The Wells map for abelian extensions of 3-Lie algebras

Youjun Tan; Senrong Xu

Czechoslovak Mathematical Journal (2019)

  • Volume: 69, Issue: 4, page 1133-1164
  • ISSN: 0011-4642

Abstract

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The Wells map relates automorphisms with cohomology in the setting of extensions of groups and Lie algebras. We construct the Wells map for some abelian extensions 0 → A ↪ L → π B → 0 of 3-Lie algebras to obtain obstruction classes in H 1 ( B , A ) for a pair of automorphisms in Aut ( A ) × Aut ( B ) to be inducible from an automorphism of L . Application to free nilpotent 3-Lie algebras is discussed.

How to cite

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Tan, Youjun, and Xu, Senrong. "The Wells map for abelian extensions of 3-Lie algebras." Czechoslovak Mathematical Journal 69.4 (2019): 1133-1164. <http://eudml.org/doc/294653>.

@article{Tan2019,
abstract = {The Wells map relates automorphisms with cohomology in the setting of extensions of groups and Lie algebras. We construct the Wells map for some abelian extensions $0\rightarrow A\hookrightarrow L\stackrel\{\pi \}\{\rightarrow \} B\rightarrow 0$ of 3-Lie algebras to obtain obstruction classes in $H^1(B,A)$ for a pair of automorphisms in $\{\rm Aut\}(A)\times \{\rm Aut\}(B)$ to be inducible from an automorphism of $L$. Application to free nilpotent 3-Lie algebras is discussed.},
author = {Tan, Youjun, Xu, Senrong},
journal = {Czechoslovak Mathematical Journal},
keywords = {automorphisms of 3-Lie algebras; representations of 3-Lie algebras; abelian extensions; cohomology; free nilpotent 3-Lie algebras},
language = {eng},
number = {4},
pages = {1133-1164},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The Wells map for abelian extensions of 3-Lie algebras},
url = {http://eudml.org/doc/294653},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Tan, Youjun
AU - Xu, Senrong
TI - The Wells map for abelian extensions of 3-Lie algebras
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 4
SP - 1133
EP - 1164
AB - The Wells map relates automorphisms with cohomology in the setting of extensions of groups and Lie algebras. We construct the Wells map for some abelian extensions $0\rightarrow A\hookrightarrow L\stackrel{\pi }{\rightarrow } B\rightarrow 0$ of 3-Lie algebras to obtain obstruction classes in $H^1(B,A)$ for a pair of automorphisms in ${\rm Aut}(A)\times {\rm Aut}(B)$ to be inducible from an automorphism of $L$. Application to free nilpotent 3-Lie algebras is discussed.
LA - eng
KW - automorphisms of 3-Lie algebras; representations of 3-Lie algebras; abelian extensions; cohomology; free nilpotent 3-Lie algebras
UR - http://eudml.org/doc/294653
ER -

References

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  1. Baer, R., 10.1007/bf01170643, Math. Z. 38 German (1934), 375-416. (1934) Zbl0009.01101MR1545456DOI10.1007/bf01170643
  2. Bardakov, V. G., Singh, M., 10.1142/s0219498817501626, J. Algebra Appl. 16 (2017), Article ID 1750162, 15 pages. (2017) Zbl06745716MR3661629DOI10.1142/s0219498817501626
  3. Daletskii, Y. L., Takhtajan, L. A., 10.1023/a:1007316732705, Lett. Math. Phys. 39 (1997), 127-141. (1997) Zbl0869.58024MR1437747DOI10.1023/a:1007316732705
  4. Filippov, V. T., 10.1007/bf00969110, Sib. Math. J. 26 (1985), 879-891 translation from Sibirsk. Mat. Zh. 26 1985 126-140. (1985) Zbl0594.17002MR0816511DOI10.1007/bf00969110
  5. Hilton, P. J., Stammbach, U., 10.1007/978-1-4419-8566-8, Graduate Texts in Mathematics 4, Springer, New York (1997). (1997) Zbl0863.18001MR1438546DOI10.1007/978-1-4419-8566-8
  6. Jin, P., 10.1016/j.jalgebra.2007.03.009, J. Algebra 312 (2007), 562-569. (2007) Zbl1131.20037MR2333172DOI10.1016/j.jalgebra.2007.03.009
  7. Jin, P., Liu, H., 10.1016/j.jalgebra.2010.04.034, J. Algebra 324 (2010), 1219-1228. (2010) Zbl1211.20044MR2671802DOI10.1016/j.jalgebra.2010.04.034
  8. Kasymov, Sh. M., 10.1007/bf02009328, Algebra Logic 26 (1987), 155-166 translation from Algebra Logika 26 1987 277-297. (1987) Zbl0658.17003MR0962883DOI10.1007/bf02009328
  9. Passi, I. B. S., Singh, M., Yadav, M. K., 10.1016/j.jalgebra.2010.03.029, J. Algebra 324 (2010), 820-830. (2010) Zbl1209.20021MR2651570DOI10.1016/j.jalgebra.2010.03.029
  10. Robinson, D. J. S., 10.1285/i15900932v33n1p121, Note Mat. 33 (2013), 121-129. (2013) Zbl1286.20029MR3071316DOI10.1285/i15900932v33n1p121
  11. Takhtajan, L., 10.1007/bf02103278, Commun. Math. Phys. 160 (1994), 295-315. (1994) Zbl0808.70015MR1262199DOI10.1007/bf02103278
  12. Takhtajan, L. A., Higher order analog of Chevalley-Eilenberg complex and deformation theory of n -gebras, St. Petersbg. Math. J. 6 (1995), 429-438 translation from Algebra Anal. 6 1994 262-272. (1995) Zbl0833.17021MR1290830
  13. Wells, C., 10.2307/1995472, Trans. Am. Math. Soc. 155 (1971), 189-194. (1971) Zbl0221.20054MR0272898DOI10.2307/1995472
  14. Xu, S., 10.1142/S0219498819501305, J. Algebra Appl. 18 (2019), Article ID 1950130, 26 pages. (2019) MR3977791DOI10.1142/S0219498819501305

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