We propose a new method to construct degenerate $3$-$(\alpha,\delta)$-Sasakian manifolds as fiber products of Boothby-Wang bundles over hyperk\"ahler manifolds. Subsequently, we study homogeneous $3$-$(\alpha, \delta)$-Sasakian and prove that no non-trivial compact examples exist well there is exactly one family nilpotent Lie groups with this geometry, the quaternionic Heisenberg groups.

The reductions for the first order linear systems of the type: Lψ(x, λ) ≡ ( i d dx + q(x)− λJ ) ψ(x, λ) = 0 , J ∈ h , q(x) ∈ gJ are studied. This system generalizes the Zakharov–Shabat system and the systems studied by Caudrey, Beals and Coifman (CBC systems). Here J is a regular complex constant element of the Cartan subalgebra h ⊂ g of the simple Lie algebra g and the potential q(x) takes val...

‎‎we exhibit an explicit construction for the second cohomology group‎ ‎$h^2(l‎, ‎a)$ for a lie ring $l$ and a trivial $l$-module $a$‎. ‎we show how the elements of $h^2(l‎, ‎a)$ correspond one-to-one to the‎ ‎equivalence classes of central extensions of $l$ by $a$‎, ‎where $a$‎ ‎now is considered as an abelian lie ring‎. ‎for a finite lie‎ ‎ring $l$ we also show that $h^2(l‎, ‎c^*) cong m(l)$‎...

in this paper we construct the category of coverings of fundamental generalized lie group-groupoid associatedwith a connected generalized lie group. we show that this category is equivalent to the category of coverings of aconnected generalized lie group. in addition, we prove the category of coverings of generalized lie groupgroupoidand the category of actions of this generalized lie group-gro...

By means of the spherical functions associated to Gelfand pair \((\mathbb {H}_{n},U(n))\) we define operator \(L+\alpha |T|\), where L denotes Heisenberg sublaplacian and T central element Lie algebra, establish a notion fundamental solution explicitly compute in terms Gauss hypergeometric function. For \(\alpha <n\) use Integral Representation Theorem obtain more detailed expression. Finally, ...

Suppose mu and v are arcs whose union is an arc alpha and whose intersection is an arc beta. Suppose further that mu and v, respectively, lie in the boundaries of the disks M and N. It is shown that either alpha lies on the boundary of a disk or mu and v, respectively, lie in the boundaries of disks M' and N' which meet only in the arc beta. The construction is basic to recent results exploring...

The theory of crystal bases introduced by Kashiwara in [4] to study the category of integrable representations of quantized Kac–Moody Lie algebras has been a major development in the combinatorial approach to representation theory. In particular Kashiwara defined the tensor product of crystal bases and showed that it corresponded to the tensor product of representations. Later, in [5] he define...

let x be a banach space of dimx > 2 and b(x) be the space of bounded linear operators on x. if l : b(x) → b(x) be a lie higher derivation on b(x), then there exists an additive higher derivation d and a linear map τ : b(x) → fi vanishing at commutators [a, b] for all a, b ∈ b(x) such that l = d + τ

The theory of crystal bases introduced by Kashiwara in [6] to study the category of integrable representations of quantized Kac–Moody Lie algebras has been a major development in the combinatorial approach to representation theory. In particular Kashiwara defined the tensor product of crystal bases and showed that it corresponded to the tensor product of representations. Later, in [7] he define...

The following integrability theorem for vertex operator algebras V satisfying some finiteness conditions (C2-cofinite and CFT-type) is proved: the vertex operator subalgebra generated by a simple Lie subalgebra g of the weight one subspace V1 is isomorphic to the irreducible highest weight ĝ-module L(k, 0) for a positive integer k, and V is an integrable ĝ-module. The case in which g is replace...