|
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 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. ==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 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''. 〔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.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bratteli diagram」の詳細全文を読む スポンサード リンク
|