‘az + b’ is the group of affine transformations of complex plane C. The coefficients a, b ∈ C. In quantum version a, b are normal operators such that ab = q2ba, where q is the deformation parameter. We shall assume that q is a root of unity. More precisely q = e 2πi N , where N is an even natural number. To construct the group we write an explicit formula for the Kac Takesaki operator W . It is...

The Murphy operators in the Hecke algebra Hn of type A are explicit commuting elements whose sum generates the centre. They can be represented by simple tangles in the Homfly skein theory version of Hn. In this paper I present a single tangle which represents their sum, and which is obviously central. As a consequence it is possible to identify a natural basis for the Homfly skein of the annulu...

We show that the tree-level part of a theory of finite type invariants of 3-manifolds (based on surgery on objects called claspers, Y-graphs or clovers) is essentially given by classical algebraic topology in terms of the Johnson homomorphism and Massey products, for arbitrary 3-manifolds. A key role of our proof is played by the notion of a homology cylinder, viewed as an enlargement of the ma...

Let G be the planar Galilean conformal algebra and G˜ its universal central extension. Then (resp. G˜) admits a triangular decomposition: G=G+?G0?G? G˜=G˜+?G˜0?G˜?). In this paper, we study generic Whittaker G-modules G˜-modules) of type ?, where ?:G+?G˜+?C is Lie homomorphism. We classify isomorphism classes modules. Moreover, show that module ? simple if only nonsingular. For nonsingular case...

Using various finite dimensional approximation properties, four convex subsets of the tracial space of a unital C∗-algebra are defined. One subset is characterized by Connes’ hypertrace condition. Another is characterized by hyperfiniteness of GNS representations. The other two sets are more mysterious but are shown to be intimately related to Elliott’s classification program. Applications of t...

A monadic (Boolean) algebra is a Boolean algebra A together with an operator 3 on A (called an existential quantifier, or, simply, a quantifier) such that 30=0, pfk 3p, and 3(^A 3q) = 3p* 3g whenever p and q are in A. Most of this note uses nothing more profound about monadic algebras than the definition. The reader interested in the motivation for and the basic facts in the theory of monadic a...

Let K be a Banach space, B a unital C∗-algebra, and π : B → L(K) an injective, unital homomorphism. Suppose that there exists a function γ : K×K → R+ such that, for all k, k1, k2 ∈ K, and all b ∈ B, (a) γ(k, k) = ‖k‖2, (b) γ(k1, k2) ≤ ‖k1‖ ‖k2‖, (c) γ(πbk1, k2) = γ(k1, πb∗k2). Then for all b ∈ B, the spectrum of b in B equals the spectrum of πb as a bounded linear operator on K. If γ satisfies ...

This paper introduces the concept of subalgebra and discusses some properties subalgebra. We also examine different types ideals how they behave. describe a homomorphism. Finally, we present quotient algebra discuss isomorphism theorems.

We consider Fréchet algebras which are subalgebras of the algebra F = C [[X]] of formal power series in one variable and of Fn = C [[X1, . . . , Xn]] of formal power series in n variables where n ∈ N. In each case, these algebras are taken with the topology of coordinatewise convergence. We begin with some basic definitions about Fréchet algebras, (F )-algebras, and other topological algebras, ...

We exhibit a Poisson module restoring a twisted Poincaré duality between Poisson homology and cohomology for the polynomial algebra R = C[X1, . . . , Xn] endowed with Poisson bracket arising from a uniparametrised quantum affine space. This Poisson module is obtained as the semiclassical limit of the dualising bimodule for Hochschild homology of the corresponding quantum affine space. As a coro...