Brauer algebras of type B

For each n>0, we define an algebra having many properties that one might expect to hold for a Brauer algebra of type Bn. It is defined by means of a presentation by generators and relations. We show that this algebra is a subalgebra of the Brauer algebra of type Dn+1 and point out a cellular structure in it. This work is a natural sequel to the introduction of Brauer algebras of type Cn, which are subalgebras of classical Brauer algebras of type A2n-1 and differ from the current ones for n>2.Comment: 5 figure

Extremal elements in Lie Algebras

The main result discussed in this lecture is an elementary proof of the following theorem: If L is a simple Lie algebra over F of characteristic distinct from 2 and 3 having an extremal element that is not a sandwich, then either F has characteristic 5 and L is isomorphic to the 5-dimensional Witt algebra W_1,1(5), or L is generated by extremal elements. We will also pay attention to the following theorem: If L is a simple Lie algebra generated by extremal elements that are not sandwiches, then it is classical, i.e., essentially a Lie algebra of Chevalley type. This result, of which various geometric proofs are emerging (mainly thanks to Cuypers, Fleischmann, Roberts, and Shpectorov), gives a new proof of the classi cation of classical simple Lie algebras of characteristic distinct from 2 and 3. This is joint work with G abor Ivanyos and Dan Roozemond. For the full paper, see [7]Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Covers of Point-Hyperplane Graphs

We construct a cover of the non-incident point-hyperplane graph of projective dimension 3 for fields of characteristic 2. If the cardinality of the field is larger than 2, we obtain an elementary construction of the non-split extension of SL_4 (F) by F^6.Comment: 10 pages, 3 figure

On a Theorem of Cooperstein

A theorem by Cooperstein that partially characterizes the natural geometry An,d(F) of subspaces of rank d − 1 in a projective space of finite rank n over a finite field F, is somewhat strengthened and generalized to the case of an arbitrary division ring F.Moreover, this theorem is used to provide characterizations of An,2(F) and A5,3(F) which will be of use in the characterization of other (exceptional) Lie group geometries

Computing in unipotent and reductive algebraic groups

The unipotent groups are an important class of algebraic groups. We show that techniques used to compute with finitely generated nilpotent groups carry over to unipotent groups. We concentrate particularly on the maximal unipotent subgroup of a split reductive group and show how this improves computation in the reductive group itself.Comment: 22 page

Lie algebras generated by extremal elements

We study Lie algebras generated by extremal elements (i.e., elements spanning inner ideals of L) over a field of characteristic distinct from 2. We prove that any Lie algebra generated by a finite number of extremal elements is finite dimensional. The minimal number of extremal generators for the Lie algebras of type An, Bn (n>2), Cn (n>1), Dn (n>3), En (n=6,7,8), F4 and G2 are shown to be n+1, n+1, 2n, n, 5, 5, and 4 in the respective cases. These results are related to group theoretic ones for the corresponding Chevalley groups.Comment: 28 page

Tangle and Brauer Diagram Algebras of Type Dn

A generalization of the Kauffman tangle algebra is given for Coxeter type Dn. The tangles involve a pole or order 2. The algebra is shown to be isomorphic to the Birman-Murakami-Wenzl algebra of the same type. This result extends the isomorphism between the two algebras in the classical case, which in our set-up, occurs when the Coxeter type is of type A with index n-1. The proof involves a diagrammatic version of the Brauer algebra of type Dn in which the Temperley-Lieb algebra of type Dn is a subalgebra.Comment: 33 page