Geodesic
Conjugacy of Cartan subalgebras in EALAs with a~non-fgc
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 78 (2017) no. 2, pp. 283-309.

Voir la notice de l'article provenant de la source Math-Net.Ru

We establish the conjugacy of Cartan subalgebras for extended affine Lie algebras whose centreless core is “of type A”, i. e., $\ell \times \ell$-matrices over a quantum torus $\mathcal{Q}$ whose trace lies in the commutator space of $\mathcal{Q}$. This settles the last outstanding part of the conjugacy problem for Extended Affine Lie Algebras that remained open. 
@article{MMO_2017_78_2_a4,
     author = {V. Chernousov and E. Neher and A. Pianzola},
     title = {Conjugacy of {Cartan} subalgebras in {EALAs} with a~non-fgc},
     journal = {Trudy Moskovskogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {283--309},
     publisher = {mathdoc},
     volume = {78},
     number = {2},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/MMO_2017_78_2_a4/}
}
TY  - JOUR
AU  - V. Chernousov
AU  - E. Neher
AU  - A. Pianzola
TI  - Conjugacy of Cartan subalgebras in EALAs with a~non-fgc
JO  - Trudy Moskovskogo matematičeskogo obŝestva
PY  - 2017
SP  - 283
EP  - 309
VL  - 78
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MMO_2017_78_2_a4/
LA  - en
ID  - MMO_2017_78_2_a4
ER  - 
%0 Journal Article
%A V. Chernousov
%A E. Neher
%A A. Pianzola
%T Conjugacy of Cartan subalgebras in EALAs with a~non-fgc
%J Trudy Moskovskogo matematičeskogo obŝestva
%D 2017
%P 283-309
%V 78
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MMO_2017_78_2_a4/
%G en
%F MMO_2017_78_2_a4
V. Chernousov; E. Neher; A. Pianzola. Conjugacy of Cartan subalgebras in EALAs with a~non-fgc. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 78 (2017) no. 2, pp. 283-309. http://geodesic.mathdoc.fr/item/MMO_2017_78_2_a4/

[1] B. Allison, “Some isomorphism invariants for Lie tori”, J. Lie Theory, 22 (2012), 163–204 | MR | Zbl

[2] B. Allison, S. Azam, S. Berman, Y. Gao, A. Pianzola, Extended affine Lie algebras and their root systems, Mem. AMS, 126, no. 603, 1997 | MR | Zbl

[3] B. Allison, S. Berman, J. Faulkner, A. Pianzola, “Multiloop realization of extended affine Lie algebras and Lie tori”, Trans. AMS, 361 (2009), 4807–4842 | DOI | MR | Zbl

[4] B. Allison, S. Berman, Y. Gao, A. Pianzola, “A characterization of affine Kac–Moody Lie algebras”, Comm. Math. Phys., 185 (1997), 671–688 | DOI | MR | Zbl

[5] V. A. Artamonov, “Serre's quantum problem”, Uspekhi Mat. Nauk, 53:4(322) (1998), 3–76 | DOI | MR | Zbl

[6] G. Benkart, E. Neher, “The centroid of extended affine and root graded Lie algebras”, J. Pure Appl. Algebra, 205 (2006), 117–145 | DOI | MR | Zbl

[7] S. Berman, Y. Gao, Y. S. Krylyuk, “Quantum tori and the structure of elliptic quasi-simple Lie algebras”, J. Funct. Anal., 135 (1996), 339–389 | DOI | MR | Zbl

[8] S. Berman, Y. Gao, Y. Krylyuk, E. Neher, “The alternative torus and the structure of elliptic quasi-simple Lie algebras of type $A_2$”, Trans. AMS, 347 (1995), 4315–4363 | MR | Zbl

[9] N. Bourbaki, Éléments de mathématique, Chap. VII: Sous-algèbres de Cartan, éléments réguliers. Chap. \rm VIII: Algèbres de Lie semi-simples déployées, v. XXXVIII, Actualités Scientifiques et Industrielles, Groupes et algèbres de Lie, Hermann, Paris, 1975, 271 pp. | MR

