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TRIPLES, ALGEBRAS AND COHOMOLOGY
 REPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES
, 2003
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H.P.Petersson: Groups of outer type E6 with trivial Tits algebras, Transformation Groups
"... Abstract. In two 1966 papers, J. Tits gave a construction of exceptional Lie algebras (hence implicitly exceptional algebraic groups) and a classification of possible indexes of simple algebraic groups. For the special case of his construction that gives groups of type E6, we connect the two papers ..."
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Abstract. In two 1966 papers, J. Tits gave a construction of exceptional Lie algebras (hence implicitly exceptional algebraic groups) and a classification of possible indexes of simple algebraic groups. For the special case of his construction that gives groups of type E6, we connect the two papers by answering the question: Given an Albert algebra A and a separable quadratic field extension K, what is the index of the resulting algebraic group? This article links two 1966 papers by Jacques Tits—namely, [Ti 66a] and [Ti 66b]—concerning simple linear algebraic groups over a field k. In the first paper, he gave a construction that takes a composition algebra K and a degree 3 Jordan algebra A and produces a simple algebraic group G(A, K). The KillingCartan type of the resulting group is given by the famous magic square: dim A
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, 2005
"... Abstract. In two 1966 papers, J. Tits gave a construction of exceptional Lie algebras (hence implicitly exceptional algebraic groups) and a classification of possible indexes of simple algebraic groups. For the special case of his construction that gives groups of type E6, we connect the two papers ..."
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Abstract. In two 1966 papers, J. Tits gave a construction of exceptional Lie algebras (hence implicitly exceptional algebraic groups) and a classification of possible indexes of simple algebraic groups. For the special case of his construction that gives groups of type E6, we connect the two papers by answering the question: Given an Albert algebra A and a separable quadratic field extension K, what is the index of the resulting algebraic group?
The Burnside Problem (also known as the Ordinary Burnside Problem) : Is it true
"... that every finitely generated group of bounded exponent is finite? The General Burnside Problem: Is it true that every finitely generated periodic group is finite? After many unsuccessful attempts to obtain a proof in the late 30searly 40s the following weaker version of The Burnside Problem was st ..."
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that every finitely generated group of bounded exponent is finite? The General Burnside Problem: Is it true that every finitely generated periodic group is finite? After many unsuccessful attempts to obtain a proof in the late 30searly 40s the following weaker version of The Burnside Problem was studied: Is it true that there are only finitely many 7??generated finite groups of exponent nl In other words the question is whether there exists a universal finite mgenerated group of exponent n having all other finite mgenerated groups of exponent n as homomorphic images. Later (thanks to W. Magnus [35]) this question became known as The Restricted Burnside Problem. In 1964 E. S. Golod gave a negative answer to The General Burnside Problem (cf. [9]). Since then a considerable array of infinitely generated periodic groups was constructed by other authors (cf. Alyoshin [2], Suschansky [44], Grigorchuk [ll],GuptaSidki [54]). In 1968 P. S. Novikov and S. I. Adian [39] constructed counterexamples
EXCEPTIONAL LIE ALGEBRAS AND RELATED ALGEBRAIC AND GEOMETRIC STRUCTURES
"... Certain algebraic structures, most notably associative, alternative, and Jordan algebras are strongly linked via construction and classification to simple Lie algebras and to interesting geometries. These geometries are in turn linked to simple Lie algebras via their groups of collineations. These l ..."
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Certain algebraic structures, most notably associative, alternative, and Jordan algebras are strongly linked via construction and classification to simple Lie algebras and to interesting geometries. These geometries are in turn linked to simple Lie algebras via their groups of collineations. These linkages serve to illustrate how
SOME SPECIAL FEATURES OF SPECIAL MOUFANG SETS
"... Abstract. We prove Timmesfeld’s conjecture that special abstract rank one groups are quasisimple. We show that in a special Moufang set the root groups are characterized on the one hand by being regular and normal in the point stabilizer, and on the other hand a normal transitive nilpotent subgroup ..."
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Abstract. We prove Timmesfeld’s conjecture that special abstract rank one groups are quasisimple. We show that in a special Moufang set the root groups are characterized on the one hand by being regular and normal in the point stabilizer, and on the other hand a normal transitive nilpotent subgroup of the point stabilizer is a root group. We prove that if a root group of a special Moufang set contains an involution, then it is of exponent 2. Furthermore we show that the root groups are abelian if and only if the socalled µmaps are involutions.
A course on Moufang sets
"... A Moufang set is essentially a doubly transitive permutation group such that the point stabilizer contains a normal subgroup which is regular on the remaining points. These regular normal subgroups are called the root groups and they are assumed to be conjugate and to generate the whole group. Moufa ..."
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A Moufang set is essentially a doubly transitive permutation group such that the point stabilizer contains a normal subgroup which is regular on the remaining points. These regular normal subgroups are called the root groups and they are assumed to be conjugate and to generate the whole group. Moufang sets play an significant role in the theory of buildings, they provide a tool to study linear algebraic groups of relative rank one, and they have (surprising) connections with other algebraic structures. In these course notes we try to present the current approach to Moufang sets. We include examples, connections with related areas of mathematics and some proofs where we think it is instructive and within the scope of
SPECIAL MOUFANG SETS, THEIR ROOT GROUPS AND THEIR µMAPS
"... Abstract. We prove Timmesfeld’s conjecture that special abstract rank one groups are quasisimple. We show that in a special Moufang set the root groups are characterized on the one hand by being regular and normal in the point stabilizer, and on the other hand a normal transitive nilpotent subgroup ..."
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Abstract. We prove Timmesfeld’s conjecture that special abstract rank one groups are quasisimple. We show that in a special Moufang set the root groups are characterized on the one hand by being regular and normal in the point stabilizer, and on the other hand a normal transitive nilpotent subgroup of the point stabilizer is a root group. We prove that if a root group of a special Moufang set contains an involution, then it is of exponent 2. We also show that the root groups are abelian if and only if the socalled µmaps are involutions.
IDENTITIES IN MOUFANG SETS
"... Abstract. Moufang sets were introduced by Jacques Tits as an axiomatization of the buildings of rank one that arise from simple algebraic groups of relative rank one. These fascinating objects have a simple definition and yet their structure is rich, while it is rigid enough to allow for (at least ..."
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Abstract. Moufang sets were introduced by Jacques Tits as an axiomatization of the buildings of rank one that arise from simple algebraic groups of relative rank one. These fascinating objects have a simple definition and yet their structure is rich, while it is rigid enough to allow for (at least partial) classification. In this paper we obtain two identities that hold in any Moufang set. These identities are closely related to the axioms that define a quadratic Jordan algebra. We apply them in the case when the Moufang set is socalled special and has abelian root groups. In addition we push forward the theory of special Moufang sets. 1.