Crystallographic actions on Lie groups and post-Lie algebra structures

Dietrich Burde

Communications in Mathematics (2021)

  • Issue: 1, page 67-89
  • ISSN: 1804-1388

Abstract

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This survey on crystallographic groups, geometric structures on Lie groups and associated algebraic structures is based on a lecture given in the Ostrava research seminar in 2017 .

How to cite

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Burde, Dietrich. "Crystallographic actions on Lie groups and post-Lie algebra structures." Communications in Mathematics (2021): 67-89. <http://eudml.org/doc/298279>.

@article{Burde2021,
abstract = {This survey on crystallographic groups, geometric structures on Lie groups and associated algebraic structures is based on a lecture given in the Ostrava research seminar in $2017$.},
author = {Burde, Dietrich},
journal = {Communications in Mathematics},
keywords = {Crystallographic groups; Pre-Lie algebras; Post-Lie algebras},
language = {eng},
number = {1},
pages = {67-89},
publisher = {University of Ostrava},
title = {Crystallographic actions on Lie groups and post-Lie algebra structures},
url = {http://eudml.org/doc/298279},
year = {2021},
}

TY - JOUR
AU - Burde, Dietrich
TI - Crystallographic actions on Lie groups and post-Lie algebra structures
JO - Communications in Mathematics
PY - 2021
PB - University of Ostrava
IS - 1
SP - 67
EP - 89
AB - This survey on crystallographic groups, geometric structures on Lie groups and associated algebraic structures is based on a lecture given in the Ostrava research seminar in $2017$.
LA - eng
KW - Crystallographic groups; Pre-Lie algebras; Post-Lie algebras
UR - http://eudml.org/doc/298279
ER -

