Geodesic
On the Theorem of the Primitive Element with Applications to the Representation Theory of Associative and Lie Algebras
Canadian mathematical bulletin, Tome 57 (2014) no. 4, pp. 735-748

Voir la notice de l'article provenant de la source Cambridge University Press

We describe all finite dimensional uniserial representations of a commutative associative (resp. abelian Lie) algebra over a perfect (resp. sufficiently large perfect) field. In the Lie case the size of the field depends on the answer to following question, considered and solved in this paper. Let $K/F$ be a finite separable field extension and let $x,\,y\,\in \,K$ . When is $F\left[ x,\,y \right]\,=\,F\left[ \alpha x\,+\,\beta y \right]$ for some nonzero elements $\alpha ,\,\beta \,\in \,F?$
DOI : 10.4153/CMB-2013-046-9
Mots-clés : 17B10, 13C05, 12F10, 12E20, uniserial module, Lie algebra, associative algebra, primitive element 
Cagliero, Leandro; Szechtman, Fernando. On the Theorem of the Primitive Element with Applications to the Representation Theory of Associative and Lie Algebras. Canadian mathematical bulletin, Tome 57 (2014) no. 4, pp. 735-748. doi: 10.4153/CMB-2013-046-9
@article{10_4153_CMB_2013_046_9,
     author = {Cagliero, Leandro and Szechtman, Fernando},
     title = {On the {Theorem} of the {Primitive} {Element} with {Applications} to the {Representation} {Theory} of {Associative} and {Lie} {Algebras}},
     journal = {Canadian mathematical bulletin},
     pages = {735--748},
     year = {2014},
     volume = {57},
     number = {4},
     doi = {10.4153/CMB-2013-046-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2013-046-9/}
}
TY  - JOUR
AU  - Cagliero, Leandro
AU  - Szechtman, Fernando
TI  - On the Theorem of the Primitive Element with Applications to the Representation Theory of Associative and Lie Algebras
JO  - Canadian mathematical bulletin
PY  - 2014
SP  - 735
EP  - 748
VL  - 57
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2013-046-9/
DO  - 10.4153/CMB-2013-046-9
ID  - 10_4153_CMB_2013_046_9
ER  - 
%0 Journal Article
%A Cagliero, Leandro
%A Szechtman, Fernando
%T On the Theorem of the Primitive Element with Applications to the Representation Theory of Associative and Lie Algebras
%J Canadian mathematical bulletin
%D 2014
%P 735-748
%V 57
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2013-046-9/
%R 10.4153/CMB-2013-046-9
%F 10_4153_CMB_2013_046_9

[AF] [AF] Anderson, F.W. and Fuller, K. R., Rings and categories of modules. Second ed., Graduate Texts in Mathematics, 13, Springer-Verlag, New York, 1992. Google Scholar

[Ar] [Ar] Artin, E., Galois theory. Dover, New York, 1998. Google Scholar

[ARS] [ARS] Auslander, M., Reiten, I., and Smalø, S. O., Representation theory of Artin algebras. Cambridge Studies in Advanced Mathematics, 36, Cambridge University Press, Cambridge, 1995. Google Scholar

[BM] [BM] Becker, M. F. and MacLane, S., The minimum number of generators for inseparable algebraic extensions.Bull. Amer. Math. Soc. 46 (1940) 182–186. Google Scholar | DOI

[B] [B] Bourbaki, N., Algebra II, Springer-Verlag, Berlin, 1990. Google Scholar

[BDS] [BDS] Browkin, J., Diviš, B., and Schinzel, A., Addition of sequences in general fields.Monatsh. Math. 82 (1976), no. 4, 261–268. Google Scholar | DOI

[CS1] [CS1] Cagliero, L. and Szechtman, F., The classification of uniserial sl(2) nV(m)-modules and a new interpretation of the Racah-Wigner 6 j-symbol.J. Algebra 386 (2013) 142–175. Google Scholar | DOI

[CS2] [CS2] Cagliero, L. and Szechtman, F., Classification of linked indecomposable modules of a family of solvable Lie algebras over an arbitrary field of characteristic 0. arxiv:1407.8125 Google Scholar

[DF] [DF] Dummit, D. S. and Foote, R. M., Abstract algebra. Third ed., JohnWiley & Sons, Hoboken, NJ, 2004. Google Scholar

[DK] [DK] Drozd, Y. and Kirichenko, V., Finite-dimensional algebras. Springer-Verlag, Berlin, 1994. Google Scholar

[EG] [EG] Eisenbud, D. and Griffith, P., Serial rings.J. Algebra 17 (1971), 389–400. Google Scholar | DOI

[Fa] [Fa] Facchini, A., Module theory. Endomorphism rings and direct sum decompositions in some classes of modules. Progress in Mathematics, 167, Birkhäuser, Basel, 1998. Google Scholar

[GP] [GP] Gelfand, I. M. and Ponomarev, V. A., Remarks on the classification of a pair of commuting linear transformations in a finite dimensional vector space.Functional Anal. Appl. 3 (1969), 325–326. Google Scholar | DOI

[GS] [GS] Guerszenzvaig, N. and Szechtman, F., Generalized Artin-Schreier polynomials. arxiv:1306.3967 Google Scholar

[I] [I] Isaacs, I. M., Degrees of sums in a separable field extension.Proc. Amer. Math. Soc. 25 (1970), 638–641. Google Scholar | DOI

[K] [K] Kaplansky, I., Fields and rings. The University of Chicago Press, Chicago, IL, 1969. Google Scholar

[M] [M] McCarthy, P. J., Algebraic extensions of fields. Dover, New York, 1991. Google Scholar

[Ma] [Ma] Makedonskyi, I., On wild Lie algebras. arxiv:1202.1401v2 Google Scholar

[N1] [N1] Nagell, T., Bemerkungen über zusammengesetzte Zahlkörper, Avh. Norske Vid. Akad. Oslo (1937), 1–26. Google Scholar

[N2] [N2] Nagell, T., Bestimmung des Grades gewisser relativalgebraischer Zahlen.Monatsh. Math. Phys. 48 (1939) 61–74. Google Scholar

[Na] [Na] Nakayama, T., On Frobeniusean algebras. II. Ann. of Math. 42 (1941), 1–21. Google Scholar | DOI

[P] [P] Petrenko, B. V., On the sum of two primitive elements of maximal subfields of a finite field.Finite Fields Appl. 9 (2003), no. 1, 102–116. Google Scholar | DOI

[Pu] [Pu] Puninski, G., Serial rings. Kluwer Academic Publishers, Dordrecht, 2001. Google Scholar

[R] [R] Robinson, D. J. S., A course in the theory of groups. Graduate Texts in Mathematics, 80, Springer-Verlag, New York, 1996. Google Scholar

[SL] [SL] Shores, T. andLewis, W., Serial modules and endomorphism rings.Duke Math. J. 41 (1974), 889–909. Google Scholar | DOI

[Sr] [Sr] Srinivasan, B., On the indecomposable representations of a certain class of groups.Proc. London Math. Soc. 10 (1960), 497–513. Google Scholar | DOI

[St] [St] Steinitz, E., Algebraische Theorie der Körper.J. Reine Angew. Math. 137 (1910), 167–309. Google Scholar

[T] [T] Teichmüller, O., p-Algebren.Deutsche Mathematik, 1 (1936), 362–388. Google Scholar

[W] [W] Weintraub, S. H., Observations on primitive, normal, and subnormal elements of field extensions.Monatsh. Math. 162 (2011) no. 2, 239–244. Google Scholar | DOI

Cité par Sources :