Quantum Superalgebras at Roots of Unity and Topological Invariants of Three-manifolds
Sacha C. Blumen
备注：
255 pages, 26 figures, PhD thesis, accepted December 2005, School of Mathematics and Statistics, University of Sydney. Can also be downloaded as a pdf file from www.maths.usyd.edu.au
The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudo-modular Hopf algebras. Pseudo-modular Hopf algebras are a class of Z_2-graded ribbon Hopf algebras that generalise the concept of a modular Hopf algebra.
The quantum superalgebra U_q(osp(1|2n)) over C is considered with q a primitive N^th root of unity for all integers N >= 3. For such a q, a certain left ideal I of U_q(osp(1|2n)) is also a two-sided Hopf ideal, and the quotient algebra U_q^(N)(osp(1|2n)) = U_q(osp(1|2n)) / I is a Z_2-graded ribbon Hopf algebra.
For all n and all N >= 3, a finite collection of finite dimensional representations of U_q^(N)(osp(1|2n)) is defined. Each such representation of U_q^(N)(osp(1|2n)) is labelled by an integral dominant weight belonging to the truncated dominant Weyl chamber. Properties of these representations are considered: the quantum superdimension of each representation is calculated, each representation is shown to be self-dual, and more importantly, the decomposition of the tensor product of an arbitrary number of such representations is obtained for even N.
It is proved that the quotient algebra U_q^(N)(osp(1|2n)), together with the set of finite dimensional representations discussed above, form a pseudo-modular Hopf algebra when N >= 6 is twice an odd number.
Using this pseudo-modular Hopf algebra, we construct a topological invariant of 3-manifolds. This invariant is shown to be different to the topological invariants of 3-manifolds arising from quantum so(2n+1) at roots of unity.
We consider fundamental facts from the theory of Hopf superalgebras. We use them to construct the quantum double of the quantum superalgebra $sl(2|1)$ at roots of unity. Thus we obtain a multiplicative formula for universal $R$-matrix. Next we construct an $R$-matrix to investigate parametrized family of centralizer algebras. We give multiplication laws in particular case and describe a structure of such algebras in the general case.
We summarize the definition of the Weyl groupoid using supercategory approach in order to investigate quantum superalgebras at roots of unity. We show how the structure of a Hopf superalgebra on a quantum superalgebra is determined by the quantum Weyl groupoid. The Weyl groupoid of $\mathfrak{sl}(m|n)$ is constructed to this end as some supercategory. We prove that in this case quantum superalgebras associated with Dynkin diagrams are isomorphic as superalgebras. It is shown how these quantum superalgebras considered as Hopf superalgebras are connected via twists and isomorphisms. We explicitl…
We summarize the definition of the Weyl groupoid using supercategory approach in order to investigate quantum superalgebras at roots of unity. We show how the structure of a Hopf superalgebra on a quantum superalgebra is determined by the quantum Weyl groupoid. The Weyl groupoid of $\mathfrak{sl}(m|n)$ is constructed to this end as some supercategory. We prove that in this case quantum superalgebras associated with Dynkin diagrams are isomorphic as superalgebras. It is shown how these quantum superalgebras considered as Hopf superalgebras are connected via twists and isomorphisms. We explicitl…
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 prove the unrolled superalgebra $\mathcal{U}_ξ^{H}\mathfrak{sl}(2|1)$ has a completion which is a ribbon superalgebra in a topological sense where $ξ$ is a root of unity of odd order. Using this ribbon superalgebra we construct its universal invariant of links. We use it to construct an invariant of $3$-manifolds of Hennings type.
We define root graded Lie superalgebras and study their connection with centerless cores of extended affine Lie superalgebras; our definition generalizes the known notions of root graded Lie superalgebras.
