| Title:
|
Koszul duality for simplicial restricted Lie algebras (English) |
| Author:
|
Konovalov, Nikolay |
| Language:
|
English |
| Journal:
|
Higher Structures |
| ISSN:
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2209-0606 |
| Volume:
|
8 |
| Issue:
|
2 |
| Year:
|
2024 |
| Pages:
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248-331 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $s_0$Lie$^r$ be the category of 0-reduced simplicial restricted Lie algebras over a fixed perfect field of positive characteristic $p$. We prove that there is a full subcategory Ho$(s_0Lie_(ξ)^(r))$ of the homotopy category Ho$s_0Lie^(r)$ and an equivalence Ho$(s_0Lie_(ξ)^(r))$ $\simeq$ Ho$(s_1CoAlg^(tr))$. Here $s_1CoAlg^(tr)$ is the category of 1-reduced simplicial truncated coalgebras; informally, a coaugmented cocommutative coalgebra $C$ is truncated if $x^(p)=0$ for any $x$ from the augmentation ideal of the dual algebra C*. Moreover, we provide a sufficient and necessary condition in terms of the homotopy groups $\pi_{\infty}(L_(•))$ for $L_(•) \in Ho(s_0Lie^(r))$ to Lie in the full subcategory Ho$(s_0Lie_(ξ)^(r))$. As an application of the equivalence above, we construct and examine an analog of the unstable Adams spectral sequence of A. K. Bousfield and D. Kan in the category sLie$^r$. We use this spectral sequence to recompute the homotopy groups of a free simplicial restricted Lie algebra. (English) |
| Keyword:
|
Koszul duality |
| Keyword:
|
restricted Lie algebras |
| Keyword:
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truncated coalgebras |
| Keyword:
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$\Lambda$-algebra |
| MSC:
|
16T15 |
| MSC:
|
17B56 |
| MSC:
|
18M70 |
| MSC:
|
18N50 |
| MSC:
|
55S10 |
| idZBL:
|
Zbl 08006153 |
| idMR:
|
MR4835391 |
| DOI:
|
10.21136/HS.2024.12 |
| . |
| Date available:
|
2026-03-13T14:37:37Z |
| Last updated:
|
2026-03-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153477 |
| . |
| Reference:
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