We study a two-boundary extension of the Temperley-Lieb algebra which has recently arisen in statistical mechanics. This algebra lies in a quotient of the affine Hecke algebra of type C and has a natural diagrammatic representation. The algebra has three parameters and, for generic values of these, we determine its representation theory. We use the action of the centre of the affine Hecke algeb...

We study finite-dimensional representations of hyper loop algebras, i.e., the hyperalgebras over a field of positive characteristic associated to nontwisted affine Kac-Moody algebras, or rather, to the underlying loop algebras. The main results are the classification of the irreducible modules, a version of Steinberg’s Tensor Product Theorem, and the construction of positive characteristic anal...

We study finite-dimensional representations of hyper loop algebras, i.e., the hyperalgebras over an algebraically closed field of positive characteristic associated to the loop algebra over a complex finite-dimensional simple Lie algebra. The main results are the classification of the irreducible modules, a version of Steinberg’s Tensor Product Theorem, and the construction of positive characte...

Let $G$ be a tamely ramified reductive $p$-adic‎ ‎group‎. ‎We study distinction of a class of irreducible admissible representations‎ ‎of $G$ by the group of fixed points $H$ of an involution‎ ‎of $G$‎. ‎The representations correspond to $G$-conjugacy classes of‎ ‎pairs $(T,phi)$‎, ‎where $T$ is a‎ ‎tamely ramified maximal torus of $G$ and $phi$ is a quasicharacter‎ ‎of $T$ whose restriction t...

We describe a pair of invariants for actions of finite groups on shifts of finite type, the left-reduced and right-reduced shifts. The left-reduced shift was first constructed by U. Fiebig, who showed that its zeta function is an invariant, and in fact equal to the zeta function of the quotient dynamical system. We also give conditions for expansivity of the quotient, and applications to combin...

From an abstract algebraic perspective, an explanation for this can be given as follows: since √ D is irrational, the polynomial t −D is irreducible over Q. Since the ring Q[t] is a PID, the irreducible element t − D generates a maximal ideal (t−D), so that the quotient Q[t]/(t−D) is a field. Moreover, the map Q[ √ D]→ Q[t]/(t −D) which is the identity on Q and sends √ D 7→ t is an isomorphism ...

We study finite-dimensional representations of hyper loop algebras, i.e., the hyperalgebras over an algebraically closed field of positive characteristic associated to the loop algebra over a complex finite-dimensional simple Lie algebra. The main results are the classification of the irreducible modules, a version of Steinberg’s Tensor Product Theorem, and the construction of positive characte...

The nonstandard Hecke algebra Ȟr was defined in [11] to study the Kronecker problem. We study a quotient Ȟr,2 of Ȟr , called the nonstandard Temperley-Lieb algebra, which is simpler than Ȟr. It is defined to be the subalgebra of Hr,2⊗Hr,2 generated by Pi := C ′ i ⊗C ′ i +Ci ⊗Ci, i ∈ [r− 1], where Hr,2 is the Temperley-Lieb algebra and C i and Ci are Kazhdan-Lusztig basis elements. We give a com...

Let W be a Coxeter group with Coxeter graph Γ. Let H be theassociated Hecke algebra. We define a certain ideal I in H and study thequotient algebra H̄ = H/I. We show that when Γ is one of the infinite seriesof graphs of type E, the quotient is semi-simple. We examine the cell structuresof these algebras and construct their irreducible representations. We discussthe case where...

We construct an A ∞ -category D(C|B) from a given A ∞ -category C and its full subcategory B. The construction is similar to a particular case of Drinfeld’s construction of the quotient of differential graded categories [Dri04]. We use D(C|B) to construct an A ∞ -functor of K-injective resolutions of a complex, when the ground ring is a field. The conventional derived category is obtained as th...