Peiffer elements in simplicial groups and algebras
The main objective of this paper is to prove in full generality the following two facts: A. For an operad O in Ab, let A be a simplicial O-algebra such that A<SUB>m</SUB> is generated as an O-ideal by (∑<SUB>i = 0</SUB><SUP>m-1</SUP> s<SUB>i</SUB> (A&...
Guardado en:
| Autores principales: | , |
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| Formato: | Articulo |
| Lenguaje: | Inglés |
| Publicado: |
2008
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| Acceso en línea: | http://sedici.unlp.edu.ar/handle/10915/84202 |
| Aporte de: |
| id |
I19-R120-10915-84202 |
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| record_format |
dspace |
| institution |
Universidad Nacional de La Plata |
| institution_str |
I-19 |
| repository_str |
R-120 |
| collection |
SEDICI (UNLP) |
| language |
Inglés |
| topic |
Ciencias Exactas Matemática Peiffer elements algebras simplicial groups |
| spellingShingle |
Ciencias Exactas Matemática Peiffer elements algebras simplicial groups Castiglioni, José Luis Ladra, M. Peiffer elements in simplicial groups and algebras |
| topic_facet |
Ciencias Exactas Matemática Peiffer elements algebras simplicial groups |
| description |
The main objective of this paper is to prove in full generality the following two facts:
A. For an operad O in Ab, let A be a simplicial O-algebra such that A<SUB>m</SUB> is generated as an O-ideal by (∑<SUB>i = 0</SUB><SUP>m-1</SUP> s<SUB>i</SUB> (A<SUB>m-1</SUB>)), for m > 1, and let NA be the Moore complex of A. Then
d(N<SUB>m</SUB>A) = ∑<SUB>I</SUB>γ (Op⊗ ∩ <SUB>i∈I<sub>1</sub></SUB> ker d<SUB>i</SUB> ⊗ ⋯ ⊗ ∩ <SUB>i∈I<sub>p</sub></SUB> ker d<SUB>i</SUB>)
where the sum runs over those partitions of [m - 1], I = (I<SUB>1</SUB>, ..., I<SUB>p</SUB>), p ≥ 1, and γ is the action of O on A.
B. Let G be a simplicial group with Moore complex NG in which G<SUB>n</SUB> is generated as a normal subgroup by the degenerate elements in dimensionn > 1, then d (N<SUB>n</SUB>G) = ∏I, J [∩<SUB>i∈I</SUB> ker d<SUB>i</SUB>, ∩<SUB>i∈J</SUB> ker d<SUB>j</SUB>], for I, J ⊆ [n - 1] with I ∪ J = [n - 1].
In both cases, d<SUB>i</SUB> is the i-th face of the corresponding simplicial object. The former result completes and generalizes results from Akça and Arvasi [I. Akça, Z. Arvasi, Simplicial and crossed Lie algebras, Homology Homotopy Appl. 4 (1) (2002) 43-57], and Arvasi and Porter [Z. Arvasi, T. Porter, Higher dimensional Peiffer elements in simplicial commutative algebras, Theory Appl. Categ. 3 (1) (1997) 1-23]; the latter completes a result from Mutlu and Porter [A. Mutlu, T. Porter, Applications of Peiffer pairings in the Moore complex of a simplicial group, Theory Appl. Categ. 4 (7) (1998) 148-173]. Our approach to the problem is different from that of the cited works. We have first succeeded with a proof for the case of algebras over an operad by introducing a different description of the inverse of the normalization functor N:Ab<SUP>Δ<sup>op</sup></SUP> → Ch≥ 0. For the case of simplicial groups, we have then adapted the construction for the inverse equivalence used for algebras to get a simplicial group NG ⊠ Λ from the Moore complex N G of a simplicial group G. This construction could be of interest in itself. |
| format |
Articulo Articulo |
| author |
Castiglioni, José Luis Ladra, M. |
| author_facet |
Castiglioni, José Luis Ladra, M. |
| author_sort |
Castiglioni, José Luis |
| title |
Peiffer elements in simplicial groups and algebras |
| title_short |
Peiffer elements in simplicial groups and algebras |
| title_full |
Peiffer elements in simplicial groups and algebras |
| title_fullStr |
Peiffer elements in simplicial groups and algebras |
| title_full_unstemmed |
Peiffer elements in simplicial groups and algebras |
| title_sort |
peiffer elements in simplicial groups and algebras |
| publishDate |
2008 |
| url |
http://sedici.unlp.edu.ar/handle/10915/84202 |
| work_keys_str_mv |
AT castiglionijoseluis peifferelementsinsimplicialgroupsandalgebras AT ladram peifferelementsinsimplicialgroupsandalgebras |
| bdutipo_str |
Repositorios |
| _version_ |
1764820488468561920 |