Limit Elements in the ConfigurationAlgebra for a Cancellative Monoid 1footnote 11footnote 1  The present paper is a complete version of the announcement [Sa2] based on the preprint RIMS-726. We rewrote the introduction, left out the filtration by (p,q), divided section §10 into §10 and 11, and updated the references. The §11, 12 are newly written, where, applying the results in §2-10 to Cayley graphs (\Gamma,G) of a cancellative monoid, we introduce a fibration \Omega(\Gamma,G)\to\Omega(P_{\Gamma,G}) of our interest. In Winter semester 05-06 at RIMS, the author held a series of seminars on the present paper. He thanks to its participants Yohei Komori, Michihiko Fujii, Yasushi Yamashita, Masahiko Yoshinaga, Takefumi Kondo, and Makoto Fuchiwaki. Particular thanks go to Yohei Komori, without whose encouredgment, this paper would not have appeared. The author is also grateful to Brian Forbes and Ken Shackleton for the careful reading of the manuscript.

Limit Elements in the Configuration
Algebra for a Cancellative Monoid 111  The present paper is a complete version of the announcement [Sa2] based on the preprint RIMS-726. We rewrote the introduction, left out the filtration by , divided section §10 into §10 and 11, and updated the references. The §11, 12 are newly written, where, applying the results in §2-10 to Cayley graphs of a cancellative monoid, we introduce a fibration of our interest. In Winter semester 05-06 at RIMS, the author held a series of seminars on the present paper. He thanks to its participants Yohei Komori, Michihiko Fujii, Yasushi Yamashita, Masahiko Yoshinaga, Takefumi Kondo, and Makoto Fuchiwaki. Particular thanks go to Yohei Komori, without whose encouredgment, this paper would not have appeared. The author is also grateful to Brian Forbes and Ken Shackleton for the careful reading of the manuscript.

Kyoji Saito 222 RIMS, Kyoto University, Kyoto 606-8502, Japan.

Abstract. We introduce two spaces and of pre-partition functions and of opposite series, respectively, which are associated with a Cayley graph of a cancellative monoid with a finite generating system and with its growth function . Under mild assumptions on , we introduce a fibration equivariant with a -action, which is transitive if it is of finite order. Then, the sum of pre-partition functions in a fiber is a linear combination of residues of the proportion of two growth functions and attached to at the places of poles on the circle of the convergent radius.

Contents

§ 1 Introduction

Replacing the square lattice in the classical Ising model ([Gi][I][O][Ba]) by a Cayley graph of a cancellative monoid with a finite generating system , we introduce the space of pre-partition functions. Here, the word pre-partition function is used only in the present introduction for a reason we explain now. Namely, for any finite region of the Cayley graph, we define the free energy by the logarithm of the sum of configurations in ((5.1.5) and (6.1.1)), and then consider accumulating points set (in a suitable toplogical setting) of the sequence of free energies of balls of radius in (§11.1 Definition). In the case of , consists of a single element. By inputting the data of Boltzmann weights to it, we get the partition function: an elliptic function dependent on the parameters involved in the Boltzmann weights. This fact, inspired the author to use the pre-partition functions to construct functions on the moduli of ([Sa1,3]).

In our new setting, the space is no longer a single element set but is a compact Hausdorff space. Under mild assumptions on , we construct a fibration (11.2.12), where is another newly introduced compact space, consisting of opposite sequences of the growth function (11.2.3). The fibration is equivariant with an action . If the action is of finite order, then it is transitive and the sum of the partition functions in a fiber of is given by a linear combination of the proportions of residues of the two series and at their poles on the circle of convergent radius . We publish these results in the present paper, eventhough our original attempt is not achieved.

