Corrigendum to the paper ‘Decidability of the isomorphism problem for stationary AF-algebras and the associated ordered simple dimension groups’

2002 ◽  
Vol 22 (02) ◽  
Author(s):  
OLA BRATTELI ◽  
PALLE E. T. JORGENSEN ◽  
KI HANG KIM ◽  
FRED ROUSH
Keyword(s):  
Isomorphism Problem ◽  
Dimension Groups ◽  
Af Algebras

DIMENSION GROUPS, EMBEDDINGS AND PRESENTATIONS OF AF ALGEBRAS ASSOCIATED TO SOLVABLE LATTICE MODELS

1989 ◽  
Vol 04 (20) ◽  
pp. 1883-1890 ◽  
Author(s):  
DAVID E. EVANS ◽  
JEREMY D. GOULD
Keyword(s):  
Lattice Models ◽  
Path Algebra ◽  
Positive Cone ◽  
Point Of View ◽  
Conformal Field ◽  
Dimension Group ◽  
Dimension Groups ◽  
Ordered Ring ◽  
Af Algebras ◽  
The Ideal

If Γ is a graph, with distinguished vertex *, let A(Γ) denote the non-commutative path algebra on the space [Formula: see text] of semi-infinite paths in Γ beginning at *. Embeddings A(Γ1)→A(Γ2) of non-commutative AF algebras associated with graphs Γ1 and Γ2 are discussed from a dimension group point of view. For certain infinite T-shaped graphs, we have K0(A(Γ))≃ ℤ[t], with positive cone identified with {0}∪{P∈ℤ(t): P(λ)>0, λ∈(0,γ]}, where γ=γ(Γ)= ||Γ||−2<1/4. Hence for certain graphs there exists a unital homomorphism A(Γ1)→A(Γ2) if ||Γ1||=||Γ2||. For certain finite T-shaped graphs K0(A(Γ))≃ℤ[t]/<Q> where <Q> denotes the ideal generated by a polynomial Q=Q(Γ) which is essentially the characteristic polynomial of the graph Γ, and positive cone identified with {0}∪{f+<Q>: f(γ)>0} where γ=γ(Γ)=||Γ||−2. Hence there exists a unital homomorphism A(Γ1)→A(Γ2) if ||Γ1||=||Γ2||, and Q(Γ1) divides Q(Γ2). The structure of K0(A(Γ)) as an ordered ring is related to the fusion rules of rational conformal field theory. Moreover, for these T-shaped graphs there is an algebraic presentation which further illuminates the above embeddings. This presentation involves a new projection and a new relation in addition to those of Temperley-Lieb, and gives a rigidity above index four.

scholarly journals Non-stationarity of isomorphism between AF algebras defined by stationary Bratteli diagrams

2000 ◽  
Vol 20 (6) ◽  
pp. 1639-1656 ◽  
Author(s):  
OLA BRATTELI ◽  
PALLE E. T. JØRGENSEN ◽  
KI HANG KIM ◽  
FRED ROUSH
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Negative Integer ◽  
Matrix Elements ◽  
Integer Matrix ◽  
Incidence Matrices ◽  
Bratteli Diagrams ◽  
Dimension Groups ◽  
Necessary And Sufficient ◽  
Af Algebras

We first study situations where the stable AF algebras defined by two square primitive non-singular incidence matrices with non-negative integer matrix elements are isomorphic, even though no powers of the associated automorphisms of thecorresponding dimension groups are isomorphic. More generally we consider necessary and sufficient conditions for two such matrices to determine isomorphic dimension groups.We give several examples.

