Second cohomology of Lie rings and the Schur multiplier
We exhibit an explicit construction for the second cohomology group$H^2(L, A)$ for a Lie ring $L$ and a trivial $L$-module $A$.We show how the elements of $H^2(L, A)$ correspond one-to-one to theequivalence classes of central extensions of $L$ by $A$, where $A$now is considered as an abelian Lie rin...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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University of Isfahan
2014-06-01
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| Series: | International Journal of Group Theory |
| Subjects: | |
| Online Access: | http://www.theoryofgroups.ir/?_action=showPDF&article=3589&_ob=cf135f0fb1340cca48124481e4a34726&fileName=full_text.pdf. |
| _version_ | 1811263555107815424 |
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| author | Max Horn Seiran Zandi |
| author_facet | Max Horn Seiran Zandi |
| author_sort | Max Horn |
| collection | DOAJ |
| description | We exhibit an explicit construction for the second cohomology group$H^2(L, A)$ for a Lie ring $L$ and a trivial $L$-module $A$.We show how the elements of $H^2(L, A)$ correspond one-to-one to theequivalence classes of central extensions of $L$ by $A$, where $A$now is considered as an abelian Lie ring. For a finite Liering $L$ we also show that $H^2(L, C^*) cong M(L)$, where $M(L)$ denotes theSchur multiplier of $L$. These results match precisely the analoguesituation in group theory. |
| first_indexed | 2024-04-12T19:46:28Z |
| format | Article |
| id | doaj.art-587837b5dcfb43deb4ff32aa099a54c2 |
| institution | Directory Open Access Journal |
| issn | 2251-7650 2251-7669 |
| language | English |
| last_indexed | 2024-04-12T19:46:28Z |
| publishDate | 2014-06-01 |
| publisher | University of Isfahan |
| record_format | Article |
| series | International Journal of Group Theory |
| spelling | doaj.art-587837b5dcfb43deb4ff32aa099a54c22022-12-22T03:18:56ZengUniversity of IsfahanInternational Journal of Group Theory2251-76502251-76692014-06-0132920Second cohomology of Lie rings and the Schur multiplierMax HornSeiran ZandiWe exhibit an explicit construction for the second cohomology group$H^2(L, A)$ for a Lie ring $L$ and a trivial $L$-module $A$.We show how the elements of $H^2(L, A)$ correspond one-to-one to theequivalence classes of central extensions of $L$ by $A$, where $A$now is considered as an abelian Lie ring. For a finite Liering $L$ we also show that $H^2(L, C^*) cong M(L)$, where $M(L)$ denotes theSchur multiplier of $L$. These results match precisely the analoguesituation in group theory.http://www.theoryofgroups.ir/?_action=showPDF&article=3589&_ob=cf135f0fb1340cca48124481e4a34726&fileName=full_text.pdf.Lie ringsSchur multiplier of Lie ringscentral extensionsecond cohomology group of Lie rings |
| spellingShingle | Max Horn Seiran Zandi Second cohomology of Lie rings and the Schur multiplier International Journal of Group Theory Lie rings Schur multiplier of Lie rings central extension second cohomology group of Lie rings |
| title | Second cohomology of Lie rings and the Schur multiplier |
| title_full | Second cohomology of Lie rings and the Schur multiplier |
| title_fullStr | Second cohomology of Lie rings and the Schur multiplier |
| title_full_unstemmed | Second cohomology of Lie rings and the Schur multiplier |
| title_short | Second cohomology of Lie rings and the Schur multiplier |
| title_sort | second cohomology of lie rings and the schur multiplier |
| topic | Lie rings Schur multiplier of Lie rings central extension second cohomology group of Lie rings |
| url | http://www.theoryofgroups.ir/?_action=showPDF&article=3589&_ob=cf135f0fb1340cca48124481e4a34726&fileName=full_text.pdf. |
| work_keys_str_mv | AT maxhorn secondcohomologyoflieringsandtheschurmultiplier AT seiranzandi secondcohomologyoflieringsandtheschurmultiplier |