The self-dual concept of a quantum quasigroup offers a far-reaching unification of Hopf algebras (along with their non-associative extensions, e.g., [1–4]) and quasigroups [5]. Consider a bimagma (A,∇, ∆), an object of a strict symmetric monoidal category V with V-morphisms giving a magma structure or multiplication ∇ : A⊗ A→ A, and a comagma structure or comultiplication ∆ : A→ A⊗ A, such that...

We introduced a non-symmetric tensor product of any two states or any two representations of Cuntz-Krieger algebras associated with a certain non-cocommutative comultiplication in previous our work. In this paper, we show that a certain set of KMS states is closed with respect to the tensor product. From this, we obtain formulae of tensor product of type III factor representations of Cuntz-Krie...

In this paper we work toward the Homflypt skein module of the lens spaces L(p, 1), S(L(p, 1)), using braids. In particular, we establish the connection between S(ST), the Homflypt skein module of the solid torus ST, and S(L(p, 1)) and arrive at an infinite system, whose solution corresponds to the computation of S(L(p,1)). We start from the Lambropoulou invariant X for knots and links in ST, th...

Let H be a bialgebra and let A be an associative algebra. The algebra A is said to be an H-module-algebra if there is an H-module structure on A such that the multiplication on A becomes an H-module morphism. For example, if S denotes the Landweber-Novikov algebra [15, 21], then the complex cobordism MU(X) of a topological space X is an S module-algebra. Likewise, the singular mod p cohomology ...

Preface These notes are based on a course given at the Tata Institute of Fundamental Research in the beginning of 1990. The aim of the course was to describe the solution by O. Mathieu of some conjectures in the representation theory of semi-simple algebraic groups. These conjectures concern the inner structure of dual Weyl modules and some of their analogues. Recall that if G is a (connected, ...

An R-module M is called epi-retractable if every submodule of MR is a homomorphic image of M. It is shown that if R is a right perfect ring, then every projective slightly compressible module MR is epi-retractable. If R is a Noetherian ring, then every epi-retractable right R-module has direct sum of uniform submodules. If endomorphism ring of a module MR is von-Neumann regular, then M is semi-...

For an elliptic curve E over ℚ, putting K=ℚ(E[p]) which is the p-th division field of for odd prime p, we study ideal class group ClK K as a Gal(K/ℚ)-module. More precisely, any j with 1⩽j⩽p-2, give condition that ClK⊗Fp has symmetric power SymjE[p] E[p] its quotient Gal(K/ℚ)-module, in terms Bloch-Kato’s Tate-Shafarevich SymjVpE. Here VpE denotes rational p-adic Tate module E. This partial gen...

In this article we formalize a quotient module of Z-module and a vector space constructed by the quotient module. We formally prove that for a Z-module V and a prime number p, a quotient module V/pV has the structure of a vector space over Fp. Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattices [14]. Some...

we show that every semi-artinian module which is contained in a direct sum of finitely presented modules in $si[m]$, is weakly co-semisimple if and only if it is regular in $si[m]$. as a consequence, we observe that every semi-artinian ring is regular in the sense of von neumann if and only if its simple modules are $fp$-injective.