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In a recent article with Oleg Smirnov, we defined short Peirce (SP) graded Kantor pairs. For any such pair P, we defined a family, parameterized by the Weyl group of type BC_2, consisting of SP-graded Kantor pairs called Weyl images of P. In this article, we classify finite dimensional simple SP-graded Kantor pairs over an algebraically closed field of characteristic 0 in terms of marked Dynkin diagrams, and we show how to compute Weyl images using these diagrams. The theory is particularly attractive for close-to-Jordan Kantor pairs (which are variations of Freudenthal triple systems), and we construct the reflections of such pairs (with nontrivial gradings) starting from Jordan matrix pairs., Comment: 39 pages

Kantor pairs arise naturally in the study of 5-graded Lie algebras. In this article, we begin the study of simple Kantor pairs of arbitrary dimension. We introduce Weyl images of Kantor pairs and use them to construct examples of Kantor pairs including a new class of central simple Kantor pairs., Comment: 43 pages. In the last section of this new version, the assumption that the ring of scalars is a field is dropped. This allows the construction of many forms of split Kantor pairs of type $E_6$ over rings. To facilitate this change, some small revisions are made in earlier sections. To be published in the Canadian Journal of Mathematics

Some forms of Lie algebras of types E_6, E_7, and E_8 are constructed using the exterior cube of a rank 9 finitely generated projective module.

Centreless Lie tori have been used by E. Neher to construct all extended affine Lie algebras (EALAs). In this article, we study notions of isotopes and isotopy for centreless Lie tori, and we use these notions, along with Neher's construction, to show that there is a 1-1 correspondence between centreless Lie tori up to isotopy and families of EALAs up to isomorphism. Also, centreless Lie tori can be coordinatized by unital algebras that are in general nonassociative, and, for many types of centreless Lie tori, there are classical notions of isotopes and isotopy for the coordinate algebras. We show for those types that an isotope of the Lie torus is coordinatized by an isotope of its coordinate algebra. In writing the article, we have not assumed prior knowledge of the theories of EALAs, Lie tori or isotopy. In fact, we hope that this article will help to introduce the reader to these theories and their interconnections., Comment: Minor corrections

An important theorem in the theory of infinite dimensional Lie algebras states that any affine Kac-Moody algebra can be realized (that is to say constructed explicitly) using loop algebras. In this paper, we consider the corresponding problem for a class of Lie algebras called extended affine Lie algebras (EALAs) that generalize affine algebras. EALAs occur in families that are constructed from centreless Lie tori, so the realization problem for EALAs reduces to the realization problem for centreless Lie tori. We show that all but one family of centreless Lie tori can be realized using multiloop algebras (in place of loop algebras). We also obtain necessary and sufficient for two centreless Lie tori realized in this way to be isotopic, a relation that corresponds to isomorphism of the corresponding families of EALAs., Comment: Marginal notes (for tex labels) deleted

The classification of structurable tori with nontrivial involution, which was begun by Allison and Yoshii, is completed. New examples of structurable tori are obtained using a construction of structurable algebras from a semilinear version of cubic forms satisfying the adjoint identity. The classification uses techniques borrowed from quadratic forms over Z_2 and from the geometry of generalized quadrangles. Since structurable tori are the coordinate algebras for the centreless cores of extended affine Lie algebras of type BC_1, the results of this paper provide a classification and new examples for this class of Lie algebras.

Multiloop algebras determined by $n$ commuting algebra automorphisms of finite order are natural generalizations of the classical loop algebras that are used to realize affine Kac-Moody Lie algebras. In this paper, we obtain necessary and sufficient conditions for a $Z^n$-graded algebra to be realized as a multiloop algebra based on a finite dimensional simple algebra over an algebraically closed field of characteristic 0. We also obtain necessary and sufficient conditions for two such multiloop algebras to be graded-isomorphic, up to automorphism of the grading group. We prove these facts as consequences of corresponding results for a generalization of the multiloop construction. This more general setting allows us to work naturally and conveniently with arbitrary grading groups and arbitrary base fields., Comment: 31 pages. Corrected typos and added minor clarifications. Accepted in Forum Mathematicum

This booklet is designed to make research findings about swimming available with interpretations for practical application. Chapter 1, "Physical Characteristics of Swimmers," discusses somatotyping, body composition, and growth. Chapter 2, "Physiological Characteristics of Swimmers," discusses resting rate, vital capacity, effects of water immersion, pulmonary ventilation, aerobic capacity, temperature regulation, warmup and swimming performance, underwater swimming, and muscle strength and endurance. Chapter 3, "Water Resistance and Energy Expenditure," concludes that comprehensive data are needed on the caloric and heart rate cost of different interval training procedures and of distance swimming to provide more accurate estimates of what swimmers can and are doing physiologically. Chapter 4, "Sociological and Psychological Aspects of Swimming," discusses psychological characteristics, social factors, psychology of coaching, motivation, and level of anticipation. Chapter 5, "Evaluation of Performance," presents information regarding Craig's velocity-duration curve, pulse rate after repeats or races, pulse rate and velocity, propulsive force, energy expenditure and efficiency, and caloric cost of swimming practices. A bibliography is included at the end of each chapter. (PD)