We show a relation of the KMS state of a certain C∗Algebra U with the Gibbs state of Thermodynamic Formalism. More precisely, we consider here the shift T : X → X acting on the Bernoulli space X = {1, 2, ..., k} and μ a Gibbs state defined by a Holder continuous potential p : X → R, and L(μ) the associated Hilbert space. Consider the C∗-Algebra U = U(μ), which is a sub-C∗-Algebra of the C∗-Alge...

We define a nontrivial homomorphism from the Hecke algebra of type B onto a reduced algebra of the Hecke algebra of type A at roots of unity. We use this homomorphism to describe semisimple quotients of the Hecke algebra of type B at roots of unity. Using these quotients we determine subfactors obtained from the inclusion of Hecke algebra of type A into Hecke algebras of type B. We also study i...

We consider the double affine Hecke algebra H = H(k0, k1, k∨ 0 , k ∨ 1 ; q) associated with the root system (C∨ 1 , C1). We display three elements x, y, z in H that satisfy essentially the Z3-symmetric Askey–Wilson relations. We obtain the relations as follows. We work with an algebra Ĥ that is more general than H, called the universal double affine Hecke algebra of type (C∨ 1 , C1). An advanta...

Many interesting C∗-algebras can be viewed as quantizations of Poisson manifolds. I propose that a Poisson manifold may be quantized by a twisted polarized convolution C∗-algebra of a symplectic groupoid. Toward this end, I define polarizations for Lie groupoids and sketch the construction of this algebra. A large number of examples show that this idea unifies previous geometric constructions, ...

In this paper, we introduce the notionofWajsberg implicative ideal(WI-ideal)of lattice Wajsberg algebra.Further, we definelattice H-Wajsberg algebra, implication homomorphismand lattice implication homomorphism oflattice Wajsberg algebra. Finally, we give kernel of implication homomorphism and obtain some oftheir properties.

A pro-C∗-algebra is a (projective) limit of C∗-algebras in the category of topological ∗algebras. From the perspective of non-commutative geometry, pro-C∗-algebras can be seen as non-commutative k-spaces. An element of a pro-C∗-algebra is bounded if there is a uniform bound for the norm of its images under any continuous ∗-homomorphism into a C∗-algebra. The ∗-subalgebra consisting of the bound...

If H is a commutative connected graded Hopf algebra over a commutative ring k, then a certain canonical k-algebra homomorphism H → H⊗QSymk is defined, where QSymk denotes the Hopf algebra of quasisymmetric functions. This homomorphism generalizes the “internal comultiplication” on QSymk, and extends what Hazewinkel (in §18.24 of his “Witt vectors”) calls the Bernstein homomorphism. We construct...

There are also four appendices. Let K be a field of characteristic 0, and let C be a commutative K-algebra. which makes C into a Lie algebra, and is a biderivation (i.e. a derivation in each argument). The pair C, {−, −} is called a Poisson algebra. Poisson brackets arise in several ways. Example 1.1. Classical Hamiltonian mechanics. Here K = R, X is an even dimensional differentiable manifold ...

A Riemann-Lie algebra is a Lie algebra G such that its dual G∗ carries a Riemannian metric compatible (in the sense introduced by the author in C. R. Acad. Sci. Paris, t. 333, Série I, (2001) 763–768) with the canonical linear Poisson structure of G∗ . The notion of Riemann-Lie algebra has its origins in the study, by the author, of Riemann-Poisson manifolds (see Differential Geometry and its A...

Definition 1.1. A Poisson algebra is an associative algebra A over a field K (fixed, of characteristic zero), equipped with a Lie bracket {−,−} such that {x,−} is a derivation for any x ∈ A, i.e. {x, yz} = {x, y}z + y{x, z}. Definition 1.2. A Poisson structure on a manifold M is a Poisson bracket {−,−} on the algebra C∞(M). Example 1.3. On T ∗Rn with position coordinates q1, ..., qn and momentu...