We recall the relation between the Lie superalgebra $osp(1/2n)$ and para-Bose operators. The quantum superalgebra $U_q[osp(1/2n)]$, defined as usual in terms of its Chevalley generators, is shown to be isomorphic to an associative algebra generated by so-called pre-oscillator operators satisfying a number of relations. From these relations, and the analogue with the non-deformed case, one can interpret these pre-oscillator operators as deformed para-Bose operators. Some consequences for $U_q[osp(1/2n)]$ (Cartan-Weyl basis, Poincaré-Birkhoff-Witt basis) and its Hopf subalgebra $U_q[gl(n)]$ are …
We consider integrable vertex models whose Boltzmann weights (R-matrices) are trigonometric solutions to the graded Yang-Baxter equation. As is well known the latter can be generically constructed from quantum affine superalgebras $U_{q}(\hat g)$. These algebras do not form a symmetry algebra of the model for generic values of the deformation parameter $q$ when periodic boundary conditions are imposed. If $q$ is evaluated at a root of unity we demonstrate that in certain commensurate sectors one can construct non-abelian subalgebras which are translation invariant and supercommute with the tra…
It is shown that the superalgebra of observables of the rational Calogero model based on the root system of I_2(n) type possesses [(n+1)/2] supertraces. Model with three-particle interaction based on the root system G_2 belongs to this class of models and its superalgebra of observables has 3 independent supertraces.
It is shown that the Clifford superalgebra Cl(n|m) generated by m pairs of Bose operators (odd elements) anticommuting with n pairs of Fermi operators (even elements) can be deformed to Cl_q(n|m) such that the latter is a homomorphic image of the quantum superalgebra U_q[osp(2n+1|2m)]. The Fock space F(n|m) of Cl_q(n|m) is constructed. At q being a root of unity (q=\exp ({iπl/k})) q-bosons (and q-fermions) are operators acting in a finite-dimensional subspace F_{l/k}(n|m) of F(n|m). Each F_{l/k}(n|m) is turned through the above mentioned homomorphism into an irreducible (root of unity) U_q[osp…
Brundan and Kleshchev showed that some parts of the representation theory of the affine Hecke-Clifford superalgebras and its finite-dimensional "cyclotomic" quotients are controlled by the Lie theory of type $A^{(2)}_{2l}$ when the quantum parameter $q$ is a primitive $(2l+1)$-th root of unity. We show in this paper that similar theorems hold when $q$ is a primitive $4l$-th root of unity by replacing the Lie theory of type $A^{(2)}_{2l}$ with that of type $D^{(2)}_{l}$.
In this article we construct link invariants and 3-manifold invariants from the quantum group associated with Lie superalgebra $\mathfrak{sl}(2|1)$. This construction based on nilpotent irreducible finite dimensional representations of quantum group $\mathcal{U}_ξ\mathfrak{sl}(2|1)$ where $ξ$ is a root of unity of odd order. These constructions use the notion of modified trace and relative $\mathit{G}$-modular category.
The purposes of this paper is to investigate the properties of the quantum extended strange superalgebra $\tilde{P}_{Q}(n)$ when his deformation parameter $Q$ goes to a root of unity.
We show how to define invariants of graphs related to quantum $\mathfrak{sl}(2)$ when the graph has more then one connected component and components are colored by blocks of representations with zero quantum dimensions.
The unitarizable irreps of the deformed para-Bose superalgebra $pB_q$, which is isomorphic to $U_q[osp(1/2)]$, are classified at $q$ being root of 1. New finite-dimensional irreps of $U_q[osp(1/2)]$ are found. Explicit expressions for the matrix elements are written down.
We obtain a family of skein identities in the Kauffman bracket skein module which relate Frobenius elements to Jones-Wenzl projectors at roots of unity. We view these skein identities as certain incarnations of Steinberg tensor product formulae from the theory of tilting modules of the quantum group $U_q(sl_2)$. We show that the simplest skein identities yield a short new proof of the existence of the Chebyshev-Frobenius homomorphism of Bonahon-Wong.