References

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  1. Abels, H., Margulis, G. A., Soifer, G. A., 10.1016/S0764-4442(99)80356-5, C. R. Acad. Sci. Paris, 324, 253-258, (253) MR1438395DOI10.1016/S0764-4442(99)80356-5
  2. Abels, H., Margulis, G. A., Soifer, G. A., 10.1007/BF02775430, Israel J. Math., 148, 2005, 11-21, (2005) MR2191222DOI10.1007/BF02775430
  3. Auslander, L., 10.1016/0040-9383(64)90012-6, Topology, 3, 1964, 131-139, (1964) MR0161255DOI10.1016/0040-9383(64)90012-6
  4. Auslander, L., 10.1090/S0002-9939-1965-0185012-7, Proceedings of the American Mathematical Society, 16, 6, 1965, 1230-1236, (1965) MR0185012DOI10.1090/S0002-9939-1965-0185012-7
  5. Auslander, L., 10.2307/2373867, Amer. J. Math., 99, 4, 1977, 809-826, (1977) MR0447470DOI10.2307/2373867
  6. Baues, O., 10.1090/S0002-9947-99-02315-6, Trans. Amer. Math. Soc., 351, 7, 1999, 2979-2996, (1999) MR1608273DOI10.1090/S0002-9947-99-02315-6
  7. Benkart, G., Kostrikin, A., Kuznetsov, M., 10.1006/jabr.1995.1041, J. Algebra, 171, 3, 1995, 894-916, (1995) MR1315926DOI10.1006/jabr.1995.1041
  8. Benzécri, J. P., Variétés localement affines, Thèse, Princeton Univ. , Princeton, N. J., 1955, (1955) MR2612352
  9. Bieberbach, L., 10.1007/BF01564500, Math. Ann., 70, 3, 1911, 297-336, (1911) MR1511623DOI10.1007/BF01564500
  10. Bieberbach, L., 10.1007/BF01456724, Math. Ann., 72, 3, 1912, 400-412, (1912) MR1511704DOI10.1007/BF01456724
  11. Benoist, Y., Une nilvariété non affine, J. Diff. Geom., 41, 1995, 21-52, (1995) MR1316552
  12. Brown, H., Bülow, R., Neubüser, J., Wondratschek, H., Zassenhaus, H., Crystallographic groups of four-dimensional space, 1978, Wiley Monographs in Crystallography, Wiley-Interscience, New York-Chichester-Brisbane, (1978) MR0484179
  13. Burde, D., 10.1006/jabr.1994.1275, Journal of Algebra, 169, 1, 1994, 112-138, (1994) MR1296585DOI10.1006/jabr.1994.1275
  14. Burde, D., Grunewald, F., 10.1016/0022-4049(94)00002-Z, Journal of Pure and Applied Algebra, 99, 3, 1995, 239-254, (1995) MR1332900DOI10.1016/0022-4049(94)00002-Z
  15. Burde, D., 10.1006/jabr.1996.0151, Journal of Algebra, 181, 3, 1996, 884-902, (1996) MR1386584DOI10.1006/jabr.1996.0151
  16. Burde, D., 10.1142/S0129167X96000323, International Journal of Mathematics, 7, 5, 1996, 599-616, (1996) MR1411303DOI10.1142/S0129167X96000323
  17. Burde, D., 10.1007/s000130050173, Archiv der Mathematik, 70, 2, 1998, 118-127, (1998) MR1491457DOI10.1007/s000130050173
  18. Burde, D., Dekimpe, K., Deschamps, S., 10.1007/s00208-004-0623-1, Mathematische Annalen, 332, 1, 2005, 161-176, (2005) MR2139256DOI10.1007/s00208-004-0623-1
  19. Burde, D., 10.2478/s11533-006-0014-9, Central European Journal of Mathematics, 4, 3, 2006, 323-357, (2006) MR2233854DOI10.2478/s11533-006-0014-9
  20. Burde, D., Deschamps, K. Dekimpe and S., 10.1090/conm/491/09612, Contemporary Mathematics, 491, 2009, 125-140, (2009) MR2537054DOI10.1090/conm/491/09612
  21. Burde, D., Eick, B., Graaf, W. A. de, 10.1016/j.jalgebra.2009.04.041, Journal of Algebra, 22, 3, 2009, 602-612, (2009) MR2531213DOI10.1016/j.jalgebra.2009.04.041
  22. Burde, D., Dekimpe, K., Vercammen, K., 10.1515/jgt.2010.018, Journal of Group Theory, 13, 5, 2010, 703-719, (2010) MR2720199DOI10.1515/jgt.2010.018
  23. Burde, D., Moens, W. A., 10.1016/j.jalgebra.2010.09.028, Journal of Algebra, 325, 1, 2011, 440-460, (2011) MR2745549DOI10.1016/j.jalgebra.2010.09.028
  24. Burde, D., Vercammen, K. Dekimpe and K., 10.1016/j.laa.2012.04.007, Linear Algebra and its Applications, 437, 5, 2012, 1250-1263, (2012) MR2942346DOI10.1016/j.laa.2012.04.007
  25. Burde, D., Dekimpe, K., 10.17323/1609-4514-2013-13-1-1-18, Moscow Mathematical Journal, 13, 1, 2013, 1-18, (2013) MR3112213DOI10.17323/1609-4514-2013-13-1-1-18
  26. Burde, D., Dekimpe, K., 10.1016/j.jalgebra.2016.05.026, Journal of Algebra, 464, 2016, 226-245, (2016) MR3533429DOI10.1016/j.jalgebra.2016.05.026
  27. Burde, D., Moens, W. A., 10.1016/j.jalgebra.2016.07.030, Journal of Algebra, 467, 2016, 183-201, (2016) MR3545958DOI10.1016/j.jalgebra.2016.07.030
  28. Burde, D., Globke, W., 10.1016/j.jalgebra.2017.06.009, Journal of Algebra, 487, 2017, 200-216, (2017) MR3671190DOI10.1016/j.jalgebra.2017.06.009
