We introduce a quantum double quasitriangular quasi-Hopf algebra D(H) associated to any quasi-Hopf algebra H. The algebra structure is a cocycle double cross product. We use categorical reconstruction methods. As an example, we recover the quasi-Hopf algebra of Dijkgraaf, Pasquier and Roche as the quantum double D(G) associated to a finite group G and group 3-cocycle φ. We also discuss D(Ug) as...

G∞-structure is shown to exist on the deformation complex of a morphism of associative algebras. The main step of the construction is extension of a B∞-algebra by an associative algebra. Actions of B∞-algebras on associative and B∞-algebras are analyzed, extensions of B∞-algebras by associative and B∞-algebras, that they act upon, are constructed. The resulting G∞-algebra on the deformation com...

With slight modifications in the zero modes contributions, the positive and negative screening currents for the quantum deformed W -algebra Wq,p(g) can be put together to form a single algebra which can be regarded as an elliptic deformation of the universal enveloping algebra of ĝ, where g is any classical simply-laced Lie algebra. Recently, various deformations of the classical and quantum Vi...

It is shown that the collection of weakly almost periodic functionals on the convolution algebra of a commutative Hopf von Neumann algebra is a C∗-algebra. This implies that the weakly almost periodic functionals on M(G), the measure algebra of a locally compact group G, is a C∗-subalgebra of M(G)∗ = C0(G) ∗∗. The proof builds upon a factorisation result, due to Young and Kaiser, for weakly com...

I summarize some recent results obtained in collaboration with P. Bouwknegt and K. Pilch on the spectrum of physical states in W 3 gravity coupled to c = 2 matter. In particular, it is shown that the algebra of operators corresponding to physical states – defined as a semi-infinite (or BRST) cohomology of the W 3 algebra – carries the structure of a G-algebra. This G-algebra has a quotient whic...

We prove a graded version of Alev-Polo’s rigidity theorem: the homogenization of the universal enveloping algebra of a semisimple Lie algebra and the Rees ring of the Weyl algebras An(k) cannot be isomorphic to their fixed subring under any finite group action. We also show the same result for other classes of graded regular algebras including the Sklyanin algebras. 0. Introduction The invarian...

Let G be a connected Lie group acting by algebra automorphisms on a finite-dimensional complex central simple algebra A. The algebra A is isomorphic to the endomorphism algebra of a projective representation V of G. We study the invariant subalgebras of A. In particular, we show that if V is irreducible, then the invariant subalgebras appear in dual pairs arising from factorizations of V . We a...

For Φ,Ψ ∈ A′′, define 〈Φ Ψ, λ〉 = 〈Φ, Ψ · λ〉 (λ ∈ A′) , and similarly for ♦. Thus (A′′, ) and (A′′,♦) are Banach algebras each containingA as a closed subalgebra. The Banach algebra A is Arens regular if and ♦ coincide on A′′, and A is strongly Arens irregular if and ♦ coincide only on A. A subspace X of A′ is left-introverted if Φ · λ ∈ X whenever Φ ∈ A′′ and λ ∈ X . There has been a great deal...

we construct a family of two parameter quantum grou-ps‎ ‎$u_{r,s}(mathfrak{g})$ associated with a generalized kac-moody‎ ‎algebra corresponding to symmetrizable admissible borcherds cartan‎ ‎matrix‎. ‎we also construct the $textbf{a}$-form $u_{textbf{a}}$ and‎ ‎the classical limit of $u_{r,s}(mathfrak{g})$‎. ‎furthermore‎, ‎we‎ ‎display the equitable presentation for a subalgebra‎ ‎$u_{r,s}^{b-...