The paper is divided into two parts. In the first part §2-10, we develop a general frame work on a topological Hopf algebra called the configuration algebra, where necessary concepts such as the configuration sums, the free energies (called equally dividing points) etc. are introduced. The algebra is equipped with two (one adic and the other classical) topologies in order to discuss carefully limit process in it. Then, inside its subspace of Lie-like elements at infinity, the set of all accumulation points of all free energies is introduced. In the second half, §11-12, we consider a Cayley graph of a monoid. Then, the set of pre-partition functions is defined as the subset of of all accumulating points of the sequence of free energies of the balls of radius in . We also introduce another limit set , called the space of opposite sequences, depending only on the Poincare series of (see (11.2.1-4) and (11.2.6)). The space is the key to relate the space with the singularities of the Poincare series on the circle of convergence (§11 Theorems 1-5). Then, comparing the two limit spaces and , we arrive at the goal: a residual presentation of pre-partition functions (§11 Theorem 6.)

Let us explain the contents of the present paper in more detail.

The isomorphism class of a colored oriented finite graph is called a configuration. The set of all configurations with fixed bounds of valency and colors, denoted by , has an additive monoid structure generated by , isomorphism classes of connected graphs (by taking the disjoint union as the product) and a partial ordering structure (§2.3). In §2.4, we introduce the basic invariant for and , called a covering coefficient. We denote by the completion of the semigroup ring with respect to the grading , called the configuration algebra (§3), where is the ring of coefficients. The algebra carries a topological Hopf algebra structure by taking the covering coefficients as structure constants (§4).

For a configuration , let be the sum of all its subgraphs, the configuration sum. We put , then forms a basis of the Lie-like space of the non-complete bi-algebra (§5 and 6). However, this is not a topological basis of the Lie-like space of the algebra . Therefore, we introduce a topological basis, denoted by , of . The coefficients of the transformation matrix between and are described by kabi-coefficients, introduced in §7. The base-change induces a linear map, called the kabi-map, from to a formal module spanned by . The kernel of the kabi-map is denoted by and is called the Lie-like space at infinity (§8).

The group-like elements of the configuration algebra is isomorphic to the fractional group of the monoid by the correspondence (§9). Thus, it contains a positive cone spanned by . We are interested in the equal division points () of the lattice points in the positive cone, and the set of their accumulation points with respect to the classical topology by specializing the coefficient to . In §10, by taking their logarithms333 The logarithm , which we call the logarithmic equal dividing point, is called the (Helmholz) free energy in statistical mechanics ([Gi][I][O][Ba]). , we define their accumulation set in . The set decomposes into a join of the infinite simplex spanned by the vertices for and a compact subset of (§10).

From §11, we fix a monoid with a finite generating system . The sequence of the logarithmic equal division points for the sequence of balls of radius in the Cayley graph accumulates to a compact subset of , called the space of limit elements for . This is the main object of interest of the present article. If is a group of polynomial growth, then due to results of Gromov [Gr1] and Pansu [P], for any generating system , is simple accumulating, i.e. .

In order to study the multi-accumulating cases, we introduce in §11 (11.2.3) another compact accumulating set : the space of opposite series of the growth series . Under mild Assumptions 1 in §11.1 and 2 in §11.2 on , we show that a natural proper surjective map equivariant with an action of exists (see 11.2 Theorems 1,2,3 and 4.), where i) is a forgetful map which remembers only the portion () of the limit elements (here of subgraphs of isomorphic to ) and ii) the -action on is generated by the map : the limit element the limit element . This -action, up to an initial constant factor, has an interpretation by the fattening action on : (here, is the equivalence class of for a representative of ) in the level of , and interpretation by the degree shift action in the level of .

Subsections 11.3 and 11.4 are devoted to the study of the space of opposite sequences for a power series (11.2.1) in general with a constraint on the growth of coefficients. The main concern is to clarify a certain duality between the set and the set of singularities of P(t) on the boundary of the disc of radius of convergence. It asks intricate analysis, and, in the present paper, we have clarified only when is a finite set. Actually, if is finite, then the -action becomes a cyclic and simple-transitive action. We explicitly determine as a set of rational functions in the opposit variable of (i.e. ). In particular, their common denominator , which is a factor of , has the degree equals to the rank of the space spanned by . If, further, is meromorphic in a neighborhood of the convergent disc, then the top part of the denominator of on the convergent circle of radius (see (11.4.1)) and the opposite denominator are related by the opposite transformation (11.4 Theorem 5).

