The Bershadsky--Polyakov algebras are the subregular quantum hamiltonian reductions of the affine vertex operator algebras associated with $\mathfrak{sl}_3$. In arXiv:2007.00396 [math.QA], we realised these algebras in terms of the regular reduction, Zamolodchikov's W$_3$-algebra, and an isotropic lattice vertex operator algebra. We also proved that a natural construction of relaxed highest-weight Bershadsky--Polyakov modules gives modules that are generically irreducible. Here, we prove that this construction, when combined with spectral flow twists, gives a complete set of irreducible weight modules whose weight spaces are finite-dimensional. This gives a simple independent proof of the main classification theorem of arXiv:2007.03917 [math.RT] for nondegenerate admissible levels and extends this classification to a category of weight modules. We also deduce the classification for the nonadmissible level $\mathsf{k}=-\frac{7}{3}$, which is new.
Anna Beliakova, Christian Blanchet, Thang T. Q. Le
For every rational homology 3-sphere with 2-torsion only we construct a unified invariant (which takes values in a certain cyclotomic completion of a polynomial ring), such that the evaluation of this invariant at any odd root of unity provides the SO(3) Witten-Reshetikhin-Turaev invariant at this root and at any even root of unity the SU(2) quantum invariant. Moreover, this unified invariant splits into a sum of the refined unified invariants dominating spin and cohomological refinements of quantum SU(2) invariants. New results on the Ohtsuki series and the integrality of quantum invariants are the main applications of our construction.
The universal enveloping algebra of any simple Lie algebra g contains a family of commutative subalgebras, called the quantum shift of argument subalgebras math.RT/0606380, math.QA/0612798. We prove that generically their action on finite-dimensional modules is diagonalizable and their joint spectra are in bijection with the set of monodromy-free opers for the Langlands dual group of G on the projective line with regular singularity at one point and irregular singularity of order two at another point. We also prove a multi-point generalization of this result, describing the spectra of commuting Hamiltonians in Gaudin models with irregular singulairity. In addition, we show that the quantum shift of argument subalgebra corresponding to a regular nilpotent element of g has a cyclic vector in any irreducible finite-dimensional g-module. As a byproduct, we obtain the structure of a Gorenstein ring on any such module. This fact may have geometric significance related to the intersection cohomology of Schubert varieties in the affine Grassmannian.
Twisting process for quantum linear spaces is defined. It consists in a particular kind of globally defined deformations on finitely generated algebras. Given a quantum space (A_1,A), a multiplicative cosimplicial quasicomplex C[A_1] in the category Grp is associated to A_1, in such a way that for every n a subclass of linear automorphisms of A^{\otimes n} is obtained from the groups C^n[A_1]. Among the elements of this subclass, the counital 2-cocycles are those which define the twist transformations. In these terms, the twisted internal coHom objects, constructed in a previous paper (cf. math.QA/0112233), can be described as twisting of the proper coHom objects, enabling us in turn to generalize the mentioned construction. The quasicomplexes C[V], V a vector space, are studied in detail, showing for instance that, when V is a coalgebra, the quasicomplexes related to Drinfeld twisting, corresponding to bialgebras generated by V, are subobjects of C[V].
The values at positive integers of the polyzeta functions are solutions of the polynomial equations arising from Drinfeld's associators, which have numerous applications in quantum algebra. Considered as iterated integrals they become periods of the motivic fundamental groupoid of $\mathbb{P}^1\setminus\{0,1,\infty\}$. From there comes a fundamental, yet no more explicit, system of algebraic relations; it implies the system of associators. We focus here on the combinatorics properties of another system of relations, the ``double shuffles'', which comes from elementary series and integrals manipulations. We show that it shares a torsor property with associators and ``motivic'' relations, is implied by the latter and defines a polynomial algebra over $\mathbb{Q}$ (a result first due to J. Ecalle). We obtain these results for more general numbers: values of Goncharov's multiple polylogarithms at roots of unity. These results were previously announced in math.QA:0012024. Here is the detailed proof.
Let $U$ and $A$ be algebras over a field $k$. We study algebra structures $H$ on the underlying tensor product $U{\otimes}A$ of vector spaces which satisfy $(u{\otimes}a)(u'{\otimes}a') = uu'{\otimes}aa'$ if $a = 1$ or $u' = 1$. For a pair of characters $ρ\in \Alg(U, k)$ and $χ\in \Alg(A, k)$ we define a left $H$-module $L(ρ, χ)$. Under reasonable hypotheses the correspondence $(ρ, χ) \mapsto L(ρ, χ)$ determines a bijection between character pairs and the isomorphism classes of objects in a certain category ${}_H\underline{\mathcal M}$ of left $H$-modules. In many cases the finite-dimensional objects of ${}_H\underline{\mathcal M}$ are the finite-dimensional irreducible left $H$-modules. In math.QA/0603269 we apply the results of this paper and show that the finite-dimensional irreducible representations of a wide class of pointed Hopf algebras are parameterized by pairs of characters.
