Inductive Construction of Nilpotent Modules of Quantum Groups at Roots of Unity

素数单位根处的有色 Khovanov 同调理论

Colored Khovanov homology theories at prime roots of unity

本文为有色 Khovanov 同调引入次数 2 导子，在特征 p>2 域上得到 p-幂零微分，实现 2p 次单位根处 gl(2) 同调的范畴化。

Quantum-sl(2) action on a divided-power quantum plane at even roots of unity

We describe a nonstandard version of the quantum plane, the one in the basis of divided powers at an even root of unity $q=e^{iπ/p}$. It can be regarded as an extension of the "nearly commutative" algebra $C[X,Y]$ with $X Y =(-1)^p Y X$ by nilpotents. For this quantum plane, we construct a Wess--Zumino-type de Rham complex and find its decomposition into representations of the $2p^3$-dimensional quantum group $U_q sl(2)$ and its Lusztig extension; the quantum group action is also defined on the algebra of quantum differential operators on the quantum plane.

On the automorphism groups of q-enveloping algebras of nilpotent Lie algebras

We investigate the automorphism group of the quantised enveloping algebra U of the positive nilpotent part of certain simple complex Lie algebras g in the case where the deformation parameter q \in \mathbb{C}^* is not a root of unity. Studying its action on the set of minimal primitive ideals of U we compute this group in the cases where g=sl_3 and g=so_5 confirming a Conjecture of Andruskiewitsch and Dumas regarding the automorphism group of U. In the case where g=sl_3, we retrieve the description of the automorphism group of the quantum Heisenberg algebra that was obtained independently by A…

Evaluable Jones-Wenzl idempotents at root of unity and modular representation on the center of U_qsl(2)

Let p an integer. We define a family of idempotents (and nilpotents) in the Temperley - Lieb algebras at 4p-th roots of unity which generalize the usual Jones-Wenzl idempotents. These new idempotents correspond to finite dimentional simple and projective indecomposable representations of the restricted quantum group Uqsl(2), where q is a 2p-th root of unity. In the manner of the [BHMV95] topological quantum field theorie (TQFT), they provide a canonical basis in colored skein modules to define mapping class groups representations. In the torus case, this basis allows us to obtain a partial mat…

Cohomology of subregular tilting modules for small quantum groups

Let $U$ be the quantum group with divided powers in $l-$th root of unity and let $u\subset U$ be the Frobenius kernel. V.Ginzburg and S.Kumar proved that the cohomology algebra of $u$ with trivial coefficients is isomorphic to the functions algebra of the nilpotent cone of the corresponding Lie algebra. In this note we show that there exists tilting module $T$ such that the cohomology of $u$ with coefficients in $T$ is isomorphic to the functions algebra of the closure of the subregular nilpotent orbit.

Anomaly-free TQFTs from the super Lie algebra $\mathfrak{sl}(2|1)$

It is known that the category $\mathscr{C}^H$ of nilpotent weight modules over the quantum group associated with the super Lie algebra $\mathfrak{sl}(2|1)$ is a relative pre-modular $G$-category. Its modified trace enables to define an invariant of $3$-manifolds. In this article we show that the category $\mathscr{C}^H$ is a relative modular $G$-category which allows one to construct a family of non-semi-simple extended topological quantum field theories which surprisingly are anomaly free. The quantum group associated with $\mathfrak{sl}(2|1)$ is considered at odd roots of unity.

双参数量化代数 $U^+_{r,s}(B_2)$ 的单模与 PI 结构

Simple Modules and PI Structure of the Two-Parameter Quantized Algebra $U^+_{r,s}(B_2)$

本文研究单位根处双参数量子代数 $U^+_{r,s}(B_2)$ 的 PI 结构，借助子代数 $B$ 与正规元完整分类其有限维单模。

Support theory for the small quantum group and the Springer resolution

We consider the small quantum group u_q(G), for an almost-simple algebraic group G over the complex numbers and a root of unity q of sufficiently large order. We show that the Balmer spectrum for the small quantum group in type A admits a continuous surjection P(Ñ) \to Spec(stab u_q(G)) from the (projectivized) Springer resolution. This surjection is shown to be a homeomorphism over a dense open subset in the spectrum. In type A_1 we calculate the Balmer spectrum precisely, where it is shown to be the projectivized nilpotent cone. Our results extend to arbitrary Dynkin type provided certain co…

The Hall algebra of a cyclic quiver and canonical bases of the Fock space

We prove that the Hall algebra U^-_n of the cyclic quiver of type A^(1)_{n-1} decomposes as a direct product of the quantum negative nilpotent subalgebra U_q^-(\hat{sl}_{n)) and C[q,q^{-1},z_1,z_2...]. We use this to prove a conjecture of Varagnolo and Vasserot in math/9803023 relating the canonical basis of U^-_n and the canonical basis of the level 1 Fock space representation of U_q(\hat{sl}_n) introduced by Leclerc and Thibon. This yields a proof of the positivity conjecture of Lascoux, Leclerc and Thibon, and a q-analogue of the Lusztig character formula for simple modules of the quantum g…

Non semi-simple TQFTs, Reidemeister torsion and Kashaev's invariants

We construct and study a new family of TQFTs based on nilpotent highest weight representations of quantum sl(2) at a root of unity indexed by generic complex numbers. This extends to cobordisms the non-semi-simple invariants defined in (arXiv:1202.3553) including the Kashaev invariant of links. Here the modular category framework does not apply and we use the ``universal construction''. Our TQFT provides a monoidal functor from a category of surfaces and their cobordisms into the category of graded finite dimensional vector spaces and their degree 0-morphisms and depends on the choice of a roo…

Quantum Groups, the loop Grassmannian, and the Springer resolution

We establish equivalences of derived categories of the following 3 categories: (1) Principal block of representations of the quantum at a root of 1; (2) G-equivariant coherent sheaves on the Springer resolution; (3) Perverse sheaves on the loop Grassmannian for the Langlands dual group. The equivalence (1)-(2) is an `enhancement' of the known expression for quantum group cohomology in terms of nilpotent variety, due to Ginzburg-Kumar. The equivalence (2)-(3) is a step towards resolving an old mystery surrounding the existense of two completely different realizations of the affine Hecke algebra…