The Norm Functor over Schemes (with Philippe Gille and Cameron Ruether).
An introduction to norm (= corestriction) over fields and rings, given in the European Non-Associative Algebra Seminar (2024-06-10),
Youtube version of my talk
(Talk on Steinberg Groups for Jordan pairs at the workshop Nonassociative Algebras and Geometry, August 2019, Bonne Bay Marine Station, Newfoundland)
Steinberg groups for Jordan pairs - an introduction with open problems, to appear in Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification: In Honor of Vyjayanthi Chari on the Occasion of her 60th Birthday', Progress in Mathematics, Birkhäuser.
Lectures on extended affine Lie algebras, given at the Fields Institute Summer School on Geometric Representation Theory and Extended Affine Lie Algebras, Ottawa June 2009. Published as Extended affine Lie algebras -- an introduction to their structure theory, pages 107-167
of a Fields Institute Communications series volume on Geometric Representation Theory and Extended Affine Lie Algebras, edited by E. Neher, A. Savage and W. Wang.
Reflection systems and partial root systems (with Ottmar Loos, May 2009, 43 pages), published in Forum Math. 23 (2011), 349–411. This preprint is also available in a previous long version (50 pages). The long version contains more details, but the same results. The additional text is marked in the form >> ...(additional text)...<<.
Abstract: We develop a general theory of reflection systems and, more specifically, partial root systems which provide a unifying framework for finite root systems, Kac-Moody root systems, extended affine root systems and various generalizations thereof. Nilpotent and prenilpotent subsets are studied in this setting, based on commutator sets and the descending central series. We show that our notion of a prenilpotent pair coincides, for Kac-Moody root systems, with the one defined by Tits in terms of positive systems and the Weyl group.
Extended affine Lie algebras and other generalizations of affine Lie algebras -- a survey (June 2008) This is a survey on extended affine Lie algebras and related types of Lie algebras, which generalize affine Lie algebras. It appeared in Developments and trends in infinite-dimensional Lie theory, 53-126, Progr. Math., 288, Birkhäuser Boston, Inc., Boston, MA, 2011,
editors: K.-H. Neeb and A. Pianzola.
Nondegeneracy for Lie triple systems and Kantor pairs (with Esther García and Miguel Gómez Lozana), Canad. Math. Bull. 54(3), 2011, 442-455. Abstract: We study the transfer of nondegeneracy between Lie triple systems and their standard Lie algebra envelopes as well as between Kantor pairs, their associated Lie triple systems and their Lie algebra envelopes. We also show that simple Kantor pairs and Lie triple systems in characteristic 0 are nondegenerate.
Graded-simple Lie algebras of type B_2 and Jordan systems covered by a triangle (with Maribel Tocón, preprint March 2007). This is a research announcement which appeared in the proceedings of the satellite conference of the ICM 2006 ``From Lie Algebras to Quantum Groups", held at the University of Coimbra (Portugal), June 28-30, 2006. Abstract: We announce a classification of graded-simple Jordan systems covered by a compatible triangle, under some natural assumptions on the abelian group, in order to get the corresponding classification of
graded-simple Lie algebras of type B2.
A construction of gradings of Lie algebras (with Antonio Fernández López,
Esther García and Miguel Gómez Lozano); Int. Math. Res. Not. IMRN, 2007, no. 16, Art. ID rnm051, 34 pages; the published version has some strange formatting. Abstract: In this paper we present a method to construct gradings of Lie algebras. It requires the existence of an
abelian inner ideal B of the Lie algebra whose subquotient, a Jordan pair, is covered by a finite grid, and it produces a grading of the Lie algebra L by the weight lattice of the root system associated to the covering grid. As a corollary one obtains a finite Z-grading of L in the form L=L_{-n} + ... + L_n such that B=L_n. In particular, our assumption on B holds for abelian inner ideals of finite length in nondegenerate Lie algebras.
The centroid of extended affine and root graded Lie algebras
(with Georgia Benkart, Journal of Pure and Applied Algebra, vol. 205 (2006), no.1, 117--154. Abstract: We develop general results on centroids of Lie algebras and apply them to determine the centroid of extended affine Lie algebras, loop-like and Kac-Moody Lie algebras, and Lie algebras graded by finite root systems.
Extended affine Lie Algebras
(August 2004, 11 pages)
This is a research announcement. A shortened version has been published in two parts in the Mathematical Reports of the Academy of Science of the Royal Society of Canada: Lie Tori, C. R. Math. Rep. Acad. Sci. Canada Vol. 26, (3), 2004 pp. 84-89 and Extended affine Lie algebras, C. R. Math. Rep. Acad. Sci. Canada Vol. 26, (3), 2004 pp. 90-96.
Locally finite root systems
(with Ottmar Loos), published as Memoirs of the Amer. Math. Soc. vol. 171, number
811 (2004). If you would like a paper copy please contact me.
An introduction to universal central extensions of Lie superalgebras (July 2002, 19 pages)
A re-formatted version has appeared in the conference proceedings of Groups, rings, Lie and Hopf algebras (St. John's, NF, 2001, 141--166, (St. John's, NF, 2001), Math. Appl., 555, Kluwer Acad, Publ, Dordrecht, 2003.