W. Plesken found a simple but interesting construction of a Lie algebra from a finite group. Cohen and Taylor posed themselves the question of what the Plesken Lie algebra, which is the Lie subalgebra of the group algebra k[G] generated by the elements g − g−1, could be. The result is very fascinating: It turns out that the Lie algebra decomposition of the Plesken Lie algebra into simple Lie al...

A new geometric proof is given for the standard description of subgroups in the Chevalley group G = G(F4, R) of type F4 over a commutative ring R that are normalized by the elementary subgroup E(F4, R). There are two major approaches to the proof of such results. Localization proofs (Quillen, Suslin, Bak) are based on a reduction in the dimension. The first proofs of this type for exceptional g...

N. L. Gordeev proved that a generalized group identity holds in Chevalley groups with multiply laced root systems. It was also shown that a stronger identity is valid for the Chevalley groups of types Bl and Cl. In the present paper, it is proved that this strong identity is fulfilled in Chevalley groups of type F4 and fails to be true in Chevalley groups of type G2. The main result of the pape...

The paper is devoted to a detailed study of some remarkable semisimple elements of (extended) Chevalley groups that are diagonalizable over the ground field — the weight elements. These are the conjugates of certain semisimple elements hω(ε) of extended Chevalley groupsG = G(Φ, K), where ω is a weight of the dual root system Φ∨ and ε ∈ K∗. In the adjoint case the hω(ε)’s were defined by Chevall...

This work is concerned with multi-party stabilizer states in the sense of quantum information theory. We investigate the homological invariants for states of which each party holds a large equal number N of quantum bits. We show that in many cases there is a generic expected value of the invariants : for large N it is approximated with arbitrarily high probability if a stabilizer state is chose...

Let A be a symmetrizable generalized Cartan matrix. Let g be the corresponding Kac-Moody algebra. Let G(R) be Tits’ Kac-Moody group functor over commutative rings R associated to g. Then G(R) is given by five axioms KMG1-KMG5 which are natural extensions of the properties of Chevalley-Demazure group schemes. We give a construction of the Tits functor for symmetrizable Kac-Moody groups G using i...