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

top
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

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.