‎in this paper‎, ‎we introduce the structure of a groupoid associated to a vector field‎ ‎on a smooth manifold‎. ‎we show that in the case of the $1$-dimensional manifolds‎, ‎our‎ ‎groupoid has a‎ ‎smooth structure such that makes it into a lie groupoid‎. ‎using this approach‎, ‎we associated to‎ ‎every vector field an equivalence‎ ‎relation on the lie algebra of all vector fields on the smooth...

All coboundary Lie bialgebras and their corresponding Poisson–Lie structures are constructed for the oscillator algebra generated by {N,A+, A−,M}. Quantum oscillator algebras are derived from these bialgebras by using the Lyakhovsky and Mudrov formalism and, for some cases, quantizations at both algebra and group levels are obtained, including their universal R–matrices.

We define a semi-Hopf algebra which is more general than a Hopf algebra. Then we construct the supersymmetry algebra via the adjoint action on this semi-Hopf algebra. As a result we have a supersymmetry theory with quantum gauge group, i.e., quantised enveloping algebra of a simple Lie algebra. For the example, we construct the Lagrangian N =1 and N =2 supersymmetry. ∗email : [email protected] 1

The aim of this article is to compute the Hochschild and cyclic homology groups of the Yang-Mills algebras YM(n) (n ∈ N≥2) defined by A. Connes and M. Dubois-Violette in [CD1], continuing thus the study of these algebras that we have initiated in [HS]. The computation involves the use of a spectral sequence associated to the natural filtration on the universal enveloping algebra YM(n) provided ...

Because they play a role in our understanding of the symmetric group algebra, Lie idempotents have received considerable attention. The Klyachko idempotent has attracted interest from combinatorialists, partly because its definition involves the major index of permutations. For the symmetric group Sn, we look at the symmetric group algebra with coefficients from the field of rational functions ...

Given a simple, simply laced, complex Lie algebra g corresponding to the Lie group G, let n+ be the subalgebra generated by the positive roots. In this paper we construct a BV-algebra BV[g] whose underlying graded commutative algebra is given by the cohomology, with respect to n+, of the algebra of regular functions on G with values in ∧ (n+\g). We conjecture that BV[g] describes the algebra of...

The Higgs sector of the low–energy physics of n of coincident D–branes contains the necessary elements for constructing noncommutative manifolds. The coordinates orthogonal to the coincident branes, as well as their conjugate momenta, take values in the Lie algebra of the gauge group living inside the brane stack. In the limit when n → ∞ (and in the absence of orientifolds), this is the unitary...

In this thesis we investigate the existence of complex-valued harmonic morphisms on Lie groups and homogeneous Hadamard manifolds. The Lie groups that we are interested in have a particular decomposition of their Lie algebra. This decomposition allows us to define harmonic morphisms to R, n ≥ 2. Any homogeneous Hadamard manifold is isometric to a solvable Lie group S with a left-invariant metri...

SUANMALI, SUTHATHIP. On the Relationship between the Class of a Lie Algebra and the Classes of its Subalgebras. (Under the direction of Ernie L. Stitzinger.) A classical nilpotency result considers finite p-groups whose proper subgroups all have class bounded by a fixed number n. We consider the analogous property in nilpotent Lie algebras. In particular, we investigate whether this condition p...

‎highest weight modules of the double affine lie algebra $widehat{widehat{mathfrak{sl}}}_{2}$ are studied under a‎ ‎new triangular decomposition‎. ‎singular vectors of verma modules are‎ ‎determined using a similar condition with horizontal affine lie‎ ‎subalgebras‎, ‎and highest weight modules are described under the‎ ‎condition $c_1>0$ and $c_2=0$.