The goal of this paper is to study cohomological theory n-Lie algebras with derivations. We define the representation an n-LieDer pair and consider its cohomology. Likewise, we verify that a cohomology could be derived from associated LeibDer pair. Furthermore, discuss (n−1)-order deformations Nijenhuis operator pairs. central extensions pairs are also investigated in terms first groups coeffic...

a unital $c^*$ -- algebra $mathcal a,$ endowed withthe lie product $[x,y]=xy- yx$ on $mathcal a,$ is called a lie$c^*$ -- algebra. let $mathcal a$ be a lie $c^*$ -- algebra and$g,h:mathcal a to mathcal a$ be $bbb c$ -- linear mappings. a$bbb c$ -- linear mapping $f:mathcal a to mathcal a$ is calleda lie $(g,h)$ -- double derivation if$f([a,b])=[f(a),b]+[a,f(b)]+[g(a),h(b)]+[h(a),g(b)]$ for all ...

For a perfect Lie algebra h we classify all Lie algebras containing h as a subalgebra of codimension 1. The automorphism groups of such Lie algebras are fully determined as subgroups of the semidirect product h n (k∗ × AutLie(h)). In the non-perfect case the classification of these Lie algebras is a difficult task. Let l(2n + 1, k) be the Lie algebra with the bracket [Ei, G] = Ei, [G,Fi] = Fi, ...

Quantum duality principle is applied to study classical limits of quantum algebras and groups. For a certain type of Hopf algebras the explicit procedure to construct both classical limits is presented. The canonical forms of quantized Lie-bialgebras are proved to be two-parametric varieties with two classical limits called dual. When considered from the point of view of quantized symmetries su...

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 develop a categorical approach to the dynamical Yang-Baxter equation (DYBE) for arbitrary Hopf algebras. In particular, we introduce a notion of dynamical extension of a monoidal category, which provides a natural environment for quantum dynamical R-matrices. In this context, we define dynamical associative algebras and show that such algebras give a quantization of vector bundles on coadjoi...

The affine Toda field theories based on the non simply-laced Lie algebras are discussed. By rewriting the S-matrix formulae found by Delius et al, a universal form for the coupling-constant dependence of these models is obtained, and related to various general properties of the classical couplings. This is illustrated via the S-matrix associated with the dual pair of algebras f (1) 4 and e (2) 6 .

We study Lagrangian and orthogonal splittings\textbf{\ }of quadratic vector spaces establishing an equivalence with complex product structures. Then we show that a Manin triple equipped generalized metric $\mathcal{G}+% \mathcal{B}$ such $\mathcal{B}$ is $\mathcal{O}$-operator extension $\mathcal{G}$ of mass -1 can be turned in another admits also splitting in\textbf{\ }Lie ideals. Conversely, ...

Braided m-Lie algebras induced by multiplication are introduced, which generalize Lie algebras, Lie color algebras and quantum Lie algebras. The necessary and sufficient conditions for the braided m-Lie algebras to be strict Jacobi braided Lie algebras are given. Two classes of braided m-Lie algebras are given, which are generalized matrix braided m-Lie algebras and braided m-Lie subalgebras of...