Low cohomogeneity and polar actions on exceptional compact Lie groups

We study isometric Lie group actions on the compact exceptional groups E₆, E₇, E₈, F₄ and G₂ endowed with a bi-invariant metric. We classify polar actions on these groups, in particular, we show that all polar actions are hyperpolar. We determine all isometric actions of cohomogeneity less than three on E₆, E₇, F₄ and all isometric actions of cohomogeneity less than 20 on E₈. Moreover, we determine the principal isotropy algebras for all isometric actions on G₂.     

Derived Equivalence Classification of the Cluster-Tilted Algebras of Dynkin Type E

We obtain a complete derived equivalence classification of the cluster-tilted algebras of Dynkin type E. There are 67, 416, 1574 algebras in types E ₆, E ₇ and E ₈ which turn out to fall into 6, 14, 15 derived equivalence classes, respectively. This classification can be achieved computationally and we outline an algorithm which has been implemented to carry out this task. We also make the classification explicit by giving standard forms for each derived equivalence class as well as complete lists of the algebras contained in each class; as these lists are quite long they are provided as supplementary material to this paper. From a structural point of view the remarkable outcome of our classification is that two cluster-tilted algebras of Dynkin type E are derived equivalent if and only if their Cartan matrices represent equivalent bilinear forms over the integers which in turn happens if and only if the two algebras are connected by a sequence of “good” mutations. This is reminiscent of the derived equivalence classification of cluster-tilted algebras of Dynkin type A, but quite different from the situation in Dynkin type D where a far-reaching classification has been obtained using similar methods as in the present paper but some very subtle questions are still open.     

The J-invariant and Tits algebras for groups of inner type E 6

A connection between the indices of the Tits algebras of a split linear algebraic group G and the degree one parameters of its motivic J-invariant was introduced by Quéguiner-Mathieu, Semenov and Zainoulline through use of the second Chern class map in the Riemann-Roch theorem without denominators. In this paper we extend their result to higher Chern class maps and provide applications to groups of inner type E ₆.     

Bedingungen für die Zweispiegeligkeit der Automorphismengruppen von Cayleyalgebren

In this paper we derive necessary and sufficient conditions for the bireflectionality of the automorphism group of a Cayley algebra over a field of characteristic not 2. These are of particular interest for split Cayley algebras since their automorphism groups are the Chevalley groups of type G ₂. As an application we show the bireflectionality of the automorphism groups of Cayley algebras over real closed fields.     

Some Forms of Exceptional Lie Algebras

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.     

Quasitriangular Pointed Hopf Algebras Constructed by Ore Extensions

We study the quasitriangular structures for a family of pointed Hopf algebras which is big enough to include Taft's Hopf algebras H _n ², Radford's Hopf algebras H _N,n,q, and E(n). We give necessary and sufficient conditions for the Hopf algebras in our family to be quasitriangular. For the case when they are, we determine completely all the quasitriangular structures. Also, we determine the ribbon elements of the quasitriangular Hopf algebras and the quasi-ribbon elements of their Drinfel'd double.     

PI non-equivalence in positive characteristic

The algebras A _a,b appeared in the study of the tensor products of verbally prime PI algebras. They are in-between the well known algebras M _n (E) and ${M_{a,b}(E)\otimes E}$ , see the definitions below. Here E is the Grassmann algebra. The main result of this note consists in showing that the algebras A _a,b and M _a+b (E) are not PI equivalent in characteristic p > 2.     

Grassmann algebras over finite fields

We study an analogue of a problem of procesi about matrices [9, page 185(e)]: are there non-trivial polynomials over Z which become identities over Z _p - for grassmann algebras E? when 1 ? E, we show that such polynomials do not exist, but when 1 ?,E such polynomials exist - also for matrices over E. these results are deduced from a careful study of the various codimensions of these algebras.     

Cochains and homotopy type

Finite type nilpotent spaces are weakly equivalent if and only if their singular cochains are quasi-isomorphic as E_∞ algebras. The cochain functor from the homotopy category of finite type nilpotent spaces to the homotopy category of E_∞ algebras is faithful but not full.     

Commutator Leavitt Path Algebras

For any field K and directed graph E, we completely describe the elements of the Leavitt path algebra L _K (E) which lie in the commutator subspace [L _K (E), L _K (E)]. We then use this result to classify all Leavitt path algebras L _K (E) that satisfy L _K (E)?=?[L _K (E),L _K (E)]. We also show that these Leavitt path algebras have the additional (unusual) property that all their Lie ideals are (ring-theoretic) ideals, and construct examples of such rings with various ideal structures.     

Super-Clifford Gravity,Higher Spins,Generalized Supergeometry and Much More

An extended Orthogonal-Symplectic Clifford Algebraic formalism is developed which allows the novel construction of a graded Clifford gauge field theory of gravity. It has a direct relationship to higher spin gauge fields, bimetric gravity, antisymmetric metrics and biconnections. In one particular case it allows a plausible mechanism to cancel the cosmological constant contribution to the action. The possibility of embedding these Orthogonal-Symplectic Clifford algebras into an infinite dimensional algebra, coined Super-Clifford Algebra is described. Finally, some physical applications of the geometry of Super-Clifford spaces to Generalized Supergeometries, Double Field Theories, U-duality, 11D supergravity, M-theory, and E ₇, E ₈, E ₁₁ algebras are outlined.     

