‎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.

We have discovered that the gauge invariant observables of matrix models invariant under U(N) form a Lie algebra, in the planar large-N limit. These models include Quantum Chromodynamics and the M(atrix)-Theory of strings. We study here the gauge invariant states corresponding to open strings ('mesons'). We find that the algebra is an extension of a remarkable new Lie algebra V Λ by a product o...

Let gl(n, R) be the Lie algebra consisting of all n × n matrices over a commutative ring R with identity 1. In this paper, we prove that every generalized Lie triple derivation of gl(n, R)(n ≥ 2) is the sum of a Lie triple derivation and a homothety.

In categorified symplectic geometry, one studies the categorified algebraic and geometric structures that naturally arise on manifolds equipped with a closed nondegenerate (n + 1)-form. The case relevant to classical string theory is when n = 2 and is called ‘2-plectic geometry’. Just as the Poisson bracket makes the smooth functions on a symplectic manifold into a Lie algebra, there is a Lie 2...

We have discovered that the gauge invariant observables of matrix models invariant under U(N) form a Lie algebra, in the planar large-N limit. These models include Quantum Chromodynamics and the M(atrix)-Theory of strings. We study here the gauge invariant states corresponding to open strings ('mesons'). We find that the algebra is an extension of a remarkable new Lie algebra V Λ by a product o...

In these notes, we demonstrate how to compute the eigenvalue of the quadratic Casimir operator and the second-order index for an irreducible representation of a simple Lie algebra. Explicit results for the fundamental and adjoint representations of su(n), so(n) and sp(n) are given. The relation of these results to the dual Coxeter number is clarified. Finally, the dependence on the normalizatio...

In categorified symplectic geometry, one studies the categorified algebraic and geometric structures that naturally arise on manifolds equipped with a closed non-degenerate n + 1-form. The case relevant to classical string theory is when n = 2 and is called ‘2-plectic geometry’. Just as the Poisson bracket makes the smooth functions on a symplectic manifold into a Lie algebra, there is a Lie 2-...

This paper studies certain relations among vertex algebras, vertex Lie algebras and vertex poisson algebras. In this paper, the notions of vertex Lie algebra (conformal algebra) and vertex poisson algebra are revisited and certain general construction theorems of vertex poisson algebras are given. A notion of filtered vertex algebra is formulated in terms of a notion of good filtration and it i...

We examine a natural supersymmetric extension of the bosonic covariant 3-algebra model for M-theory proposed in [1]. It possesses manifest SO(1,10) symmetry and is constructed based on the Lorentzian Lie 3-algebra associated with the U(N) Lie algebra. There is no ghost related to the Lorentzian signature in this model. It is invariant under 64 supersymmetry transformations although the supersym...

We introduce a new class of simple Lie algebras W (n, m) (see Definition 1) that generalize the Witt algebra by using " exponential " functions, and also a subalgebra W * (n, m) thereof; and we show each derivation of W * (1, 0) can be written as a sum of an inner derivation and a scalar derivation (Theorem. 2) [10]. The Lie algebra W (n, m) is Z-graded and is infinite growth [4].