Following the ideas of our previous works math.QA/0008232 (joint with Andruskiewitsch) and math.QA/0101049, we study families of triangular Hopf algebras obtained by twisting finite supergroups by a twist lying entirely in the odd part. These families are parametrized by data (G,V,u,B), where G is a finite group, V its finite dimensional representation, u a central element of G of order 2 acting by -1 on V, and B an element of S^2V. We fix the discrete data G,V,u, and find the set of isomorphism classes of the members of the family as Hopf algebras, in terms of the continuous parameter B. This set is often infinite, which provides examples of nontrivial continuous families of triangular Hopf algebras. The lowest dimension in which such a family occurs is 32, in which case we get 3 families which are dual to the 3 families of pointed Hopf algebras of dimension 32 constructed recently by Grana. Furthermore, we show that if (S^2V)^G=0 then such continuous families are nontrivial not only up to a Hopf algebra isomorphism, but also up to twisting of the multiplication. Thus, they provide counterexamples to Masuoka's weakened Kaplansky's 10th conjecture, which claims that up to twisting, there are finitely many types of Hopf algebras in each dimension. Finally, we study the algebra structure of the duals of our families, and show they are direct sums of Clifford algebras. Since there are finitely many types of Clifford algebras in each dimension, this allows us to construct nontrivial families of rigid tensor structures on the abelian category of modules over a finite dimensional algebra, with a fixed Grothendieck ring.
The Hopf algebra generated by the l-functionals on the quantum double C_q[G] \bowtie C_q[G] is considered, where C_q[G] is the coordinate algebra of a standard quantum group and q is not a root of unity. It is shown to be isomorphic to C_q[G]^op \bowtie U_q(g). This was conjectured by T. Hodges in [Ho]. As an algebra it can be embedded into U_q(g) \otimes U_q(g). Here it is proven that there is no bialgebra structure on U_q(g) \otimes U_q(g), for which this embedding becomes a homomorphism of bialgebras. In particular, it is not an isomorphism. As a preliminary a lemma of [Ho] concerning the structure of l-functionals on C_q[G] is generalized. For the classical groups a certain choice of root vectors is expressed in terms of l-functionals. A formula for their coproduct is derived.
Pavel Etingof, Olivier Schiffmann, Alexander Varchenko
In this paper we study twisted traces of products of intertwining operators for quantum affine algebras. They are interesting special functions, depending on two weights lambda, mu, three scalar parameters q, omega, k, and spectral parameters z_1,...,z_N, which may be regarded as q-analogs of conformal blocks of the Wess-Zumino-Witten model on an elliptic curve. It is expected that in the rank 1 case they essentially coincide with the elliptic hypergeometric functions defined in math.QA/0110081. Our main result is that after a suitable renormalization the traces satisfy four systems of difference equations -- the Macdonald-Ruijsenaars equation, the q-Knizhnik-Zamolodchikov-Bernard equation, and their dual versions. We also show that in the case when the twisting automorphism is trivial, the trace functions are symmetric under the permutation lambda <--> mu, k <--> omega. Thus, our results here generalize our previous results, dealing with the case q = 1 and the finite dimensional case.
We motivate and prove a series of identities which form a generalization of the Euler's pentagonal number theorem, and are closely related to specialized Macdonald's identities for powers of the Dedekind $η$--function. More precisely, we show that what we call ``denominator formula'' for the Virasoro algebra has ``higher analogue'' for all $c_{s,t}$-minimal models. We obtain one identity per series which is in agreement with features of conformal field theory such as {\em fusion} and {\em modular invariance} that require all the irreducible modules of the series. In particular, in the case of $c_{2,2k+1}$--minimal models we give a new proof of a family of specialized Macdonald's identities associated with twisted affine Lie algebras of type $A^{(2)}_{2k}, k \geq 2$ (i.e., $BC_k$-affine root system) which involve $(2k^2-k)$-th powers of the Dedekind $η$-function. Our paper is in many ways a continuation of math.QA/0309201.
The "quantum duality principle" states that a quantisation of a Lie bialgebra provides also a quantisation of the dual formal Poisson group and, conversely, a quantisation of a formal Poisson group yields a quantisation of the dual Lie bialgebra as well. We extend this to a much more general result: namely, given any principal ideal domain R, for each prime h in R we establish sort of an "inner" Galois' correspondence on the category HA of torsionless Hopf algebras over R, via the definition of two functors (from HA to itself) such that the image of the first, resp. of the second, is the full subcategory of those Hopf algebras which are commutative, resp. cocommutative, modulo h (i.e. they are "quantum function algebras" (=QFA), resp. "quantum universal enveloping algebras" (=QUEA), at h). In particular we provide a machine to get two quantum groups - a QFA and a QUEA - out of any Hopf algebra H over a field k: just plug in a parameter x and apply the functors to H[x] for h = x. Several relevant examples are studied in full detail: the trivial quantisations, the semisimple groups, the Euclidean group, the Heisenberg group, and the Kostant-Kirillov structure on any Lie algebra; furthermore, an interesting application to renormalisation theory in quantum electro-dynamics is studied, as a sample of application of the principle to a quite large class of problems. This work is a far-reaching "evolution" of the same author's preprint math.QA/9912186: the present paper is entirely self-contained, is more general from the mathematical point of view, and contains additional examples. WARNING: This preprint has been overtaken by a new, deeply enhanced and improved version, available as {\tt math.QA/0303019}; the interested reader is kindly asked to refer to that new preprint.
