In the paper \cite{BK} we defined categories of equivariant quantum $\mathcal{O}_q$-modules and $\mathcal{D}_q$-modules on the quantum flag variety of $G$. We proved that the Beilinson-Bernstein localization theorem holds at a generic $q$. Here we prove that a derived version of this theorem holds at the root of unity case. Namely, the global section functor gives a derived equivalence between category of $U_q$-modules and $\mathcal{D}_q$-modules on the quantum flag variety.
For this we first prove that $\mathcal{D}_q$ is an Azumaya algebra over an open subset ofthe cotangent bundle $T^\star X$ of the classical (char 0) flag variety $X$. This way we get a derived equivalence between representations of $U_q$ and certain $\mathcal{O}_{T^\star X}$-modules.
In the paper \cite{BMR} similar results were obtained for a Lie algebra $\g_p$ in char $p$. Hence, representations of $\g_p$ and of $U_q$ (when $q$ is a p'th root of unity) are related via the cotangent bundles $T^\star X$ in char 0 and in char $p$, respectively.
In this article, the quantum spatial ageing algebra due to V. V. Bavula and T. Lu has been studied and a full classification of simple modules are given at root of unity
We will classify finite dimensional irreducible modules for affine quantum Schur algebras at roots of unity and generalize \cite[(6.5f) and (6.5g)]{Gr80} to the affine case in this paper.
We will give an explicit description of the center of the De Concini-Kac type specialization of a quantized enveloping algebra at an even root of unity. The case of an odd root of unity was already dealt with by De Concini-Kac-Procesi. Our description in the even case is similar to but a little more complicated than the odd case.
In this paper, we consider the necessary and sufficient conditions for the tensor product of the fundamental representations for the restricted quantum loop algebras of type A at roots of unity to be irreducible.
We show that the quantized flag manifold at a root of unity has natural affine open covering parametrized by the elements of the Weyl group. In particular, the quantized flag manifold turns out to be a quasi-scheme in the sense of Rosenberg.
The purpose of this paper is to prove that we can construct all finite dimensional irreducible nilpotent modules of type 1 inductively by using Schnizer homomorphisms for quantum algebra at roots of unity of type A, B, C, D or G.
Specializing properly the parameters contained in the maximal cyclic representation of the non-restricted A-type quantum algebra at roots of unity, we find the unique primitive vector in it. We show that the submodule generated by the primitive vector can be identified with an irreducble highest weight module of the finite dimensional A-type quantum algebra which is defined as the subalgebra of the restricted quantum algebra at roots of unity.
Properly specializing the parameters in ``Schnizer modules'', for type A,B,C and D, we get its unique primitive vector. Then we show that the module generated by the primitive vector is an irreducible highest weight module of finite dimensional classical quantum groups at roots of unity.
In this article the quantized matrix algebras as in the title have been studied at a root of unity. A full classification of simple modules over such quantized matrix algebras of rank $2$ along with a class of finite dimensional indecomposable modules are given.
The purpose of this paper is to compute the Drinfel'd polynomials for two types of evaluation representations of quantum affine algebras at roots of unity and construct those representations as the submodules of evaluation Schnizer modules. Moreover, we obtain the necessary and sufficient condition for that the two types of evaluation representations are isomorphic to each other.
For a finite dimensional semisimple Lie algebra and a root of unity, Lusztig defined an infinite dimensional quantum group of divided powers. Under certain restrictions on the order of the root of unity, he constructed a Frobenius homomorphism with finite dimensional Hopf kernel and with image the universal enveloping algebra. In this article we define and describe the Frobenius homomorphism for arbitrary roots of unity by systematically using the theory of Nichols algebras. In several new exceptional cases the Frobenius-Lusztig kernel is associated to a different Lie algebra than the initial …
We study the bimodule structure of the quantum function algebra at roots of 1 and prove that it admits an increasing filtration with factors isomorphic to the tensor products of the dual of Weyl modules $V_λ^* \otimes V_{- ω_0 λ}^*$. As an application we compute the 0-th Hochschild cohomology of the function algebra at roots of 1.
We give a representation-theoretic interpretation of the Langlands character duality of Frenkel and Hernandez, and show that the "Langlands branching multiplicities" for symmetrizable Kac-Moody Lie algebras are equal to certain tensor product multiplicities. For finite type quantum groups, the connection with tensor products can be explained in terms of tilting modules.
