__Steinberg groups for Jordan pairs - an introduction with open problems__, submitted__Conjugacy of Cartan subalgebras in EALAs with a non-fgc centreless core__(with Vladimir Chernousov and Arturo Pianzola); to appear in Trans. Moscow Math. Soc. (Vinberg's volume)__Integrable representations of root-graded Lie algebras__(with Nathan Manning and Hadi Salmasian); Journal of Algebra**500**(2018), 253-302.__On conjugacy of Cartan subalgebras in non-fgc Lie tori__(with Vladimir Chernousov and Arturo Pianzola); Transformation Groups**21**(2016), 1003-1037.__On conjugacy of Cartan subalgebras in extended affine Lie algebras__(with Vladimir Chernousov, Arturo Pianzola, and Uladzimir Yahorau); Advances in Mathematics**290**(2016), 260-292__Extensions and block decompositions for finite-dimensional representations of equivariant map algebras__(with Alistair Savage, Transformation Groups**20**(2015), 183-228)__Invariant bilinear forms of algebras given by faithfully flat descent__(with Arturo Pianzola, Daniel Prelat and Claudia Sepp), Communications in Contemporary Mathematics**17**(2015), 1450009 (37 pages), DOI: 10.1142/S0219199714500096 (Talk at the CMS winter meeting 2013, Ottawa)__Étale Descent of Derivations__(with Arturo Pianzola), Transformation Groups**18**(2013), 1189-1205.- A survey of equivariant map algebras with open problems (with Alistair Savage,
Contemporary Mathematics vol.
**602**(2013), 165-182) - Universal central extensions of direct limits of Lie superalgebras
(with Jie Sun),
Journal of Algebra
**368**(2012), 169-181. - Basic polynomial invariants, fundamental representations and the Chern class map (with Sanghoon Baek and Kirill Zainoulline), Documenta Math. 17 (2012), 135-150.
- Lie tori of type B_2 and graded-simple Jordan structures covered by a
triangle (with Maribel Tocón), Journal of Algebra 344 (2011), 78-113, contains the proofs of the
research announcement
*Graded-simple Lie algebras of type B_2 and Jordan systems covered by a triangle*, posted below. - Irreducible finite-dimensional representations of equivariant map algebras (with Alistair Savage and Prasad Senesi), Trans. Amer. Math. Soc. 364 (2012), 2619-2646
(
__talk__) at the CMS Winter meeting 2009, Windsor. -
__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, to appear in April 2011. - 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. - Invertible and nilpotent elements in the group algebra of a unique product group (published in Acta Appl. Math. 108 (2009), 135-139)

*Abstract*: We describe the nilpotent and invertible elements in group algebras k[G] for k a commutative associative unital ring and G a unique product group, for example an ordered 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), preprint Oct. 2007, published in Canad. Math. Bull.
**54**(3), 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. - An introduction to the theory of extended affine Lie algebras , based on a lecture at the Oberwolfach meeting on "Infinite dimensional Lie Theory", Dec. 10-16, 2006. Appeared in Oberwolfach Reports.
- A construction of gradings of Lie algebras (with Antonio Fernández López,
Esther García and Miguel Gómez Lozano, preprint version, Dec. 2006); published in
Int. Math. Res. Not. IMRN, 2007, no. 16, Art. ID rnm051, 34 pages; the
published version has some strange formatting.

*Abstract* - 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.
- Gelfand-Kirillov dimension and local finiteness of Jordan superpairs covered by grids and their associated Lie superalgebras
(with Esther García, Feb. 2002, 25 pages).
Published in Communications in Algebra, 32 (2004), 2149--2176.

- Semiprime, prime and simple Jordan superpairs covered by grids,
(with Esther García), published in the Journal of Algebra 273 (2004), 1--32.

- Tits-Kantor-Koecher Superalgebras of Jordan superpairs covered by grids
(with Esther García, 2001). Published in Communications in Algebra, 31 (2003), no. 7, 3335--3375.

- Quadratic Jordan superpairs covered by grids
(Nov. 2002, 43 pages) Published in the Journal of Algebra, 269 (2003), 28-73.

- 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.

- Derivations and invariant forms of Jordan and alternative tori (with Yoji Yoshii). Published in Trans. Amer. Math. Soc. 33 (2003), 1079--1108.

- Transformation groups of the Andersson-Perlman cone. Published in the Journal of Lie Theory, 9 (1999), no. 1, 203--213.

- Polynomial Identities and non-identities of split Jordan pairs. Published in the Journal of Algebra, 211 (1999), no. 1, 206--224.