A Bratteli diagram is given by the following objects:
A sequence of sets Vn labeled by positive integer set N. In some literature each element v of Vn is accompanied by a positive integer bv > 0.
A sequence of sets En labeled by N, endowed with maps s: En → Vn and r: En → Vn+1, such that:
* For each v in Vn, the number of elements e in En with s = v is finite.
* So is the number of e ∈ En−1 with r = v.
* When the vertices have markings by positive integers bv, the number av, v ' of the edges with s = v and r = v' for v ∈ Vn and v' ∈ Vn+1 satisfies bvav, v' ≤ bv'.
A customary way to pictorially represent Bratteli diagrams is to align the vertices according to their levels, and put the number bv beside the vertexv, or use that number in place of v, as in An ordered Bratteli diagram is a Bratteli diagram together with a partial order on En such that for any v ∈ Vn the set 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 Emax and the set of all minimal edges Emin. A Bratteli diagram with a unique infinitely long path in Emax and Emin is called essentially simple.
Sequence of finite-dimensional algebras
Any semisimple algebra over the complex numbersC of finite dimension can be expressed as a direct sum ⊕k Mnk 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=1iMnk into ⊕l=1jMml may be represented by a collection of positive numbersak, l satisfying ∑ nkak, l ≤ ml. This can be illustrated as a bipartite graph having the vertices marked by numbers k on one hand and the ones marked by lon the other hand, and having ak, l edges between the vertex nk and the vertex ml. Thus, when we have a sequence of finite-dimensional semisimple algebras An and injective homomorphisms φn : An' → An+1: between them, we obtain a Bratteli diagram by putting , marked by the size of matrices.
Sequence of split semisimple algebras
Any semisimple algebra is one whose modules are completely reducible, i.e. they decompose into the direct sum of simple modules. Let be a chain of split semisimple algebras, and let be the indexing set for the irreducible representations of. Denote by the irreducible module indexed by. Because of the inclusion, any -module restricts to a -module. Let denote the decomposition numbers The Bratteli diagram for the chain is obtained by placing one vertex for every element of on level and connecting a vertex on level to a vertex on level with edges.
Examples
If, the ith symmetric group, the corresponding Bratteli diagram is the same as Young's lattice. If is the Brauer algebra or the Birman–Wenzl algebra on i strands, then the resulting Bratteli diagram has partitions of i–2k 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. If is the Temperley–Lieb algebra on i strands, the resulting Bratteli has integers i–2k with one edge between integers on adjacent levels if one can be obtained from the other by adding or subtracting 1.