  29. Burde, D., Globke, W., Minchenko, A., 10.1007/s00031-018-9483-8, Transformation Groups, 24, 3, 2019, 769-780, (2019) MR3989690DOI10.1007/s00031-018-9483-8
  30. Burde, D., Dekimpe, K., Moens, W. A., 10.1016/j.jalgebra.2019.02.003, Journal of Algebra, 526, 2019, 12-29, (2019) MR3914193DOI10.1016/j.jalgebra.2019.02.003
  31. Burde, D., Ender, C., Moens, W. A., 10.1142/S0218196718500406, International Journal of Algebra and Computation, 28, 5, 2018, 915-933, (2018) MR3844707DOI10.1142/S0218196718500406
  32. Burde, D., Gubarev, V., 10.1080/00927872.2018.1536206, Communications in Algebra, 47, 5, 2019, 2280-2296, (2019) MR3977738DOI10.1080/00927872.2018.1536206
  33. Burde, D., Zusmanovich, P., 10.1080/00927872.2019.1612426, Communications in Algebra, 47, 12, 2019, 5218-5226, (2019) MR4019336DOI10.1080/00927872.2019.1612426
  34. Burde, D., Ender, C., 10.1016/j.laa.2019.09.010, Linear Algebra and its Applications, 584, 2020, 107-126, (2020) MR4009591DOI10.1016/j.laa.2019.09.010
  35. Burde, D., Gubarev, V., 10.1142/S0218196720500071, International Journal of Algebra and Computation, 30, 3, 2020, 451-466, (2020) MR4082417DOI10.1142/S0218196720500071
  36. Buser, P., A geometric proof of Bieberbach's theorems on crystallographic groups, Enseign. Math., 31, 1-2, 1985, 137-145, (1985) MR0798909
  37. Dekimpe, K., Igodt, P., 10.1007/s002220050160, Invent. Math., 129, 1, 1997, 121-140, (1997) MR1464868DOI10.1007/s002220050160
  38. Dekimpe, K., 10.1016/S0040-9383(02)00030-7, Topology, 42, 4, 2003, 821-832, (2003) MR1958530DOI10.1016/S0040-9383(02)00030-7
  39. Ender, C., Post-Lie algebra structures on classes of Lie algebras, PhD-thesis, University of Vienna, 2019, (2019) 
  40. Fried, D., Goldman, W., 10.1016/0001-8708(83)90053-1, Adv. Math., 47, 1983, 1-49, (1983) MR0689763DOI10.1016/0001-8708(83)90053-1
  41. Goldman, W. M., Two papers which changed my life: Milnor's seminal work on flat manifolds and bundles, Frontiers in complex dynamics, Princeton Math. Ser., 51, 2014, 679-703, (2014) MR3289925
  42. Gromov, M., Piatetski-Shapiro, I., 10.1007/BF02698928, Inst. Hautes Études Sci. Publ. Math., 66, 1988, 93-103, (1988) MR0932135DOI10.1007/BF02698928
  43. Helmstetter, J., 10.5802/aif.764, Ann. Inst. Fourier, 29, 1979, 17-35, (1979) MR0558586DOI10.5802/aif.764
  44. Hilbert, D., 10.1090/S0002-9904-1902-00923-3, Bulletin of the AMS, 87, 10, 1902, 437-479, (1902) MR1557926DOI10.1090/S0002-9904-1902-00923-3
  45. Jacobson, N., 10.1090/S0002-9939-1955-0068532-9, Proc. Am. Math. Soc., 6, 1955, 281-283, (1955) MR0068532DOI10.1090/S0002-9939-1955-0068532-9
  46. Margulis, G.A., 10.1007/BF01104978, J. Soviet Math., 36, 1987, 129-139, (1987) DOI10.1007/BF01104978
  47. Milnor, J., 10.1016/0001-8708(77)90004-4, Advances in Math., 25, 1977, 178-187, (1977) MR0454886DOI10.1016/0001-8708(77)90004-4
  48. Plesken, W., Schulz, T., 10.1080/10586458.2000.10504417, Experiment. Math., 9, 3, 2000, 407-411, (2000) MR1795312DOI10.1080/10586458.2000.10504417
  49. Schwarzenberger, R.L.E., 10.1017/S0305004100048696, Proc. Cambridge Philos. Soc., 76, 1974, 23-32, (1974) MR0349867DOI10.1017/S0305004100048696
  50. Segal, D., 10.1007/BF01444735, Math. Ann., 293, 1992, 569-578, (1992) MR1170527DOI10.1007/BF01444735
  51. Thurston, W.P., Three-dimensional geometry and topology. Vol. 1, 35, 1997, Edited by Silvio Levy. Princeton Mathematical Series, Princeton University Press, Princeton, NJ, (1997) MR1435975
  52. Tomanov, G., Properly discontinuous group actions on affine homogeneous spaces (English summary) Reprinted in Proc. Steklov Inst. Math., Algebra, Geometriya i Teoriya Chisel, 292, 1, 2016, 268-279, (2016) MR3628466
  53. Vallette, B., 10.1016/j.jpaa.2006.03.012, J. Pure and Applied Algebra, 208, 2, 2007, 699-725, (2007) MR2277706DOI10.1016/j.jpaa.2006.03.012
  54. (ed.), E. B. Vinberg, Geometry II, 29, 1991, Encyclopedia of Mathematical Sciences, Springer Verlag, (1991) 
  55. Zassenhaus, H., 10.1007/BF02568029, Comment. Math. Helv., 21, 1948, 117-141, (1948) MR0024424DOI10.1007/BF02568029

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