We propose a notion of a quantum universal enveloping algebra for an arbitrary Lie algebra defined by generators and relations which is based on the quantum Lie operation concept. This enveloping algebra has a PBW basis that admits the Kashiwara crystalization. We describe all skew primitive elements of the quantum universal enveloping algebra for the classical nilpotent algebras of the infinit...

The orbit method conjectures a close relationship between the set of irreducible unitary representations of a Lie group G, and admissible coadjoint orbits in the dual of the Lie algebra. We define admissibility for nilpotent coadjoint orbits of p-adic reductive Lie groups, and compute the set of admissible orbits for a range of examples. We find that for unitary, symplectic, orthogonal, general...

To each complex semisimple Lie algebra $$ \mathfrak{g} and regular element a ∈ reg, one associates Mishchenko–Fomenko subalgebra \mathcal{F} ⊆ ℂ[ ]. This amounts to completely integrable system on the Poisson variety , as such has bifurcation diagram Σa Spec( a). We prove that codimension in a) if reg is not nilpotent, it or two nilpotent. In nilpotent case, we show of possible codimensions be ...

If a Lie algebra g can be generated by M of its elements E1, . . . , EM , and if any other Lie algebra generated by M other elements F1, . . . , FM is a homomorphic image of g under the map Ei → Fi, we say that it is the free Lie algebra on M generators. The free nilpotent Lie algebra gM,r on M generators of rank r is the quotient of the free Lie algebra by the ideal gr+1 generated as follows: ...

We show that a left-invariant metric g on a nilpotent Lie group N is a soliton metric if and only if a matrix U and vector v associated the manifold (N, g) satisfy the matrix equation Uv = [1], where [1] is a vector with every entry a one. We associate a generalized Cartan matrix to the matrix U and use the theory of Kac-Moody algebras to analyze the solution spaces for such linear systems. We ...

In this paper, we begin a quantization program for nilpotent orbits OR of a real semisimple Lie group GR. These orbits arise naturally as the coadjoint orbits of GR which are stable under scaling, and thus they have a canonical symplectic structure ω where the GR-action is Hamiltonian. These orbits and their covers generalize the oscillator phase space T R, which occurs here when GR = Sp(2n,R) ...

We propose a notion of a quantum universal enveloping algebra for any Lie algebra defined by generators and relations which is based on the quantum Lie operation concept. This enveloping algebra has a PBW basis that admits a monomial crystallization by means of the Kashiwara idea. We describe all skew primitive elements of the quantum universal enveloping algebras for the classical nilpotent al...

that g* has the same derived algebra as g itself and that every ideal in g is also an ideal in t*. Let g be any algebraic Lie algebra. Denote by b the radical of g (i.e., the largest solvable ideal in g) and by n the largest ideal of g composed only of nilpotent matrices. By Levi's theorem, g is the direct sum of t and of a semi-simple subalgebra J. It can be proved that f is the direct sum of ...

In this article, we generalize the concept of the Lie algebra of vector fields to the set of smooth sections of a T-bundle which is by definition a canonical generalization of the concept of a tangent bundle. We define a Lie bracket multiplication on this set so that it becomes a Lie algebra. In the particular case of tangent bundles this Lie algebra coincides with the Lie algebra of vector fie...

Let $K$ be a field of characteristic zero, $X_n=\{x_1,\dots,x_n\}$ set variables, $K[X_n]$ the polynomial algebra and $F_n$ free metabelian Lie rank $n$ generated by $X_n$ over base $K$. Well known result Weitzenb\"ock states that $K[X_n]^\delta=\big \{u\in K[X_n] \big\vert\ \delta(u)=0\big \}$ is finitely as an algebra, where $\delta$ locally nilpotent linear derivation $K[X_n]$. Extending thi...