Bratteli diagram

In mathematics, a Bratteli diagram is a graph composed of vertices having positive integer 'level's and unoriented edges between vertices having levels differing by one. It is used in operator algebra to describe directed sequences of finite dimensional algebras. The notion was introduced by Ola Bratteli^[1] in 1972, and played an important role in Elliott's classification of AF-algebras and the theory of subfactors.

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

1 = 2 − 3 − 4 ...

\ 1 ∠ 1 ∠ 1 ... .

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. ^[2]

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 numbers a_{k, l} satisfying ∑ n_k a_k, l ≤ m_l (the equality holds if and only if the homomorphism is unital). 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.

References

^ Bratteli, Ola; Inductive limits of finite dimensional C^*-algebras. Trans. Amer. Math. Soc. 171 (1972), 195–234.
^ Herman, Richard H. and Putnam, Ian F. and Skau, Christian F.Ordered Bratteli diagrams, dimension groups and topological dynamics. International Journal of Mathematics, volume 3, number 6. 1992, pp. 827-864.

K. Davidson; C*-algebras by example
Mikael Rørdam, Flemming Larsen, Niels Laustsen; An introduction to K-theory for C*-algebras‎