We examine two different m-traces in the category of representations over the quantum Lie superalgebra associated to $\mathfrak{sl}(m|n)$ at root of unity. The first m-trace is on the ideal of projective modules and leads to new Extended Topological Quantum Field Theories. The second m-trace is on the ideal of perturbative typical modules. We consider the quotient with respect to negligible morphisms coming from this m-trace and show that in the case of $\mathfrak{sl}(2|1)$ this quotient leads to 3-manifolds invariants. We conjecture that the quotient category of perturbatives over quantum $\m…
We give shuffle algebra realization of positive part of quantum affine superalgebra $U_{v}(\widehat{\mathfrak{D}}(2,1;θ))$ associated to any simple root systems. We also determine the shuffle algebra associated to $\widehat{\mathfrak{sl}}(2|1)$ with odd root system when $v$ is a primitive root of unity of even order, generalizing results in \cite{FJMMT03}.
New solutions of the quantum Yang-Baxter equation, depending in general on three arbitrary parameters, are written down. They are based on the root of unity representations of the quantum orthosymplectic superalgebra \\U, which were found recently. Representations of the braid group $B_N$ are defined within any $N^{th}$ tensorial power of root of 1 \\U modules.
The interaction of a Lie algebra $\LL,$ having a weight space decomposition with respect to a nonzero toral subalgebra, with its corresponding root system forms a powerful tool in the study of the structure of $\LL.$ This, in particular, suggests a systematic study of the root system apart from its connection with the Lie algebra. Although there have been a lot of researches in this regard on Lie algebra level, such an approach has not been considered on Lie superalgebra level. In this work, we introduce and study extended affine root supersystems which are a generalization of both affine refl…
We introduce the notion of extended affine Lie superalgebras and investigate the properties of their root systems. Extended affine Lie algebras, invariant affine reflection algebras, finite dimensional basic classical simple Lie superalgebras and affine Lie superalgebras are examples of extended affine Lie superalgebras.
We consider various specializations of the non-twisted quantum affine algebras at roots of unity. We define and study the q-characters of their finite-dimensional representations.
We prove the abelian category of the modules over triplet VOA W(p) is category equivalent to the abelian category of the modules over quantum algebra of type sl_2 at root of unity.
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.
In a recent article, the coordinate ring of the nodal cubic was given the structure of a quantum homogeneous space. Here the corresponding coalgebra Galois extension is expressed in terms of quantum groups at roots of unity, and is shown to be cleft. Furthermore, the minimal quotient extensions are determined.
Inspired by a profound observation on the Racah--Wigner coefficients of $U_q(\mathfrak{sl}_2)$, the Askey--Wilson algebras were introduced in the early 1990s. A universal analog $\triangle_q$ of the Askey--Wilson algebras was recently studied. For $q$ not a root of unity, it is known that $Z(\triangle_q)$ is isomorphic to the polynomial ring of four variables. A presentation for $Z(\triangle_q)$ at $q$ a root of unity is displayed in this paper. As an application, a presentation for the center of the double affine Hecke algebra of type $(C_1^\vee,C_1)$ at roots of unity is obtained.
We discuss the relationship between Hopf superalgebras and Hopf algebras. We list the braided vector spaces of diagonal type with generalized root system of super type and give the defining relations of the corresponding Nichols algebras.
In this paper, we study finite dimensional quotients of the Hopf algebra U_{q}(sl_{2}) at fourth root of unity. We give explicitly all the corresponding automorphisms, real forms and idempotents.
The structure of the tensor product representation v_{λ_1}(x)\otimes V_{λ_2}(y) of U_q(\hat sl_2) is investigated at roots of unity. A polynomial identity is derived as an outcome. Also, new bases of v_{λ_1}(x)\otimes V_{λ_2}(y) are established under certain conditions.