Using vertex operators, we construct explicitly Lusztig's $\mathbb Z[q, q^{-1}]$-lattice for the level one irreducible representations of quantum affine algebras of ADE type. We then realize the level one irreducible modules at roots of unity and show that the q-dimension is still given by the Weyl-Kac character formula. As a consequence we also answer the corresponding question of realizing the affine Kac-Moody Lie algebras of simply laced type at level one in finite characteristic.
A conjecture of De Concini Kac and Procesi provides a bound on the minimal possible dimension of an irreducible module for quantized enveloping algebras at an odd root of unity. We pose the problem of the existence of modules whose dimension equals this bound. We show that this question cannot have a positive answer in full generality and discuss variants of this question.
We construct a p-DG structure on an algebra Koszul dual to a zigzag algebra used by Khovanov and Seidel to construct a categorical braid group action. We show there is a braid group action in this p-DG setting.
Noncompact forms of the Drinfeld-Jimbo quantum groups U_q(g) with (H_i)* = H_i, (X_i^{+-})* = s_i X_i^{-+} for s_i= +-1 are studied at roots of unity. This covers g = so(n,2p), su(n,p), so*(2l), sp(n,p), sp(l,R), and exceptional cases. Finite-dimensional unitary representations are found for all these forms, for even roots of unity. Their classical symmetry induced by the Frobenius-map is determined, and the meaning of the extra quasi-classical generators appearing at even roots of unity is clarified. The unitary highest weight modules of the classical case are recovered in the limit q -> 1.
We discuss different invariants of knots and links that depend on a primitive root of unity. We clarify the definitions of existing invariants with the Reshetikhin-Turaev method, present the generalization of ADO invariants to $\mathcal{U}_q(sl_N)$ and highlight the connections between different invariants.
There is a $p$-differential on the triply-graded Khovanov--Rozansky homology of knots and links over a field of positive characteristic $p$ that gives rise to an invariant in the homotopy category finite-dimensional $p$-complexes. A differential on triply-graded homology discovered by Cautis is compatible with the $p$-differential structure. As a consequence we get a categorification of the colored Jones polynomial evaluated at a $2p$th root of unity.
Under the assumption that the quantum parameter $q$ is an $l$-th primitive root of unity with $l$ odd in a field $F$ of characteristic 0 and $m+n\geq r$, we obtained a complete classification of irreducible modules of the $q$-Schur superalgebra introduced H. Rui and the first Author.
We give a diagrammatic presentation of the category of $\textbf{U}_q(\mathfrak{sl}_2)$-tilting modules $\mathfrak{T}$ for $q$ being a root of unity and introduce a grading on $\mathfrak{T}$. This grading is a "root of unity phenomenon" and might lead to new insights about link and $3$-manifold invariants deduced from $\mathfrak{T}$. We also give a diagrammatic category for the (graded) projective endofunctors on $\mathfrak{T}$, indicate how our results could generalize and collect some "well-known" facts to give a reasonably self-contained exposition.
Following ideas of Geck and Rouquier, we show that there exists a ``canonical basic set'' of Specht modules in bijection with the simple modules of Ariki-Koike algebras at roots of unity. Moreover, we determine the parametrization of this set and we give the consequences of these results on the representation theory of Ariki-Koike algebras.
In this paper we study finite p-solvable groups having irreducible complex characters chi in Irr(G) which take roots of unity values on the p-singular elements of G.
Given a finite set of roots of unity, we show that all power sums are non-negative integers iff the set forms a group under multiplication. The main argument is purely combinatorial and states that for an arbitrary finite set system the non-negativity of certain alternating sums is equivalent to the set system being a filter. As an application we determine all discrete Fourier pairs of $\{0,1\}$-matrices. This technical result is an essential step in the classification of $R$-matrices of quantum groups.
We construct quantizations of multiplicative hypertoric varieties using an algebra of q-difference operators on affine space, where q is a root of unity in C. The quantization defines a matrix bundle (i.e. Azumaya algebra) over the multiplicative hypertoric variety and admits an explicit finite étale splitting. The global sections of this Azumaya algebra is a hypertoric quantum group, and we prove a localization theorem. We introduce a general framework of Frobenius quantum moment maps and their Hamiltonian reductions; our results shed light on an instance of this framework.