DIMENSION GROUPS AND EMBEDDINGS OF GRAPH ALGEBRAS

1994 ◽  
Vol 05 (03) ◽  
pp. 291-327 ◽  
Author(s):  
DAVID E. EVANS ◽  
JEREMY D. GOULD
Keyword(s):  
Path Algebra ◽  
Positive Cone ◽  
Point Of View ◽  
Graph Algebras ◽  
Conformal Field ◽  
Dimension Group ◽  
Dimension Groups ◽  
Ordered Ring ◽  
Af Algebras ◽  
The Ideal

If Γ is a graph, with distinguished vertex *, let A(Γ) denote the non-commutative path algebra on the space [Formula: see text] of semi-infinite paths in Γ beginning at *. We discuss embeddings A(Γ1) → A(Γ2) of AF algebras associated with graphs Γ1 and Γ2 from a dimension group point of view. For certain infinite T-shaped graphs, we have K0(A(Γ)) ≅ ℤ [t], with positive cone identified with {0}∪ {P ∈ ℤ [t]: P (λ) > 0, λ ∈ (0, γ]}, where γ = γ (Γ) =||Γ||−2 < 1/4. Hence for certain graphs there exists a unital homomorphism A(Γ1) → A(Γ2) if ||Γ1|| ≤ ||Γ2||. For certain finite T-shaped graphs K0 (A(Γ)) ≅ ℤ [t]/<Q> where <Q> denotes the ideal generated by a polynomial Q=Q(Γ) which is essentially the characteristic polynomial of the graph Γ and positive cone identified with {0}∪ {f + <Q>: f(γ) > 0} where γ = γ(Γ) = ||Γ||-2. Hence there exists a unital homomorphism A(Γ1) → A(Γ2) if ||Γ1|| = ||Γ2||, and Q(Γ2) divides Q(Γ1). The structure of K0(A(Γ)) as an ordered ring is related to the fusion rules of rational conformal field theory.

scholarly journals The isomorphism problem for k-trees is complete for logspace

2012 ◽  
Vol 217 ◽  
pp. 1-11 ◽  
Author(s):  
V. Arvind ◽  
Bireswar Das ◽  
Johannes Köbler ◽  
Sebastian Kuhnert
Keyword(s):  
Isomorphism Problem

scholarly journals MET: a Java package for fast molecule equivalence testing

2020 ◽  
Vol 12 (1) ◽  
Author(s):  
Jördis-Ann Schüler ◽  
Steffen Rechner ◽  
Matthias Müller-Hannemann
Keyword(s):  
Chemical Properties ◽  
Graph Isomorphism ◽  
Isomorphism Problem ◽  
Equivalence Classes ◽  
Equivalence Testing ◽  
Equivalence Test ◽  
Graph Isomorphism Problem ◽  
2D Structure ◽  
Node Labels ◽  
Labelled Graphs

AbstractAn important task in cheminformatics is to test whether two molecules are equivalent with respect to their 2D structure. Mathematically, this amounts to solving the graph isomorphism problem for labelled graphs. In this paper, we present an approach which exploits chemical properties and the local neighbourhood of atoms to define highly distinctive node labels. These characteristic labels are the key for clever partitioning molecules into molecule equivalence classes and an effective equivalence test. Based on extensive computational experiments, we show that our algorithm is significantly faster than existing implementations within , and . We provide our Java implementation as an easy-to-use, open-source package (via GitHub) which is compatible with . It fully supports the distinction of different isotopes and molecules with radicals.

scholarly journals Isomorphism, canonization, and definability for graphs of bounded rank width

2021 ◽  
Vol 64 (5) ◽  
pp. 98-105
Author(s):  
Martin Grohe ◽  
Daniel Neuen
Keyword(s):  
Fixed Point ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Graph Isomorphism ◽  
Isomorphism Problem ◽  
Descriptive Complexity ◽  
Graph Isomorphism Problem ◽  
Structural Graph ◽  
Graph Classes ◽  
Bounded Rank

We investigate the interplay between the graph isomorphism problem, logical definability, and structural graph theory on a rich family of dense graph classes: graph classes of bounded rank width. We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3 k + 4) is a complete isomorphism test for the class of all graphs of rank width at most k. A consequence of our result is the first polynomial time canonization algorithm for graphs of bounded rank width. Our second main result addresses an open problem in descriptive complexity theory: we show that fixed-point logic with counting expresses precisely the polynomial time properties of graphs of bounded rank width.

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