‎In this paper Lie symmetry analysis is applied in order to find new solutions for Fokker Plank equation of Ornstein-Uhlenbeck process‎. ‎This analysis classifies the solutions format of the Fokker Plank equation by using the Lie algebra of the symmetries of our considered stochastic process‎.

It is well-known that the second homology group H2(stn(R)) of the Steinberg Lie algebra stn(R) is trivial when n ≥ 5. In this paper, we will work out H2(stn(R)) explicitly for n = 3, 4 which are not necessarily trivial. Consequently, we obtained H2(sln(R)) for n = 3, 4. Introduction Steinberg Lie algebras stn(R) and/or their universal coverings have been studied by Bloch [Bl], Kassel-Loday [KL]...

We extend the basic fact that every ideal of a finite dimensional semisimple Lie algebra has a unique complement to the case of closed ideals of prosemisimple Lie algebras. We prove that if A is a closed ideal of a prosemisimple Lie algebra L = lim ←−−Ln (n ∈ N), where the Ln are finite dimensional semisimple Lie algebras, then there exists a unique ideal B of L such that L = A ⊕ B. Mathematics...

We consider three Lie algebras: Der C((t)), the Lie algebra of all derivations on the algebra C((t)) of formal Laurent series; the Lie algebra of all differential operators on C((t)); and the Lie algebra of all differential operators on C((t)) ⊗ Cn. We prove that each of these Lie algebras has an essentially unique nontrivial central extension. The Lie algebra of all derivations on the Laurent ...

A Lie 2-algebra is a ‘categorified’ version of a Lie algebra: that is, a category equipped with structures analogous those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry ...

Let A+(k) denote the ring l~[ t] / t k+t and let fr be a reductive complex Lie algebra with exponents m 1 . . . . . m,. This paper concerns the Lie algebra cohomology of ~| considered as a bigraded algebra (here one of the gradings is homological degree and the other, which we call weight, is inherited from the obvious grading of f f | § (k)). We conjecture that this Lie algebra cohomology is a...

‎let $l$ be a lie algebra‎, ‎$mathrm{der}(l)$ be the set of all derivations of $l$ and $mathrm{der}_c(l)$ denote the set of all derivations $alphainmathrm{der}(l)$ for which $alpha(x)in [x,l]:={[x,y]vert yin l}$ for all $xin l$‎. ‎we obtain an upper bound for dimension of $mathrm{der}_c(l)$ of the finite dimensional nilpotent lie algebra $l$ over algebraically closed fields‎. ‎also‎, ‎we classi...

in this paper, lie group structure and lie algebra structure of unit complex 3-sphere     are studied. in order to do this, adjoint representations of unit biquaternions (complexified quaternions) are obtained. also, a correspondence between the elements of     and the special complex unitary matrices    (2) is given by expressing biquaternions as 2-dimensional bicomplex numbers    .  the relat...

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A Lie algebra gQ over Q is said to be R-universal if every homomorphism from gQ to gl(n,R) is conjugate to a homomorphism into gl(n,Q) (for every n). By using Galois cohomology, we provide a short proof of the known fact that every real semisimple Lie algebra has an R-universal Q-form. We also provide a classification of the R-universal Lie algebras that are semisimple.