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Let |$R$| be a smooth affine domain of dimension |$d\geq 3$| over |$\overline{{\mathbb{F}}}_p$| with |$p\neq 2$|⁠. Let |$P$| be a projective |$R$| -module of rank |$d-1$| with trivial determinant. We prove that |$P$| splits off a free summand of rank 1 if and only if |$P$| surjects onto a complete intersection ideal of height |$d-1$|⁠. [ABSTRACT FROM AUTHOR]

We prove a uniqueness theorem for irreducible non-critical Gelfand–Tsetlin modules. The uniqueness result leads to a complete classification of the irreducible Gelfand–Tsetlin modules with 1-singularity. An explicit construction of such modules was given in Futorny et al. [ 7 ]. In particular, we show that the modules constructed in Futorny et al. [ 7 ] exhaust all irreducible Gelfand–Tsetlin modules with 1-singularity. To prove the result, we introduce a new category of modules (called Drinfeld category) related to the Drinfeld generators of the Yangian Y(gln) and define a functor from the category of non-critical Gelfand–Tsetlin modules to the Drinfeld category. [ABSTRACT FROM AUTHOR]

Over a connected geometrically unibranch scheme X of finite type over a finite field, we show, with purely geometric arguments, finiteness of the number of irreducible Qℓ-lisse sheaves, with bounded rank and bounded ramification in the sense of Drinfeld, up to twist by a character of the finite field. On X smooth, with bounded ramification in the sense of [7, Definition 3.6], this is Deligne's theorem [7, Theorem 1.1], the proof of which uses the whole strength of [3] and [6]. We also generalize Deligne's theorem [7, Theorem 1.1] from X smooth to X normal, using Deligne's theorem (not reproving it) and a few more geometric arguments. [ABSTRACT FROM AUTHOR]

We study the derived representation scheme DRepg(a) parameterizing the representations of a Lie algebra a in a reductive Lie algebra g. In our earlier work [2], we defined two canonical maps Trg(a): HC(r)∙ (a) → H∙[DRepg(a)]G and Φg(a): H∙[DRepg(a)]G → H∙[DReph(a)]W, called the Drinfeld trace and the derived Harish-Chandra homomorphism, respectively. In this paper, we give general formulas for these maps in terms of Chern-Simons forms. As a consequence, we show that, if a is an abelian Lie algebra, the composite map Φg(a) · Trg(a) is given by a canonical differential operator defined on differential forms on A = Sym(a) and depending only on the Cartan data (h,W,P), where P ∈ Sym(h*)W. We derive a combinatorial formula for this operator that plays a key role in the study of derived commuting schemes in [2]. [ABSTRACT FROM AUTHOR]

The universal R operator for the positive representations of split real quantum groups is computed, generalizing the formula of compact quantum groups Uq(g) by Kirillov-Reshetikhin and Levendorski-Soibelman, and the formula in the case of Uqq(sl(2, R)) by Faddeev, Kashaev, and Bytsko-Teschner. Several new functional relations of the quantum dilogarithm are obtained, generalizing the quantum exponential relations and the pentagon relations. The quantum Weyl element and Lusztig's isomorphism in the positive setting are also studied in detail. Finally, we introduce a C-algebraic version of the split real quantum group in the language of multiplier Hopf algebras, and consequently the definition of R is made rigorous as the canonical element of the Drinfeld's double U of certain multiplier Hopf algebra Ub. Moreover, a ribbon structure is introduced for an extension of U. [ABSTRACT FROM AUTHOR]

In this paper, we introduce certain new features of the shuffle algebra of [3] that will allow us to obtain explicit formulas for the isomorphism between its Drinfeld double and the elliptic Hall algebra of [1, 7]. These results are necessary for our work in [4, 6], where they will be applied to the study of the Hilbert scheme and to computing knot invariants. [ABSTRACT FROM PUBLISHER]

Denis associated to each Drinfeld module ϕover a global function field La canonical height function , which plays a role analogous to that of the Néron–Tate height in the context of elliptic curves. We prove that there exists a constant ε>0, depending only on the number of places at which ϕhas bad reduction, such that either x∈ϕ(L) is a torsion point of bounded order, or else , where jϕ and are analogs of the j-invariant and minimal discriminant of an elliptic curve. As an application, we make some observations about specializations of one-parameter families of Drinfeld modules. [ABSTRACT FROM PUBLISHER]

We study the ring of quasi-endomorphisms for certain infinitely definable subgroups in separably closed fields. Based on the results we obtain, we are able to prove a Mordell-Lang theorem for Drinfeld modules of finite characteristic. Using specialization arguments, we prove also a Mordell-Lang theorem for Drinfeld modules of generic characteristic. [ABSTRACT FROM AUTHOR]

We prove a vanishing theorem of the cohomology arising from the two quantized Drinfeld-Sokolov reductions (“+” and “−” reductions) introduced by FeĬgin and Frenkel, and by Frenkel, Kac, and Wakimoto. As a consequence, the vanishing conjecture of Frenkel, Kac, and Wakimoto is proved for the “−” reduction and partially for the “+” reduction. [ABSTRACT FROM AUTHOR]

The article presents a study on Drinfeld modules without prime numbers of supersingular reduction. It mentions the argument of mathematics professor Noam Elkies on elliptic curves over number fields with a real place. It also notes the ring of linear endomorphisms as the ring of linear polynomials identified by their twisted counterparts.

The article discusses the cohomology of Laumon's spaces and the intersection cohomology of Drinfeld's quasimaps' spaces. It states that the compactifications of quasimaps, quasiflags and stable maps were formulated by Drinfeld, Laumon and Kontsevich. It also mentions that the cohomology of quasiflags takes a cell decomposition and has a pure Tate-Hodge structure.

The article presents a report on the cohomology of Drinfel'd algebra. The authors describe the deformations of Drinfel'd algebras and reduces to the standard cohomology in the case of bialgebras. They also show that the existence and essential uniqueness of the various constructions involved in the definition of such a theory follow from some abstract arguments involving monoidal categories and free resolutions.

This article is divided into two parts. In the first part we endow a certain ring of “Drinfeld quasi-modular forms” for (where q is a power of a prime) with a system of “divided derivatives” (or hyperderivations). This ring contains Drinfeld modular forms as defined by Gekeler in 7, and the hyperdifferential ring obtained should be considered as a close analogue in positive characteristic of Ramanujan's famous differential system relating to the first derivatives of the classical Eisenstein series of weights 2, 4, and 6. In the second part of this article, we prove that, when q ≠ 2, 3, if is a nonzero hyperdifferential prime ideal, then it contains the Poincaré series h = Pq+1,1 of 7. This last result is the analogue of a crucial property proved by Nesterenko 12 in characteristic zero in order to establish a multiplicity estimate. [ABSTRACT FROM PUBLISHER]