Let H be a Hopf algebra which is a finite module over a central sub-Hopf algebra R. We continue the study of such algebras begun in RT/9911234, concentrating in this case on the example of $O_ε[G]$, a quantised function algebra at root of unity. In particular we determine the representation type and block structure of the family of reduced quantised function algebras, and describe many of them up to isomorphism. A series of parallel results is obtained for the quantised Borel algebras $U_ε^{\geq 0}$.
In this paper, we will fully describe the representations of the crystallographic rank two affine Hecke algebras using elementary methods, for all possible values of q. The focus is on the case when q is a root of unity of small order.
We describe the cohomology of the sheaf of twisted differential operators on the quantized flag manifold at a root of unity whose order is a prime power. It follows from this and our previous results that for the De Concini-Kac type quantized enveloping algebra, where the parameter $q$ is specialized to a root of unity whose order is a prime power, the number of irreducible modules with a certain specified central character coincides with the dimension of the total cohomology group of the corresponding Springer fiber. This gives a weak version of a conjecture of Lusztig concerning non-restrict…
For any finite group $G$, Thompson proved that, for each $χ\in {\rm Irr}(G)$, $χ(g)$ is a root of unity or zero for more than a third of the elements $g\in G$, and Gallagher proved that, for each larger than average class $g^G$, $χ(g)$ is a root of unity or zero for more than a third of the irreducible characters $χ\in {\rm Irr}(G)$. We show that in many cases "more than a third" can be replaced by "more than half".
We study the natural labeling of the one dimensional representations for Ariki-Koike algebras at roots of unity. For Hecke algebras of types A and B, some of these representations can be identified with the socle of the Steinberg representation of a finite reductive group. We here give closed formulas for them. This uses, in particular, several results concerning crystal isomorphisms and the Mullineux involution.
This paper studies the "reduction mod $p$" method, which constructs large classes of representations for a semisimple algebraic group $G$ from representations for the corresponding Lusztig quantum group $U_ζ$ at a $p^r$-th root of unity. The $G$-modules arising in this way include the Weyl modules, the induced modules, and various reduced versions of these modules. We present a relation between $\operatorname{Ext}^n_G(V,W)$ and $\operatorname{Ext}^n_{U_ζ}(V',W')$, when $V,W$ are obtained from $V',W'$ by reduction mod $p$. Since the dimensions of $\operatorname{Ext}^n$-spaces for $U_ζ$-modules …
To describe the representation theory of the quantum Weyl algebra at an $l$th primitive root $γ$ of unity, Boyette, Leyk, Plunkett, Sipe, and Talley found all nonsingular irreducible matrix solutions to the equation $yx-γxy=1$, assuming $yx\neq xy$. In this note, we complete their result by finding and classifying, up to equivalence, all irreducible matrix solutions $(X, Y)$, where $X$ is singular.
We introduce multi-colour partition algebras $P_{n,m}(δ_0, ..., δ_{m-1})$, which are generalization of both bubble algebras and partition algebras, then define the bubble algebra $T_{n,m}(δ_0, ..., δ_{m-1})$ as a sub-algebra of the algebra $P_{n,m}(δ_0, ..., δ_{m-1})$. We present general techniques to determine the structure of the bubble algebra over the complex field in the non-semisimple case.
We prove that large classes of algebras in the framework of root of unity quantum cluster algebras have the structures of maximal orders in central simple algebras and Cayley-Hamilton algebras in the sense of Procesi. We show that every root of unity upper quantum cluster algebra is a maximal order and obtain an explicit formula for its reduced trace. Under mild assumptions, inside each such algebra we construct a canonical central subalgebra isomorphic to the underlying upper cluster algebra, such that the pair is a Cayley-Hamilton algebra; its fully Azumaya locus is shown to contain a copy o…
In a previous paper we demonstrated that Bethe's equations are not sufficient to specify the eigenvectors of the XXZ model at roots of unity for states where the Hamiltonian has degenerate eigenvalues. We here find the equations which will complete the specification of the eigenvectors in these degenerate cases and present evidence that the $sl_2$ loop algebra symmetry is sufficiently powerful to determine that the highest weight of each irreducible representation is given by Bethe's ansatz.