PI (non)equivalence and Gelfand-Kirillov dimension in positive characteristic

The verbally prime algebras are well understood in characteristic 0 while over a field of positive characteristic p > 2 little is known about them. In previous papers we discussed some sharp differences between these two cases for the characteristic; we showed that the so-called Tensor Product Theorem cannot be extended for infinite fields of positive characteristic p > 2. Furthermore we studied the Gelfand-Kirillov dimension of the relatively free algebras of verbally prime and related algebras. In this paper we compute the GK dimensions of several algebras and thus obtain a new proof of the fact that the algebras M _a,a (E) ⊗ E and M _2a (E) are not PI equivalent in characteristic p > 2. Furthermore we show that the following algebras are not PI equivalent in positive characteristic: M _a,b (E) ⊗ M _c,d (E) and M _ac+bd,ad+cb (E); and M _a,b (E) ⊗ M _c,d (E) and M _{e, f} (E) ⊗ M _g,h (E) when a ≥ b, c ≥ d, e ≥ f, g ≥ h, ac + bd = eg+ f h, ad +bc = eh + fg and ac ≠ eg. Here E stands for the infinite dimensional Grassmann algebra with 1, and M _a,b (E) is the subalgebra of M _a+b (E) of the block matrices with blocks a × a and b × b on the main diagonal with entries from E ₀, and off-diagonal entries from E ₁; E = E ₀ ⊗ E ₁ is the natural grading on E. Partially supported by CNPq 620025/2006-9. This paper was written during the author’s PhD study at the UNICAMP, under the supervision of P.Koshlukov, to whom he expresses his sincere thanks.     

The graded Gelfand--Kirillov dimension of verbally prime algebras

Let E be the infinite-dimensional Grassmann algebra over a field F of characteristic 0. In this article, we consider the verbally prime algebras M _n (F), M _n (E) and M _a,b (E) endowed with their gradings induced by that of Vasilovsky, and we compute their graded Gelfand--Kirillov dimensions.     

Seminormal Representations of Weyl Groups and Iwahori-Hecke Algebras

The purpose of this paper is to describe a general procedurefor computing analogues of Young's seminormal representationsof the symmetric groups. The method is to generalize the Jucys-Murphyelements in the group algebras of the symmetric groups to arbitraryWeyl groups and Iwahori-Hecke algebras. The combinatorics ofthese elements allows one to compute irreducible representationsexplicitly and often very easily. In this paper we do thesecomputations for Weyl groups and Iwahori-Hecke algebras of typesA_n, B_n, D_n, G₂. Although these computations are in reach fortypes F₄, E₆ and E₇, we shall postpone this to another work.1991 Mathematics Subject Classification: primary 20F55, 20C15;secondary 20C30, 20G05.     

Projective planes over 2-dimensional quadratic algebras

The split version of the Freudenthal–Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras ,  and . The geometries appearing in the second row are Severi varieties [24]. We provide an easy uniform axiomatization of these geometries and related ones, over an arbitrary field. In particular we investigate the entry _A2×_A2$A_2\times A_2$ in the magic square, characterizing Hermitian Veronese varieties, Segre varieties and embeddings of Hjelmslev planes of level 2 over the dual numbers. In fact this amounts to a common characterization of “projective planes over 2-dimensional quadratic algebras”, in cases of the split and non-split Galois extensions, the inseparable extensions of degree 2 in characteristic 2 and the dual numbers.     

Benz planes of Dembowski type and groups of tangential projectivities

Using the tangential relation we introduce in Benz planes M of Dembowski type, which generalize the Benz planes over algebras of characteristic 2, the group ?? of tangential perspectivities. We prove that these groups have the same behaviour as the classical groups of projectivities if any tangential perspectivity is induced by an automorphism of M. As permutation groups of a circle onto itself the groups ?? essentially differs from the classical groups of projectivities. If M is a Laguerre plane of Dembowski type, then ?? is always sharply 3-transitive. For Minkowski planes of Dembowski type ?? is at least 2-transitive. If M is a finite Benz plane of order 2 ^s , then ?? is isomorphic to the group PGL ₂(2 ^s ) in its sharply 3-transitive representation.     

Weighted Algebras of Vector‐Valued Continuous Functions

In this paper we present a systematic study of the algebras CV (X, E) and CV₀(X, E) of continuous functions with weight conditions given by a Nachbin family V and with values in a locally convex algebra E. Some relevant examples (and counterexamples to some assertions in the literature) are provided.     

Polynomial Lie Algebras

We introduce and study a special class of infinite-dimensional Lie algebras that are finite-dimensional modules over a ring of polynomials. The Lie algebras of this class are said to be polynomial. Some classification results are obtained. An associative co-algebra structure on the rings k[x ₁,...,x _n]/(f ₁,...,f _n) is introduced and, on its basis, an explicit expression for convolution matrices of invariants for isolated singularities of functions is found. The structure polynomials of moving frames defined by convolution matrices are constructed for simple singularities of the types A,B,C, D, and E ₆.     

Zur konstruktion von Lie-Algebren aus Jordan-Tripelsystemen

In this paper we study certain Lie algebras which are constructed from the (-1)-eigenspaees of an involution of a Jordan algebra. The construction is a generalisation of the Koecher-Tits-construction. We give necessary conditions in terms of the Jordan algebras for the Lie algebras being simple. If the (-1)-spaces are Peirce-1/2-components then we obtain a close relation between the Lie algebras under consideration and the structure algebras of Jordan algebras. We finally give a list of those types of simple Lie algebras which can be formed by this construction; among them are Lie algebras of type E₆ and E₇.Of fundamental importance for our considerations is a close connection between the constructed Lie algebras and the standard imbeddings of Lie triple systems.