Here is a list of chapters: 1 Introduction 2 Notation and preliminaries Part I: Finite quantum groups 3 2x2 Matrix quantum groups and the quantum plane 4 Quantum enveloping algebras at a root of unity Part II: q-Oscillators 5 Representations of q-oscillators at a root of unity 6 qr-Oscillator at a root of unity Part III: Infinite quantum groups 7 Quantum affine algebras 8 Quantum affine algebras at a root of unity
The quantum group SL_q(2,R) at roots of unity is introduced by means of duality pairings with the quantum algebra U_q(sl(2,R)). Its irreducible representations are constructed through the universal T-matrix. An invariant integral on this quantum group is given. Endowed with that some properties like unitarity and orthogonality of the irreducible representations are discussed.
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 compute the center and Azumaya locus in the simplest non-abelian examples of quantized multiplicative quiver varieties at a root of unity: quantum Weyl algebras of rank $N$, and quantum differential operators on the quantum group $\mathrm{GL}_2$. These examples illustrate in elementary terms much more general phenomena explored further in [Ganev-Jordan-Safronov 2019].
It is shown that when q is a primitive root of unity of order not equal to 2 mod 4, A(SL_q(2)) is a free module of finite rank over the coordinate ring of the classical group SL(2). An explicit set of generators is provided.
We survey some recent results in hopfological algebra and the program of categorification at prime roots of unity. A categorical Jones-Wenzl projector at prime roots of unity is studied, and it is shown that this projector is hopfologically finite in special characteristics, while generically it is infinite.
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.
Hall-Littlewood functions indexed by rectangular partitions, specialized at primitive roots of unity, can be expressed as plethysms. We propose a combinatorial proof of this formula using A. Schilling's bijection between ribbon tableaux and ribbon rigged configurations.
A group theoretical understanding of the two dimensional fractional supersymmetry is given in terms of the quantum Poincare group at roots of unity. The fractional supersymmetry algebra and the quantum group dual to it are presented and the pseudo-unitary, irreducible representations of them are obtained. The matrix elements of these representations are explicitly constructed.
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.
The general formula for R-matrices of slq(2,C) for the highest weight repre- sentations both for general q and for q being a root of unity by generalizing G.Gomez's and G.Sierra's one for semiperiodic representations of slq(2,C) at roots of unity is presented.
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.
We give a combinatorial algorithm for computing Zelevinsky's involution of the set of isomorphism classes of irreducible representations of the affine Hecke algebra $\H_m(t)$ when $t$ is a primitive $n$th root of 1. We show that the same map can also be interpreted in terms of aperiodic nilpotent orbits of $\Zb/n\Zb$-graded vector spaces.
In this short note, we show that the Ginzburg-Vasserot map between the quantum affine algebra of type A_(n-1) and the equivariant K-theory group of the Steinberg Variety (of n-step flags in C^d) restricts and remains surjective at the level of the integral forms. In particular, this shows that the parametrization of irreducible, finite-dimensional modules and the Kazhdan-Lusztig multiplicity formulas are valid for the (restricted) specialization at roots of unity.
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.
We study the root of unity degeneration of cluster algebras and quantum dilogarithm identities. We prove identities for the cyclic dilogarithm associated with a mutation sequence of a quiver, and as a consequence new identities for the non-compact quantum dilogarithm at $b=1$.
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.
A description of the orthosymplectic Lie superalgebra $osp(2n+1/2m)$ and also of its $q-$deformed analogue $U_q[osp(2n+1/2m)]$ in terms of a new set of generators, called Green generators, is given. These generators are very different form the well known Chevalley generators. Mathematically the Green generators are the root vectors, corresponding to the positive and the negative orthogonal roots. Physically, they consist of $m$ pairs of (deformed) para-Bose operators and $n$ pairs of (deformed) para-Fermi operators.
We describe the structure of the Grothendieck ring of projective modules of basic Hopf algebras using a positive integer determined by the composition series of the principal indecomposable projective module.
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.
