The aim of this paper is to give a complete classification of irreducible finite-dimensional representations of the nonstandard q-deformation U ′ q(son) (which does not coincide with the Drinfel’d-Jimbo quantum algebra Uq(son)) of the universal enveloping algebra U(son(C)) of the Lie algebra son(C) when q is not a root of unity. These representations are exhausted by irreducible representations...

Multisymplectic geometry is a generalization of symplectic geometry suitable for n-dimensional field theories, in which the nondegenerate 2-form of symplectic geometry is replaced by a nondegenerate (n + 1)-form. The case n = 2 is relevant to string theory: we call this ‘2-plectic geometry.’ Just as the Poisson bracket makes the smooth functions on a symplectic manifold into a Lie algebra, the ...

A Lie bialgebra is a vector space endowed simultaneously with the structure of algebra and coalgebra, some compatibility condition. Moreover, brackets have skew symmetry. Because close relation between bialgebras quantum groups, it interesting to consider structures on L related Virasoro algebra. In this paper, are investigated by computing Der(L, L⊗L). It proved that all such triangular coboun...

To a hyperbolic smooth curve defined over a number-field one naturally associates an "anabelian" representation of the absolute Galois group of the base field landing in outer automorphism group of the algebraic fundamental group. In this paper, we introduce several deformation problems for Lie-algebra versions of the above representation and show that, this way we get a richer structure than t...

We study the problem of matrix Lie algebra conjugacy. Lie algebras arise centrally in areas as diverse as differential equations, particle physics, group theory, and the Mulmuley–Sohoni Geometric Complexity Theory program. A matrix Lie algebra is a set L of matrices such that M1,M2 ∈ L =⇒ M1M2 − M2M1 ∈ L. Two matrix Lie algebras are conjugate if there is an invertible matrix M such that L1 = ML...

By Grothedieck's Anabelian conjectures, Galois representations landing in outer automorphism group of the algebraic fundamental group which are associated to hyperbolic smooth curves defined over number fields encode all arithmetic information of these curves. The goal of this paper is to develope and arithmetic teichmuller theory, by which we mean, introducing arithmetic objects summarizing th...

In the present paper, we prove that if L is a nilpotent Lie algebra whose proper subalge- bras are all nilpotent of class at most n, then the class of L is at most bnd=(d 1)c, where b c denotes the integral part and d is the minimal number of generators of L.

This paper encloses a complete and explicit description of the derivations of the Lie algebra D(M) of all linear differential operators of a smooth manifold M , of its Lie subalgebra D 1 (M) of all linear first-order differential operators of M , and of the Poisson algebra S(M) = P ol(T * M) of all polynomial functions on T * M, the symbols of the operators in D(M). It turns out that, in terms ...

This is the third in a series of papers. The first two, by Yiftach Barnea and this author, study the maximal bounded Z-filtrations of the finitedimensional simple Lie algebras over the complex numbers. Those papers obtain a complete characterization for all but the five exceptional Lie algebras, namely the ones of type G2, F4, E6, E7 and E8. Here, we fill in the missing step for the algebra G2....

This paper encloses a complete and explicit description of the derivations of the Lie algebra D(M) of all linear differential operators of a smooth manifold M , of its Lie subalgebra D 1 (M) of all linear first-order differential operators of M , and of the Poisson algebra S(M) = Pol(T * M) of all polynomial functions on T * M, the symbols of the operators in D(M). It turns out that, in terms o...