and with . Stability (resp. semi-stability) implies that (resp. ) above. Therefore, is product of polynomials with positive coefficients.□
Remark: In most of the literature the stability concept we use goes by the name of positive stability, while the stability name is used also as Hurwitz stability, or Lyapunov stability.
The system of differential equations
is said to be stable if for every vector , the solution of the above system has the property that .
We recall here the celebrated.
Lyapunov criterion: The system is stable if and only if is a stable matrix, equivalently there is a real positive definite matrix such that
It is not hard to see that given , the corresponding is unique. A proof of the criterion and its equivalence to other stability conditions are considered in .
9.2 Semi-stable powers
Let be the eigenvalues of the real matrix with in polar form. Observe that , for , are the eigenvalues of and
Lemma. For a positive integer the following assertions are equivalent:
a. is a semi-stable matrix;
Proof. If is a semi-stable matrix, then has . Since is a real matrix then . Therefore
Assume that . Since for , adding up, we get
Hence we have equalities for . Then is semi-stable.□
We say that is a stable power (resp. semi-stable power) of if is a stable (resp. semi-stable) matrix.
10. Nakayama algebras
10.1 Cyclotomic Nakayama algebras
As a well-understood example the representation theory of the Nakayama algebras stands appart. Let be the quotient obtained from the linear quiver with vertices with radical of nilpotency index .
For instance, for the Cartan matrix and Coxeter matrix are:
whose characteristic polynomial is cyclotomic as we know from  or might be verified calculating , for and applying the criterion of Theorem 1. Indeed, for.
Starting with the sequence of traces repeats cyclically. Therefore, for all . Then is of cyclotomic type.
10.2 An example
We recall in some length the argument given in  for the cyclotomicity of , for all .
Consider the algebra with vertices and whose quiver is given as
with all commutative relations. The corresponding Coxeter polynomial
is a product of cyclotomic polynomials, therefore is a cyclotomic polynomial. In fact , where is the hereditary algebra associated to the linear quiver .
For odd, we consider.
The following holds for the sequence of algebras and its Coxeter polynomials :
is derived equivalent to .
, for all ;
Observe that the sequence of algebras forms an interlaced tower of algebras, that is, it is a sequence of triangular algebras , such that is a basic algebra with simple modules and, among others, the condition
is satisfied for . Moreover, is a one-point extension (or coextension) of an accessible algebra by an exceptional - module such that the perpendicular category formed in the derived category is triangular equivalent to , for .
The following was shown in : Consider an interlaced tower of algebras with . If is contained in the union of the unit circle and the semi-ray of positive real numbers then either all are of cyclotomic type or . In the latter case, .
Since we know that , for all , we conclude that , for all . That is the Nakayama algebras of the form are of cyclotomic type.
10.3 Non-cyclotomic Nakayama algebras
Calculation of for and in intervals, for data sets , yield interesting information. Namely,
Many Nakayama algebras are of cyclotomic type;
Not all Nakayama algebras are of cyclotomic type. The case illustrates this claim:
is of cyclotomic type for all except for
A canonical algebra of weight is strongly accessible if and only if , in that case, is derived-equivalent to a representation-finite algebra if and only if the weight type does not dominate .