We construct the space of vector fields on a generic quantum group. Its elements are products of elements of the quantum group itself with left invariant vector fields. We study the duality between vector fields and 1-forms and generalize the construction to tensor fields. A Lie derivative along any (also non left invariant) vector field is proposed and a puzzling ambiguity in its definition di...

We present a systematic procedure for the determination of a complete set of kth-order (k ≥ 2) differential invariants corresponding to vector fields in three variables for three-dimensional Lie algebras. In addition, we give a procedure for the construction of a system of two kth-order ODEs admitting three-dimensional Lie algebras from the associated complete set of invariants and show that th...

We completely describe the decompositions (into indecomposable submodules) of the tensor products of irreducible sl(2)-modules in characteristic 3. The answer resembles analogous decompositions for the Lie superalgebra sl(1|1).

that is, the problem of finding a function u on M whose derivative along the flow is equal to a given function f on M . Roughly speaking, the Ccohomology of the flow is the space of non-trivial obstructions to the existence of a C solution u of (1.1) for any given function f ∈ C(M). This notion is well-defined if the range of the Lie derivative operator on the space C(M) is closed. In this case...

In this note we give a definition of semiinfinite cohomology for Tate Lie algebras using the language of differential graded Lie algebroids with curvature (CDG Lie algebroids). 2000 Math. Subj. Class. 17-XX.

Bracket formulations of two kinds of dynamical systems, called incomplete and complete, are reviewed and developed, including double bracket and metriplectic dynamics. Dissipation based on the Cartan-Killing metric is introduced. Various examples of incomplete and complete dynamics are discussed, including dynamics associated with three-dimensional Lie algebras.

The space D p of differential operators of order ≤ k, from the differential forms of degree p of a smooth manifold M into the functions of M, is a module over the Lie algebra of vector fields of M, when it’s equipped with the natural Lie derivative. In this paper, we compute all equivariant i.e. intertwining operators T : D p → D l q and conclude that the preceding modules of differential opera...

The article is devoted to the problem of classification of Manin triples up to weak and gauge equivalence. The case of complex simple Lie algebras can be obtained by papers of A.Belavin, V.Drinfel'd, M.Semenov-Tian-Shanskii. Studing the action of conjugaton on complex Manin triples, we get the list of real doubles. There exists three types of the doubles. We classify all ad-invariant forms on t...

The trigonometric KZ equations associated with a Lie algebra g depend on a parameter λ ∈ h where h ⊂ g is the Cartan subalgebra. We suggest a system of dynamical difference equations with respect to λ compatible with the KZ equations. The dynamical equations are constructed in terms of intertwining operators of g -modules.

Let 9 be a reductive complex Lie algebra, with adjoint group G, Cartan subalgebra ~ and Weyl group W. Then G acts naturally on the algebra of polynomial functions &'(g) and hence on the ring of differential operators with polynomial coefficients, .97(g). Similarly, W acts on ~ and hence on .97(~). In [BC2], Harish-Chandra defined an algebra homomorphism J : .97(g)G -t .97(~)w. Recently, Wallach...