If is finite, then again the -action on becomes cyclic and simple-transitive action for some such that for the period of the -action on . Therefore, the map is equivalent to the Galois covering map . Let us call the kernel of the homomorphism the inertia group and the sum of elements in of an orbit of the inertia group a trace element. As the goal of the present paper, we express the trace elements as linear combinations of the proportions of the residues of meromorphic functions and at the places in the root of . (11.5 Theorem 6). In the proof, we essentially uses the duality theory in 11.4.

Finally in §12, we give a few concluding remarks. Since we are only at the starting of the study of the limit space , the questions are scattered in various directions of general nature or of specific nature.

As an immediate generalization of our goal Theorem 6 for the cases when is not finite, in Problem 1.1, we ask measure theoretic approach for the duality between and , and give a conjectural formula.

Another important generalization of Theorem 6 is the globalization in the following sense: in many important examples, the growth function analytically extends to a meromorphic function in a covering regions of (and same for ). Let be a pole of order of such a meromorphic function, then for (which we call a residue of depth at ) belongs to (even though it is no longer a limit element). Theorem 6 treats only the extremal case and . Therefore, we ask to study all residues at all possible poles together with a possible action of a Galois group, in particular, to clarify the meaning of the (higher) residues at .

We conjecture that hyperbolic groups and some groups of geometric significance (surface groups, mapping class groups and Artin groups for suitable choices of generators) are finite accumulating, i.e. .

§ 2 Colored graphs and covering coefficients

An isomorphism class of finite graphs with a fixed color-set and a bounded number of edges (valency) at each vertex is called a configuration. The set of all configurations carries the structure of an abelian monoid with a partial ordering. The goal of the present section is to introduce a numerical invariant, called the covering coefficient, and to show some of its basic properties.

2.1 Colored Graphs.

We first give a definition of colored graph which is used in the present paper.

Definition.

1. A pair is called a graph, if is a set and is a subset of with , where is the involution and is the diagonal subset. An element of is called a vertex and a -orbit in is called an edge. A graph is called finite if . We sometimes denote a graph by and the set of its vertices by .

2. Two graphs are isomorphic if there is a bijection of vertices inducing a bijection of edges. Any subset of carries a graph structure by taking as the set of edges for . The set equipped with this graph structure is called a subgraph (or a full subgraph) of and is denoted by the same . In the present paper, the word “subgraph” shall be used only in this sense, and the notation shall mean also that is a subgraph of associated to the subset. Hence, we have the bijection:

3. A pair of a set and an involution on (i.e. a map with ) is called a color set. For a graph , a map is called a -coloring , or -coloring, if is equivariant with respect to involutions: . The pair consisting of a graph and a -coloring is called a -colored graph. Two -colored graphs are called -isomorphic if there is an isomorphism of the graphs compatible with the colorings. Subgraphs of a -colored graph are naturally -colored.

If all points of are fixed by , then the graph is called un-oriented. If consists of one orbit of , then the graph is called un-colored.

The isomorphism class of a -colored graph is denoted by . Sometimes we will write instead of (for instance, we put , and call connected if is topologically connected as a simplicial complex).

Example. (Colored Cayley graph of a monoid with cancellation conditions).  Let be a monoid satisfying the left and right cancellation conditions: if in for then in . In the other words, for any two elements , if there exists such that (resp. ) then is uniquely determined from and , which we shall denote by (resp. ). Let be a finite generating system of with . Then, we equip with a graph structure by taking as the set of edges. Due to the left cancellation condition, it becomes a colored graph by taking as the color set and by putting for , where is a formally defined set consisting of elements of symbols for and identifying with if in (such may not always exists). Due to the right cancellation condition, for any vertex and any , vertices connected with by the edges of color (i.e.  s.t. ) is unique. Let us call the graph, denoted by or for simplicity, the colored Cayley graph of the monoid with respect to the generating system . The left action of on is a color preserving graph embedding map from to itself.

If , then is a group and the above definition coincides with the usual definition of a Cayley graph of a group.

