Bratteli diagram

In mathematics, a Bratteli diagram is a combinatorial structure: a graph composed of vertices labelled by positive integers ("level") and unoriented edges between vertices having levels differing by one. The notion was introduced by Ola Bratteli^[1] in 1972 in the theory of operator algebras to describe directed sequences of finite-dimensional algebras: it played an important role in Elliott's classification of AF-algebras and the theory of subfactors. Subsequently Anatoly Vershik associated dynamical systems with infinite paths in such graphs.^[2]

Definition

A Bratteli diagram is given by the following objects:

A sequence of sets V_n ('the vertices at level n ') labeled by positive integer set N. In some literature each element v of V_n is accompanied by a positive integer b_v > 0.
A sequence of sets E_n ('the edges from level n to n + 1 ') labeled by N, endowed with maps s: E_n → V_n and r: E_n → V_n+1, such that:
- For each v in V_n, the number of elements e in E_n with s(e) = v is finite.
- So is the number of e ∈ E_n−1 with r(e) = v.
- When the vertices have markings by positive integers b_v, the number a_v, v ' of the edges with s(e) = v and r(e) = v' for v ∈ V_n and v' ∈ V_n+1 satisfies b_v a_v, v' ≤ b_v'.

A customary way to pictorially represent Bratteli diagrams is to align the vertices according to their levels, and put the number b_v beside the vertex v, or use that number in place of v, as in

alt=E_0={a}. a is labeled 1, and has two edges to b and one to c. E_1={b,c}. b is labeled 2, and has one edge to d. c is labeled 1, and has one edge to d and one to e. E_2={d,e}. d is labeled 3, and has one edge to f. e is labeled 1, and has one edge to f and one to g. E_3={f,g}. f is labeled 4. g is labeled 1. Etc.

An ordered Bratteli diagram is a Bratteli diagram together with a partial order on E_n such that for any v ∈ V_n the set { e ∈ E_n−1 : r(e) = v } is totally ordered. Edges that do not share a common range vertex are incomparable. This partial order allows us to define the set of all maximal edges E_max and the set of all minimal edges E_min. A Bratteli diagram with a unique infinitely long path in E_max and E_min is called essentially simple. ^[3]

Sequence of finite-dimensional algebras

Any semisimple algebra over the complex numbers C of finite dimension can be expressed as a direct sum ⊕_k M_{n_k}(C) of matrix algebras, and the C-algebra homomorphisms between two such algebras up to inner automorphisms on both sides are completely determined by the multiplicity number between 'matrix algebra' components. Thus, an injective homomorphism of ⊕_k=1ⁱ M_{n_k}(C) into ⊕_l=1^j M_{m_l}(C) may be represented by a collection of positive numbers a_{k, l} satisfying Σ n_k a_k, l ≤ m_l. (The equality holds if and only if the homomorphism is unital; we can allow non-injective homomorphisms by allowing some a_k,l to be zero.) This can be illustrated as a bipartite graph having the vertices marked by numbers (n_k)_k on one hand and the ones marked by (m_l)_l on the other hand, and having a_k, l edges between the vertex n_k and the vertex m_l.

Thus, when we have a sequence of finite-dimensional semisimple algebras A_n and injective homomorphisms φ_n : A_n' → A_n+1: between them, we obtain a Bratteli diagram by putting

V_n = the set of simple components of A_n

(each isomorphic to a matrix algebra), marked by the size of matrices.

(E_n, r, s): the number of the edges between M_{n_k}(C) ⊂ A_n and M_{m_l}(C) ⊂ A_n+1 is equal to the multiplicity of M_{n_k}(C) into M_{m_l}(C) under φ_n.

Sequence of split semisimple algebras

Any semisimple algebra (possibly of infinite dimension) is one whose modules are completely reducible, i.e. they decompose into the direct sum of simple modules. Let $A_{0}\subseteq A_{1}\subseteq A_{2}\subseteq \cdots$ be a chain of split semisimple algebras, and let ${\hat {A}}_{i}$ be the indexing set for the irreducible representations of $A_{i}$ . Denote by $A_{i}^{\lambda }$ the irreducible module indexed by $\lambda \in {\hat {A}}_{i}$ . Because of the inclusion $A_{i}\subseteq A_{i+1}$ , any $A_{i+1}$ -module $M$ restricts to a $A_{i}$ -module. Let $g_{\lambda ,\mu }$ denote the decomposition numbers

A_{i+1}^{\mu }\downarrow _{A_{i}}^{A_{i+1}}=\bigoplus _{\lambda \in {\hat {A}}_{i}}g_{\lambda ,\mu }A_{i}^{\lambda }.