[10] V. Chernousov, P. Gille, A. Pianzola, “Conjugacy theorems for loop reductive group schemes and Lie algebras”, Bull. Math. Sci., 4 (2014), 281–324 | DOI | MR | Zbl

[11] V. Chernousov, E. Neher, A. Pianzola, U. Yahorau, “On conjugacy of Cartan subalgebras in extended affine Lie algebras”, Adv. Math., 290 (2016), 260–292 | DOI | MR | Zbl

[12] V. Chernousov, E. Neher, A. Pianzola, “On conjugacy of Cartan subalgebras in non-fgc Lie tori”, Transform. Groups, 21:4 (2016), 1003–1037 | DOI | MR | Zbl

[13] V. Kac, Infinite dimensional Lie algebras, 2nd ed., Cambridge Univ. Press, Cambridge, 1985 | MR | Zbl

[14] Y. Krylyuk, “On automorphisms and isomorphisms of quasi-simple Lie algebras”, J. Math. Sci. (New York), 100:1 (2000), 1944–2002 | DOI | MR | Zbl

[15] E. Neher, “Lie tori”, C. R. Math. Acad. Sci. Soc. R. Can., 26:3 (2004), 84–89 | MR | Zbl

[16] E. Neher, “Extended affine Lie algebras”, C. R. Math. Acad. Sci. Soc. R. Can., 26:3 (2004), 90–96 | MR | Zbl

[17] E. Neher, “Extended affine Lie algebras and other generalizations of affine Lie algebras — a survey”, Developments and trends in infinite-dimensional Lie theory, Progr. Math., 288, Birkhäuser Boston Inc., Boston, MA, 2011, 53–126 | MR | Zbl

[18] E. Neher, “Extended affine Lie algebras — an introduction to their structure theory”, Geometric representation theory and extended affine Lie algebras, Fields Inst. Commun., 59, AMS, Providence, RI, 2011, 107–167 | MR | Zbl

[19] E. Neher, A. Pianzola, D. Prelat, C. Sepp, “Invariant bilinear forms of algebras given by faithfully flat descent”, Commun. Contemp. Math., 2014 | DOI | MR

[20] E. Neher, Y. Yoshii, “Derivations and invariant forms of Jordan and alternative tori”, Trans. AMS, 355:3 (2003), 1079–1108 | DOI | MR | Zbl

[21] J. M. Osborn, D. S. Passman, “Derivations of skew polynomial rings”, J. Algebra, 176:2 (1995), 417–448 | DOI | MR | Zbl

[22] D. H. Peterson, V. Kac, “Infinite flag varieties and conjugacy theorems”, Proc. Nat. Acad. Sci. USA, 80:6 (1983), 1778–1782 | DOI | MR | Zbl

[23] D. Quillen, “Higher algebraic $K$-theory: I”, Algebraic $K$-theory, I: Higher $K$-theories, Proc. Conf. (Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math., 341, Springer, Berlin, 1973, 85–147 | DOI | MR

[24] J. Rosenberg, Algebraic $K$-theory and its applications, Graduate Texts in Mathematics, 147, Springer-Verlag, New York, 1994 | DOI | MR | Zbl

[25] C. Weibel, The $K$-book. An introduction to algebraic $K$-theory, Graduate Studies in Mathematics, 145, AMS, Providence, RI, 2013 | DOI | MR | Zbl

[26] Y. Yoshii, “Coordinate algebras of extended affine Lie algebras of type $A_1$”, J. Algebra, 234:1 (2000), 128–168 | DOI | MR | Zbl

[27] Y. Yoshii, “Lie tori — a simple characterization of extended affine Lie algebras”, Publ. Res. Inst. Math. Sci., 42:3 (2006), 739–762 | DOI | MR | Zbl