2.2 Configuration.

For the remainder of the paper, we fix a finite color set (i.e. ) and a non-negative integer , and consider only the -colored graphs such that the number of edges ending at any vertex (called valency) is at most . The isomorphism class of such a graph is called a -configuration (or, a configuration). The set of all (connected) configurations is defined by:

\hb@xt@.01(2.2.0)

The isomorphism class of an empty graph is contained in but not in . Sometimes it is convenient to exclude from . So put:

\hb@xt@.01(2.2.0)
Remark.

To be exact, the set of configurations (2.2) should have been denoted by . If there is a map between two color sets compatible with their involutions and an inequality , then there is a natural map . Thus, for any inductive system (i.e.  and for ), we get the inductive limit . In [S2], we used such limit set. However, in this paper, we fix and , since the key limit processes (3.2.0) and (10.1.0) can be carried out for fixed and .

2.3 Semigroup structure and partial ordering structure on .

We introduce the following two structures 1. and 2. on .

1. The set naturally has an abelian semigroup structure by putting

where is the disjoint union of graphs and representing the isomorphism classes and . The empty class plays the role of the unit and is denoted by 1. It is clear that is freely generated by . The power or denotes the class of a disjoint union of -copies of .

2. The set is partially ordered, where we define, for and ,

there exist graphs and with and .

The unit is the unique minimal element in by this ordering.

2.4 Covering coefficients.

For and , we introduce a non-negative integer:

\hb@xt@.01(2.4.0)

and call it the covering coefficient, where is defined by the following:

i) Fix any -graph with .

ii) Define a set:

\hb@xt@.01(2.4.0)

iii) Show: an isomorphism induces a bijection .

Remark.

In the definition (2.4.0), one should notice that

i) Each in (2.4.0) should be a full subgraph of (see (2.1) Def. 2.).

ii) The union of the edges of () does not have to cover all edges of .

iii) The set of vertices () may overlap the set .

Example.

Let be elements of with for . Then and

The covering coefficients are the most basic tool in the present paper. We shall give their elementary properties in 2.5 and their two basic rules: the composition rule in 2.6 and the decomposition rule in 2.7.

2.5 Elementary properties for covering coefficients.

Some elementary properties of covering coefficients, as immediate consequences of the definition, are listed below. They will be used in the study of the Hopf algebra structure on the configuration algebra in §4.

i) unless for and .

ii) is invariant by permutations of .

iii) For , one has an elimination rule:

\hb@xt@.01(2.5.0)

iv) For the case , the covering coefficients are given by

\hb@xt@.01(2.5.0)

v) For the case , the covering coefficients are given by

\hb@xt@.01(2.5.0)

vi) For the case , the covering coefficients are given by

\hb@xt@.01(2.5.0)

2.6 Composition rule.

Assertion.

For , (), one has

\hb@xt@.01(2.6.0)

Proof. If , then the formula reduces to 2.5 iii) and iv). Assume and consider the map

Here, means the subgraph of whose vertices are the union of the vertices of the () (cf. (2.1) Def. 2.) and the class is denoted by . The fiber over a point is bijective to the set so that one has the bijection

      

Note.

The LHS of (2.6.0) is a finite sum, since the only positive summands arise with due to 2.5 i).

2.7 Decomposition rule.

Assertion.

Let . For and , one has

\hb@xt@.01(2.7.0)

Here and run over all possible decompositions of in .

Proof. If , this is (2.5.0). Consider the map

One checks easily that the map is bijective.         

Note.

The RHS of (2.7.0) is a finite sum, since the only positive summands arises when and .

§ 3 Configuration algebra.

We complete the semigroup ring , where is a commutative associative unitary algebra, by use of the adic topology with respect to the grading , and call the completion the configuration algebra. It is a formal power series ring of infinitely many variables . We discuss several basic properties of the algebra, including topological tensor products.