The Bratteli diagram for the chain $A_{0}\subseteq A_{1}\subseteq A_{2}\subseteq \cdots$ is obtained by placing one vertex for every element of ${\hat {A}}_{i}$ on level $i$ and connecting a vertex $\lambda$ on level $i$ to a vertex $\mu$ on level $i+1$ with $g_{\lambda ,\mu }$ edges.

Examples

Bratteli diagram for Brauer and BMW algebras on i=0,1,2,3, and 4 strands.

(1) If $A_{i}=S_{i}$ , the ith symmetric group, the corresponding Bratteli diagram is the same as Young's lattice.^[4]

(2) If $A_{i}$ is the Brauer algebra or the Birman–Wenzl algebra on i strands, then the resulting Bratteli diagram has partitions of i–2k (for $k=0,1,2,\ldots ,\lfloor i/2\rfloor$ ) with one edge between partitions on adjacent levels if one can be obtained from the other by adding or subtracting 1 from a single part.

(3) If $A_{i}$ is the Temperley–Lieb algebra on i strands, the resulting Bratteli has integers i–2k (for $k=0,1,2,\ldots ,\lfloor i/2\rfloor$ ) with one edge between integers on adjacent levels if one can be obtained from the other by adding or subtracting 1.

References

^ Bratteli, Ola (1972). "Inductive limits of finite dimensional C^*-algebras". Transactions of the American Mathematical Society. 171: 195–234. doi:10.1090/s0002-9947-1972-0312282-2. Zbl 0264.46057.
^ Vershik, A.M. (1985). "A theorem on the Markov periodic approximation in ergodic theory". Journal of Soviet Mathematics. 28 (5): 667–674. doi:10.1007/bf02112330. Zbl 0559.47006.
^ Herman, Richard H.; Putnam, Ian F.; Skau, Christian F. (1992). "Ordered Bratteli diagrams, dimension groups and topological dynamics". International Journal of Mathematics. 3 (6): 827–864. doi:10.1142/S0129167X92000382.
^ Alcock-Zeilinger, Judith M. "The Symmetric Group, its Representations, and Combinatorics" (PDF). University of Tübingen Department of Mathematics. Th. 4.5.

Halverson, Tom; Ram, Arun (1995). "Characters of algebras containing a Jones basic construction: The Temperley-Lieb, Okada, Brauer, and Birman–Wenzl algebras". Advances in Mathematics. 116 (2): 263–321. doi:10.1006/aima.1995.1068. ISSN 0001-8708. Zbl 0856.16038.
Davidson, Kenneth R. (1996). C*-algebras by example. Fields Institute Monographs. Vol. 6. Providence, RI: American Mathematical Society. ISBN 0-8218-0599-1. Zbl 0958.46029.
Rørdam, Mikael; Larsen, Flemming; Laustsen, Niels (2000). An introduction to K-theory for C*-algebras. London Mathematical Society Student Texts. Vol. 49. Cambridge: Cambridge University Press. ISBN 0-521-78334-8. Zbl 0967.19001.
Durand, Fabien (2010). "6. Combinatorics on Bratteli diagrams and dynamical systems". In Berthé, Valérie; Rigo, Michael (eds.). Combinatorics, automata, and number theory. Encyclopedia of Mathematics and its Applications. Vol. 135. Cambridge: Cambridge University Press. pp. 324–372. ISBN 978-0-521-51597-9. Zbl 1272.37006.

[1] Bratteli, Ola (1972). "Inductive limits of finite dimensional C^*-algebras". Transactions of the American Mathematical Society. 171: 195–234. doi:10.1090/s0002-9947-1972-0312282-2. Zbl 0264.46057.

[2] Vershik, A.M. (1985). "A theorem on the Markov periodic approximation in ergodic theory". Journal of Soviet Mathematics. 28 (5): 667–674. doi:10.1007/bf02112330. Zbl 0559.47006.

[3] Herman, Richard H.; Putnam, Ian F.; Skau, Christian F. (1992). "Ordered Bratteli diagrams, dimension groups and topological dynamics". International Journal of Mathematics. 3 (6): 827–864. doi:10.1142/S0129167X92000382.

[4] Alcock-Zeilinger, Judith M. "The Symmetric Group, its Representations, and Combinatorics" (PDF). University of Tübingen Department of Mathematics. Th. 4.5.

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