| Title:
|
An explicit symmetric DGLA model of a triangle (English) |
| Author:
|
Griniasty, Itay |
| Author:
|
Lawrence, Ruth |
| Language:
|
English |
| Journal:
|
Higher Structures |
| ISSN:
|
2209-0606 |
| Volume:
|
3 |
| Issue:
|
1 |
| Year:
|
2019 |
| Pages:
|
1-16 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We give explicit formulae for a differential graded Lie algebra (DGLA) model of the triangle which is symmetric under the geometric symmetries of the cell. This follows the work of Lawrence-Sullivan on the (unique) DGLA model of the interval and of Gadish-Griniasty-Lawrence on an explicit symmetric model of the bi-gon. As in the case of the bi-gon, the essential intermediate step is the construction of a symmetric point. Although in this warped geometry of points given by solutions of the Maurer-Cartan equation and lines given by a gauge transformation by Lie algebra elements of grading zero, the medians of a triangle are not concurrent, various other geometric constructions can be carried out. The construction can similarly be applied to give symmetric models of arbitrary $k$-gons. (English) |
| Keyword:
|
DGLA |
| Keyword:
|
infinity structure |
| Keyword:
|
Maurer-Cartan |
| Keyword:
|
Baker-Campbell-Hausdorff |
| MSC:
|
17B01 |
| MSC:
|
17B55 |
| MSC:
|
55U15 |
| idZBL:
|
Zbl 1473.17048 |
| idMR:
|
MR3939044 |
| DOI:
|
10.21136/HS.2019.01 |
| . |
| Date available:
|
2026-03-10T18:11:12Z |
| Last updated:
|
2026-03-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153408 |
| . |
| Reference:
|
[1] Buijs, U., Félix, Y., Murillo, A., Tanré, D.: Maurer-Cartan elements in the Lie models of finite simplicial complexes..Canad. Math. Bull. 60, 470–477 arxiv:1606.08794 http://arxiv.org/pdf/1606.08794 [math.AT] |
| Reference:
|
[2] Buijs, U., Félix, Y., Murillo, A., Tanré, D.: Symmetric Lie models of a triangle..arxiv:1802.01121 http://arxiv.org/pdf/1802.01121 [math.AT] |
| Reference:
|
[3] Dynkin, E.: Calculation of the coefficients in the Campbell–Hausdorff formula..Dokl. Akad. Nauk USSR (in Russian) 57, 323–326 |
| Reference:
|
[4] Eichler, M.: A new proof of the Baker-Campbell-Hausdorff formula..J. Math. Soc. Japan 20, 23–25 |
| Reference:
|
[5] Gadish, N., Griniasty, I., Lawrence, R.: An explicit symmetric DGLA model of a bi-gon..arxiv:1705.08483 http://arxiv.org/pdf/1705.08483 to appear in J. Knot Th. Ramif |
| Reference:
|
[6] Lawrence, R., Sullivan, D.: A formula for topology/deformations and its significance..Fundamenta Mathematica 225 229–242 arxiv:math/0610949 http://arxiv.org/pdf/math/0610949 [math.AT] |
| Reference:
|
[7] Quillen, D.: Rational homotopy theory..Ann. of Math. (2) 90, 205–295 |
| Reference:
|
[8] Sullivan, D.: Infinitesimal computations in topology..Inst. Hautes Études Sci. Publ. Math. 47, 269–331 10.1007/BF02684341 |
| Reference:
|
[9] Tradler, T., Zeinalian, M.: Infinity structure of Poincaré duality spaces..Algebr. Geom. Topol. 7, 233–260 arxiv:math/0309455 http://arxiv.org/pdf/math/0309455 [math.AT] |
| . |
