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]

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

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]

Any semisimple algebra over the complex numbers C of finite dimension can be expressed as a direct sum ⊕_k M_nk(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_nk(C) into ⊕_l=1^j M_ml(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_nk(C) ⊂ A_n and M_ml(C) ⊂ A_n+1 is equal to the multiplicity of M_nk(C) into M_ml(C) under φ_n.

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 ${\displaystyle A_{0}\subseteq A_{1}\subseteq A_{2}\subseteq \cdots }$ be a chain of split semisimple algebras, and let ${\displaystyle {\hat {A}}_{i}}$ be the indexing set for the irreducible representations of ${\displaystyle A_{i}}$. Denote by ${\displaystyle A_{i}^{\lambda }}$ the irreducible module indexed by ${\displaystyle \lambda \in {\hat {A}}_{i}}$. Because of the inclusion ${\displaystyle A_{i}\subseteq A_{i+1}}$, any ${\displaystyle A_{i+1}}$-module ${\displaystyle M}$ restricts to a ${\displaystyle A_{i}}$-module. Let ${\displaystyle g_{\lambda ,\mu }}$ denote the decomposition numbers

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

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

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

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

• Bratteli–Vershik diagram