In this paper we give some examples of generalized Massey products, arising from deformations of A-infinity and L-infinity algebras. The generalized Massey products are given by certain graded commutative algebra structures, depending on the deformation problem considered. We show that the formal deformations of infinity algebras give rise to the same algebraic structure as the deformations of Z2-graded Lie algebras. However, the problem of deforming a Lie algebra into an L-infinity algebra, or an associative algebra into an A-infinity algebra gives rise to a new algebra. We also consider deformations of infinity algebras with a base. In math.QA/9602024, it was shown that the Jacobi identity for a bracket on the tensor product of a Lie algebra and a graded commutative algebra is equivalent to a Maurer-Cartan formula. We prove a generalized result in the case of infinity algebras.
Dror Bar-Natan, Stavros Garoufalidis, Lev Rozansky
et al.
Continuing the work started in Part I and II of this series (see q-alg/9706004 and math.QA/9801049), we prove the relationship between the Aarhus integral and the invariant $Ω$ (henceforth called LMO) defined by T.Q.T. Le, J. Murakami and T. Ohtsuki in q-alg/9512002. The basic reason for the relationship is that both constructions afford an interpretation as "integrated holonomies". In the case of the Aarhus integral, this interpretation was the basis for everything we did in Parts I and II. The main tool we used there was "formal Gaussian integration". For the case of the LMO invariant, we develop an interpretation of a key ingredient, the map $j_m$, as "formal negative-dimensional integration". The relation between the two constructions is then an immediate corollary of the relationship between the two integration theories.
We reduce certain proofs in math.RA/0108067, math.RA/0408155, and math.QA/0409589 to depth two quasibases from one side only, a minimalistic approach which leads to a characterization of Galois extensions for finite projective bialgebroids without the Frobenius extension property. We prove that a proper algebra extension is a left $T$-Galois extension for some right finite projective left bialgebroid over some algebra $R$ if and only if it is a left depth two and left balanced extension. Exchanging left and right in this statement, we have a characterization of right Galois extensions for left finite projective right bialgebroids. Looking to examples of depth two, we establish that a Hopf subalgebra is normal if and only if it is a Hopf-Galois extension. We characterize finite weak Hopf-Galois extensions using an alternate Galois canonical mapping with several corollaries: that these are depth two and that surjectivity of the Galois mapping implies its bijectivity.
In our previous paper math.QA/0412192 the Cayley-Hamilton identity for the GL(m|n) type quantum matrix algebra was obtained. Here we continue investigation of that identity. We derive it in three alternative forms and, most importantly, we obtain it in a factorized form. The factorization leads to a separation of the spectra of the quantum supermatrix into the "even" and "odd" parts. The latter, in turn, allows us to parameterize the characteristic subalgebra (which can also be called the subalgebra of spectral invariants) in terms of the supersymmetric polynomials in the eigenvalues of the quantum supermatrix. For our derivation we use two auxiliary results which may be of independent interest. First, we calculate the multiplication rule for the linear basis of the Schur functions $s_λ(M)$ for the characteristic subalgebra of the Hecke type quantum matrix algebra. The structure constants in this basis are the Littlewood-Richardson coefficients. Second, we derive a series of bilinear relations in the graded ring $Λ$ of Schur symmetric functions in countably many variables.
In this paper, we continue our study of the Hilbert polynomials of coinvariants begun in our previous work math.QA/0205324 (paper I). We describe the sl_n-fusion products for symmetric tensor representations following the method of Feigin and Feigin, and show that their Hilbert polynomials are A_{n-1}-supernomials. We identify the fusion product of arbitrary irreducible sl_n-modules with the fusion product of their resctriction to sl_{n-1}. Then using the equivalence theorem from paper I and the results above for sl_3, we give a fermionic formula for the Hilbert polynomials of a class of affine sl_2-coinvariants in terms of the level-restricted Kostka polynomials. The coinvariants under consideration are a generalization of the coinvariants studied in [FKLMM]. Our formula differs from the fermionic formula established in [FKLMM] and implies the alternating sum formula conjectured in [FL] for this case.
This paper continues the same-named article, Part I (math.QA/9812083). We give a global operator approach to the WZWN theory for compact Riemann surfaces of an arbitrary genus g with marked points. Globality means here that we use Krichever-Novikov algebras of gauge and conformal symmetries (i.e. algebras of global symmetries) instead of loop and Virasoro algebras (which are local in this context). The elements of this global approach are described in Part I. In the present paper we give the construction of conformal blocks and the projective flat connection on the bundle constituted by them.