Let d \subset d' be finite-dimensional Lie algebras, H = U(d), H'=U(d') the corresponding universal enveloping algebras endowed with cocommutative Hopf algebra structure. We show that if L is a primitive pseudoalgebra over then all finite irreducible L' Cur_H^{H'} L-modules are of form V, where V an L-module, single class exceptions. Indeed, when H(d, \chi, \omega), we introduce non current L'-...

This paper explores the new and growing field of topological data analysis (TDA). TDA is a data analysis method that provides information about the ’shape’ of data. The paper describes what types of shapes TDA detects and why these shapes having meaning. Additionally, concepts from algebraic topology, the mathematics behind TDA, will be discussed. Specifically, the concepts of persistent homolo...

The hyperbolic algebra A h , studied recently by Katavolos and Power [5], is the weak star closed operator algebra on L 2 (R) generated by H ∞ (R), as multiplication operators, and by the dilation operators V t , t ≥ 0, given by V t f (x) = e t/2 f (e t x). We show that A h is a reflexive operator algebra and that the four dimensional manifold Lat A h (with the natural topology) is the reflexiv...

Let k be a field, H a cocommutative bialgebra, A a commutative left H-module algebra, Hom(H,A) the k-algebra of the k-linear maps from H to A under the convolution product, Z(H,A) the submonoid of Hom(H,A) whose elements satisfy the cocycle condition and G any subgroup of the monoid Z(H,A). We give necessary and sufficient conditions for the projectivity and flatness over the graded ring of nor...

We construct quantum commutators on module-algebras of quasitriangular Hopf algebras. These are quantum-group covariant, and have generalized antisymmetry and Leibniz properties. If the Hopf algebra is triangular they additionally satisfy a generalized Jacobi identity, turning the modulealgebra into a quantum-Lie algebra. The purpose of this short communication is to present a quantum commutato...

We present the following reflexivity-like result concerning the automorphism group of the C∗-algebra B(H), H being a separable Hilbert space. Let φ : B(H) → B(H) be a multiplicative map (no linearity or continuity is assumed) which can be approximated at every point by automorphisms of B(H) (these automorphisms may, of course, depend on the point) in the operator norm. Then φ is an automorphism...

Let H be the quaternion algebra. Let g be a complex Lie algebra and let U(g) be the enveloping algebra of g. The quaternification g = (H ⊗ U(g), [ , ]gH ) of g is defined by the bracket [ z ⊗ X , w ⊗ Y ] gH = (z · w) ⊗ (XY ) − (w · z) ⊗ (Y X) , for z, w ∈ H and the basis vectors X and Y of U(g). Let SH be the ( non-commutative) algebra of H-valued smooth mappings over S and let Sg = SH ⊗ U(g). ...

These rings admit numerous algebraic and geometric realizations, but one of the historically first constructions, dating to the work of Steinitz in 1900 completed later by Hall, was given in terms of what is now called the classical Hall algebra H (see [Ma], Chapter II ). This algebra has a basis consisting of isomorphism classes of abelian q-groups, where q is a fixed prime power, and the stru...

Let G be a locally compact group, π : G → U(H) be a strongly continuous unitary representation, and CB(B(H)) the space of normal completely bounded maps on B(H). We study the range of the map Γπ : M(G)→ CB(B(H)), Γπ(μ) = Z G π(s)⊗ π(s)dμ(s) where we identify CB(B(H)) with the extended Haagerup tensor product B(H)⊗ B(H). We use the fact that the C*-algebra generated by integrating π to L(G) is u...

We introduce the notion of a half-ribbon Hopf algebra, which is a Hopf algebra H along with a distinguished element t ∈ H such that (H, R,C) is a ribbon Hopf algebra, where R = (t ⊗ t)∆(t) and C = t. The element t is closely related to the topological ‘half-twist’, which twists a ribbon by 180 degrees. We construct a functor from a topological category of ribbons with half-twists to the categor...