Differential geometry of group lattices
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Cites background or result from "Differential geometry of group latt..."
...in a way different from other relevant proposals (see, e.g. [19], [20], [21], [22],[ 23 ])....
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...[21], [22],[ 23 ]) the key point is different....
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...[18] and others (see, e.g. [20], [21], [22], [ 23 ], [15], [16])....
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References
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3,751 citations
"Differential geometry of group latt..." refers background in this paper
... of 1-forms is, in general, larger than the A-bimodule generated by the image of the space of functions under the action of the exterior derivative. There is a generalized digraph (“Schreier diagram” [16]) associated with such a first order differential calculus which in general has multiple links and also loops. Some further remarks are collected in section 9. 2 First order differential calculus associa...
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...set Kto a point representing a coset K′ whenever there is an h ∈S such that K′ = Kh, we obtain a digraph. This coset digraph [47] is also known as the Schreier diagram of the triple (G,S,H) (see Ref. [16], for example). In contrast to the digraphs (group lattices) considered in the previous sections, coset digraphs may have multiple arrows between two sites and even loops (i.e. arrows from a site to i...
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...a differential calculus which corresponds to the hypercubic lattice and which leads to an elegant formulation of lattice gauge theory [1]. A special class of digraphs are Cayley graphs [15] (see Refs. [16,17], for example), which are also known as group lattices in the physics literature. These are determined by a discrete group Gand a subset S. The elements of Gare the vertices of the digraph and the ele...
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...al first order differential calculus on the two disjoint parts of S 3 in this case. Another choice is S= {(12),(123)}. The corresponding digraph is shown on the righthand side of Fig. 2 (see also Refs. [16,17]). We call a group lattice bicovariant if ad(S)S⊂S. The significance of this definition will be made clear in section 3. Our previous examples of group lattices are indeed bicovariant, except for (S 3,S...
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"Differential geometry of group latt..." refers background in this paper
...Woronowicz [34] introduced the wedge product θ ∧ θ ′ = 1 2 (id− σ)(θ ⊗A θ ′) ....
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...A differential calculus which is both left and right covariant is called bicovariant [34]....
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...For a bicovariant differential calculus a bimodule isomorphism σ : Ω(1) ⊗A Ω 1 → Ω(1) ⊗A Ω 1 exists [6, 34] such that σ(θ1 ⊗A θ 2) = θ12 ⊗A θ h1 = θ0 ⊗A θ h1 (4....
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1,062 citations
"Differential geometry of group latt..." refers background in this paper
...a differential calculus which corresponds to the hypercubic lattice and which leads to an elegant formulation of lattice gauge theory [1]. A special class of digraphs are Cayley graphs [15] (see Refs. [16,17], for example), which are also known as group lattices in the physics literature. These are determined by a discrete group Gand a subset S. The elements of Gare the vertices of the digraph and the ele...
[...]
...al first order differential calculus on the two disjoint parts of S 3 in this case. Another choice is S= {(12),(123)}. The corresponding digraph is shown on the righthand side of Fig. 2 (see also Refs. [16,17]). We call a group lattice bicovariant if ad(S)S⊂S. The significance of this definition will be made clear in section 3. Our previous examples of group lattices are indeed bicovariant, except for (S 3,S...
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913 citations