Introduction. We let G denote an infinite compact group, and G its dual. We use the notation of our book [3, Chapters 7 and 8]. Recall that -4(G) denotes the Fourier algebra of G (an algebra of continuous functions on G), and J2f°°(G) denotes its dual space under the pairing ( / , # ) , ( / G A (G), 0 Ç if°°(G)). Further, note if°°(G) is identified with the G*-algebra of bounded operators on L(...

Let g be a complex semisimple Lie algebra, and let G be a complex semisimple group with trivial center whose root system is dual to that of g. We establish a graded algebra isomorphism H q (Xλ,C) ∼= Sg e/Iλ, where Xλ is an arbitrary spherical Schubert variety in the loop Grassmannian for G, and Iλ is an appropriate ideal in the symmetric algebra of g, the centralizer of a principal nilpotent in...

Let g be a complex semisimple Lie algebra, and let G be a complex semisimple group with trivial center whose root system is dual to that of g. We establish a graded algebra isomorphism H q (Xλ,C) ∼= Sg e/Iλ, where Xλ is an arbitrary spherical Schubert variety in the loop Grassmannian for G, and Iλ is an appropriate ideal in the symmetric algebra of g, the centralizer of a principal nilpotent in...

As Don Schack mentioned in his plenary talk [SDS], Gerstenhaber [MG] observed that if A is an associative algebra, then the Hochschild cohomology of A is a graded commutative algebra with an additional structure, viz., that of a Lie algebra. The two structures satisfy a compatibility condition (graded Poission structure). We agree to call such structures G–algebras (Gerstenhaber–algebras). Scha...

We give some constructions of diierential Gerstenhaber-Batalin-Vilkovisky algebras from a class of Lie algebras. In our construction, we make use of the solutions to the classical Yang-Baxter equations, and ideas from Poisson geometry. A graded commutative algebra (A; ^) with a bracket ] of degree?1 is called a G-algebra (Gerstenhaber algebra) if: (a) (A1]; ]) is a Lie algebra, where A1] is A w...

For a finite-dimensional Lie algebra $\mathfrak{L}$ over $\mathbb{C}$ with fixed Levi decomposition $\mathfrak{L} = \mathfrak{g} \oplus \mathfrak{r}$ where $\mathfrak{g}$ is semi-simple, we investigate $\mathfrak{L}$-modules which decompose, as $\mathfrak{g}$-modules, into direct sum of simple $\mathfrak{g}$-modules finite multiplicities. We call such modules $\mathfrak{g}$-Harish-Chandra modul...

We study the algebra U ζ obtained via Lusztig's 'integral' form [Lu 1, 2] of the generic quantum algebra for the Lie algebra g = sl 2 modulo the two-sided ideal generated by K l − 1. We show that U ζ is a smash product of the quantum deformation of the restricted universal enveloping algebra u ζ of g and the ordinary universal enveloping algebra U of g, and we compute the primitive (= prime) id...

We find the Hopf algebra U g,h dual to the Jordanian matrix quantum group GL g,h (2). As an algebra it depends only on the sum of the two parameters and is split in two subalgebras: U ′ g,h (with three generators) and U (Z) (with one generator). The subalgebra U (Z) is a central Hopf subalgebra of U g,h. The subalgebra U ′ g,h is not a Hopf subalgebra and its coalgebra structure depends on both...

Let D be a division algebra such that D ⊗ D is a Noetherian algebra, then any division subalgebra of D is a finitely generated division algebra. Let ∆ be a finite set of commuting derivations or automorphisms of the division algebra D, then the group Ev(∆) of common eigenvalues (i.e. weights) is a finitely generated abelian group. Typical examples of D are the quotient division algebra Frac(D(X...

We describe an algebra G of diagrams that faithfully gives a diagrammatic representation of the structures of both the Heisenberg–Weyl algebra H – the associative algebra of the creation and annihilation operators of quantum mechanics – and U(LH), the enveloping algebra of the Heisenberg Lie algebra LH. We show explicitly how G may be endowed with the structure of a Hopf algebra, which is also ...