An algebra generated by projections with relations of Temperley–Lieb type is considered. Knowledge of Gröbner basis of the ideal allows to find a linear basis of the algebra. Some questions of representation theory for this algebra were studied in [13]. Obtained in the present paper are the additional relations, which hold in all finite-dimensional irreducible ∗-representations, although they d...

Different properties of rings and fields are discussed. We introduce ring homomorphisms, their kernels and images, and prove the First Isomorphism Theorem, namely that for a homomorphism f : R −→ S we have R/ker(f) ∼= Im(f). Then we define prime and irreducible elements and show that every principal ideal domain is factorial. Finally we show that polynomial rings over fields are Euclidean and h...

We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a p-adic field, together with relevant arithmetic information about the fields generated by the irreducible factors. This carries out a program suggested by Ø. Ore. As an application, we obtain fast algorithms to compute discriminants, prime ideal decomposition and i...

For every hyperbolic group and more general hyperbolic graphs, we construct an equivariant ideal bicombing: this is a homological analogue of the geodesic flow on negatively curved manifolds. We then construct a cohomological invariant which implies that several Measure Equivalence and Orbit Equivalence rigidity results established in [MSb] hold for all non-elementary hyperbolic groups and thei...

The notions of int-soft semigroups and int-soft left (resp., right) ideals are introduced, and several properties are investigated. Using these notions and the notion of inclusive set, characterizations of subsemigroups and left (resp., right) ideals are considered. Using the notion of int-soft products, characterizations of int-soft semigroups and int-soft left (resp., right) ideals are discus...

In this paper, we obtain the Rényi entropy rate for irreducible-aperiodic Markov chains with countable state space, using the theory of countable nonnegative matrices. We also obtain the bound for the rate of Rényi entropy of an irreducible Markov chain. Finally, we show that the bound for the Rényi entropy rate is the Shannon entropy rate.

The leptonic Higgs doublet model of neutrino masses is implemented with an A4 discrete symmetry (the even permutation of 4 objects or equivalently the symmetry of the tetrahedron) which has 4 irreducible representations: 1, 1, 1, and 3. The resulting spontaneous and soft breaking of A4 provides a realistic model of charged-lepton masses as well as a nearly degenerate neutrino mass matrix. Pheno...

Fix a prime p > 2. Let ρ : Gal(Q/Q) → GL2(I) be the Galois representation coming from a non-CM irreducible component I of Hida’s p-ordinary Hecke algebra. Assume the residual representation ρ̄ is absolutely irreducible. Under a minor technical condition we identify a subring I0 of I containing Zp[[T ]] such that the image of ρ is large with respect to I0. That is, Im ρ contains ker(SL2(I0) → SL2...

Remark 0.1 (Notation). |G| denotes the order of a finite group G. [E : F ] denotes the degree of a field extension E/F. We write H ≤ G to mean that H is a subgroup of G, and N G to mean that N is a normal subgroup of G. If E/F and K/F are two field extensions, then when we say that K/F is contained in E/F , we mean via a homomorphism that fixes F. We assume the following basic facts in this set...