We use braided groups to introduce a theory of ∗-structures on general inhomogeneous quantum groups, which we formulate as quasi-∗ Hopf algebras. This allows the construction of the tensor product of unitary representations up to a quantum cocycle isomorphism, which is a novel feature of the inhomogeneous case. Examples include q-Poincaré quantum group enveloping algebras in R-matrix form appro...

Let p, q be distinct prime numbers, and k an algebraically closed field of characteristic 0. Under certain restrictions on p, q, we discuss the structure of semisimple Hopf algebras of dimension pq. As an application, we obtain the structure theorems for semisimple Hopf algebras of dimension 9q over k. As a byproduct, we also prove that odd-dimensional semisimple Hopf algebras of dimension less...

We prove a Morita reduction theorem for the cyclotomic Hecke algebras Hr,p,n(q,Q) of type G(r, p, n). As a consequence, we show that computing the decomposition numbers of Hr,p,n(Q) reduces to computing the psplittable decomposition numbers (see Definition 1.1) of the cyclotomic Hecke algebras Hr′,p′,n′ (Q ), where 1 ≤ r ≤ r, 1 ≤ n ≤ n, p | p and where the parameters Q are contained in a single...

In this paper we consider Q the class of solvable Lie algebras L with the following property: if A is a subalgebra of L, then Φ(A) ⊆ Φ(L) (where Φ(L) denotes the Frattini subalgebra of L;that is Φ(L) is the intersection of all maximal subalgebras of L). The class Q is shown to contain all solvable Lie algebras whose derived algebra is nilpotent. Necessary conditions are found such that an ideal...

We prove a Morita reduction theorem for the cyclotomic Hecke algebras Hr,p,n(q,Q) of type G(r, p, n). As a consequence, we show that computing the decomposition numbers of Hr,p,n(Q) reduces to computing the psplittable decomposition numbers (see Definition 1.1) of the cyclotomic Hecke algebras Hr′,p′,n′ (Q ), where 1 ≤ r ≤ r, 1 ≤ n ≤ n, p | p and where the parameters Q are contained in a single...

The concepts of (∈,∈ ∨ q)-fuzzy (implicative, positive implicative and fantastic) filters of BL-algebras are introduced and some related properties are investigated. Some characterizations of these generalized fuzzy filters are derived. In particular, we describe the relationships among ordinary fuzzy (implicative, positive implicative and fantastic) filters, (∈,∈∨q)-fuzzy (implicative, positiv...

In this paper we study division algebras over the function fields of curves over Qp. The first and main tool is to view these fields as function fields over nonsingular S which are projective of relative dimension 1 over the p adic ring Zp. A previous paper showed such division algebras had index bounded by n assuming the exponent was n and n was prime to p. In this paper we consider algebras o...

In [3] and [13], the authors proved the cluster multiplication theorems for finite type and affine type. We generalize their results and prove the cluster multiplication theorem for arbitrary type by using the properties of 2–Calabi–Yau (Auslander–Reiten formula) and high order associativity. Introduction Cluster algebras were introduced by Fomin and Zelevinsky in [9]. By definition, the cluste...

We extend Cuntz-Quillen’s excision theorem for algebras and pro-algebras in arbitrary Q-linear categories with tensor product.