3.1 The polynomial type configuration algebra .

The free abelian group generated by :

\hb@xt@.01(3.1.0)

naturally carries the structure of an algebra by the use of the semigroup structure on (recall 2.3), where plays the role of the unit element. It is isomorphic to the free polynomial algebra generated by , and hence is called the polynomial type configuration algebra. The algebra is graded by taking for , since one has additivity:

\hb@xt@.01(3.1.0)

3.2 The completed configuration algebra .

The polynomial type algebra (3.1.0) is not sufficiently large for our purposes, since it does not contain certain limit elements which we want to investigate (cf 4.6 Remark 3 and 6.4 Remark 2). Therefore, we localize the algebra by the completion with respect to the grading given in 3.1.

For , let us define an ideal in

\hb@xt@.01(3.2.0)

Taking as a fundamental system of neighborhoods of , we define the adic topology on (see Remark below). The completion

\hb@xt@.01(3.2.0)

will be called the completed configuration algebra, or, simply, the configuration algebra. More generally, for any commutative algebra with unit, we put

\hb@xt@.01(3.2.0)

and call it the configuration algebra over , or, simply, the configuration algebra. The augmentation ideal of the algebra is defined as

Let us give an explicit expression of an element of the configuration algebra by an infinite series. The quotient is naturally bijective to the free module of finite rank. Taking the inverse limit of the bijection, we obtain

In the other words, any element of the configuration algebra is expressed uniquely by an infinite series

\hb@xt@.01(3.2.0)

for some constants for all . The coefficient of the unit element is called the constant term of . The augmentation ideal is nothing but the collection of those having vanishing constant term.

Remark.

The topology on (except for the case ) defined above is not equal to the topology defined by taking the powers of the augmentation ideal as the fundamental system of neighborhoods of 0. More precisely, for and , the image of the product map:

\hb@xt@.01(3.2.0)

(c.f. (3.5.0) and (3.5.0)) does not generate (topologically) the target ideal on the RHS (= the closure in of the ideal ), since there exists a connected configuration with , but , as an element in , cannot be expressed as a function of elements of for . In this sense, the name “adic topology” is misused here.

The notation should not be mistaken for the algebra of formal power series in . In fact, it is the set of formal series in .

3.3 Finite type element in the configuration algebra.

The support for the series (3.2.0) is defined as

\hb@xt@.01(3.3.0)
Definition.

An element of a configuration algebra is said to be of finite type if is contained in a finitely generated semigroup in . Note that being of finite type does not mean that is a finite sum, but means that it is expressed only by a finite number of “variables”. The subset of consisting of all elements of finite type is denoted by . The polynomial type configuration algebra is a subalgebra of .

3.4 Saturated subalgebras of the configuration algebra.

The configuration algebra is sometimes a bit too large. For later applications, we introduce a class of its subalgebras, called the saturated subalgebras.

A subset is called saturated if for , any with belongs to . For a saturated set , let us define a subalgebra

\hb@xt@.01(3.4.0)

We shall call a subalgebra of the configuration algebra of the form (3.4.0) for some saturated a saturated subalgebra. A saturated algebra is characterized by the properties: i) is a closed subalgebra under the adic topology of the configuration algebra, and ii) if for then any connected component of (as a monomial) belongs to . We call the set

\hb@xt@.01(3.4.0)

the support of . Obviously, is the saturated subsemigroup of generated by . The algebra is determined from .

It is clear that if is a saturated subalgebra of then is a dense subalgebra of and that is naturally isomorphic to the completion of with respect to the induced adic topology.

Example.

We give two typical examples of saturated sets.

1. For any element , we define its saturation by

\hb@xt@.01(3.4.0)

2.  Let be a Cayley graph of an infinite monoid with respect to a finite generating system . Then, by choosing as the color set and as the bound of valence, we define a saturated subset of by

\hb@xt@.01(3.4.0)

Obviously, the saturated subalgebra consists only of finite type elements, whereas the algebra contains non-finite type elements. This makes the latter algebra interesting when we study limit elements in §11.

3.5 Completed tensor product of the configuration algebra.

The tensor product over of -copies of for is denoted by . In this section, we describe the completed tensor product of the completed configuration algebra,

Definition.

Let be a commutative algebra with unit. For , the completed -tensor product of the configuration algebra is defined by the inverse limit