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arXiv2006

Star reducible Coxeter groups

We define ``star reducible'' Coxeter groups to be those Coxeter groups for which every fully commutative element (in the sense of Stembridge) is equivalent to a product of commuting generators by a sequence of length-decreasing star operations (in the sense of Lusztig). We show that the Kazhdan--Lusztig bases of these groups have a nice projection property to the Temperley--Lieb type quotient, and furthermore that the images of the basis elements $C'_w$ (for fully commutative $w$) in the quotient have structure constants in ${\Bbb Z}^{\geq 0}[v, v^{-1}]$. We also classify the star reducible Coxeter groups and show that they form nine infinite families (types $A_n$, $B_n$, $D_n$, $E_n$, $F_n$, $H_n$, affine $A_{n-1}$ for $n$ odd, affine $C_{n-1}$ for $n$ even, and the case where the Coxeter graph is complete), with two exceptional cases (of ranks 6 and 7). This paper is the sequel to math.QA/0509362.

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arXiv2005

Formal Homotopy Quantum Field Theories, II : Simplicial Formal Maps

Simplicial formal maps were introduced in the first paper, (math.QA/0512032), of this series as a tool for studying Homotopy Quantum Field Theories with background a general homotopy 2-type. Here we continue their study, showing how a natural generalisation can handle much more general backgrounds. The question of the geometric interpretation of these formal maps is partially answered in terms of combinatorial bundles. This suggests new interpretations of HQFTs.

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arXiv2006

Representations Parameterized by a Pair of Characters

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.

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arXiv2006

New deformations of group algebras of Coxeter groups, II

In our previous paper math.QA/0409261, we defined a deformation of the group algebra of the group of even elements of a Coxeter group W, and showed that it is flat for all values of parameters if and only if all the rank 3 parabolic subgroups of W are infinite. In this paper, we study what happens in the general case. Then the deformation is flat only for some values of parameters, and the set of all such values is called the flatness locus. The main result of the paper is an explicit description of the this flatness locus as a scheme over Z. More specifically, we show that this scheme is the intersection of the flatness loci for the subalgebras corresponding to parabolic subgroups of rank 3. The latter are determined by solving the rigid multiplicative Deligne-Simpson problem. We also define additive analogs of our algebras and study their properties.

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arXiv2006

$t$--analogs of $q$--characters of quantum affine algebras of type $E_6$, $E_7$, $E_8$

We compute $t$--analogs of $q$--characters of all $l$--fundamental representations of the quantum affine algebras of type $E_6^{(1)}$, $E_7^{(1)}$, $E_8^{(1)}$ by a supercomputer. In particular, we prove the fermionic formula for Kirillov-Reshetikhin modules conjectured by Hatayama et al. (math.QA/9812022) for these classes of representations. We also give explicitly the monomial realization of the crystal of the corresponding fundamental representations of the qunatum enveloping algebras associated with finite dimensional Lie algebras of types $E_6$, $E_7$, $E_8$. These are computations of Betti numbers of graded quiver varieties, quiver varieties and determination of all irreducible components of the lagrangian subvarities of quiver varieties of types $E_6$, $E_7$, $E_8$ respectively.

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arXiv2006

Uniqueness of higher Gaudin hamiltonians

For any semisimple Lie algebra $\mathfrak{g}$, the universal enveloping algebra of the infinite-dimensional pro-nilpotent Lie algebra $\mathfrak{g}_-:=\mathfrak{g}\otimes t^{-1}\mathbb{C}[t^{-1}]$ contains a large commutative subalgebra $\mathcal{A}\subset U(\mathfrak{g}_-)$. This subalgebra comes from the center of the universal enveloping of the affine Kac--Moody algebra $\hat{\mathfrak{g}}$ at the critical level and gives rise to the construction of higher hamiltonians of the Gaudin model (due to Feigin, Frenkel and Reshetikhin). Though there are no explicit formulas for the generators of $\mathcal{A}$ known in general, the "classical analogue" of this subalgebra, i.e. the associated graded subalgebra in the Poisson algebra $A\subset S(\mathfrak{g}_-)$, can be easily described. In this note we show that the "classical" subalgebra $A\subset S(\mathfrak{g}_-)$ is the Poisson centralizer of some of its quadratic elements, and deduce from this that the "quantum" subalgebra $\mathcal{A}\subset U(\mathfrak{g}_-)$ is uniquely determined by the space of quadratic elements of the classical one. In particular, this means that some different constructions of higher Gaudin hamiltonians (namely, Feigin-Frenkel-Reshetikhin's method and Talalaev-Chervov's method), give the same family of commuting operators. The proof uses some ideas of the previous paper math.QA/0608586.

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arXiv2009

Massey products on cycles of projective lines and trigonometric solutions of the Yang-Baxter equations

We show that a nondegenerate unitary solution $r(u,v)$ of the associative Yang-Baxter equation (AYBE) for $\Mat(N,\C)$ (see math.AG/0008156) with the Laurent series at $u=0$ of the form $r(u,v)=\frac{1\ot 1}{u}+r_0(v)+...$ satisfies the quantum Yang-Baxter equation, provided the projection of $r_0(v)$ to traceless matrices has a period. We classify all such solutions of the AYBE extending the work of Schedler math.QA/0212258. We also characterize solutions coming from triple Massey products in the derived category of coherent sheaves on cycles of projective lines.

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arXiv1998

Calculating zeros of a q-zeta function numerically

The note is a continuation of the previous paper ``On q-analogues of Riemann's zeta'' (math.QA/980499). It contains an output of the computer program calculating the zeros of the ``sharp'' q-zeta function.

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arXiv1998

Unitarity of induced representations from coisotropic quantum groups

We study unitarity of the induced representations from coisotropic quantum subgroups which were introduced in math.QA/9804138. We define a real structure on coisotropic subgroups which determines an involution on the homogeneous space. We give general invariance properties for functionals on the homogeneous space which are sufficient to build a unitary representation starting from the induced one. We present the case of the one-dimensional quantum Galilei group, where we have to use in all enerality our definition of quasi-invariant functional.

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arXiv2003

The Aarhus integral of rational homology 3-spheres III: The Relation with the Le-Murakami-Ohtsuki Invariant

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.

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arXiv1998

On Function Theory in Quantum Disc: Covariance

Our previous work (math.QA/9808015) introduces the basic notions and announces some results on function theory in the quantum disc. The present paper establishes a relationship between those results and the quantum groups theory.

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arXiv1998

Infinity Algebras, Massey Products, and Deformations

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.

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arXiv1999

Quantum matrix ball: the weighted Bergman kernels

We proceed with studying the q-analogues of Cartan domains introduced in q-alg/9703005 and turn to the case of a ball in the space of complex matrices. An explicit expression for a positive $U_q\frak{su}_{nm}$-invariant integral (see math.QA/9803110) is used to define q-analogues for weighted Bergman spaces. The orthogonal projections onto these spaces are integral operators; this work presents an explicit formula for their kernels.

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arXiv1998

What is moonshine?

This is an informal write up of my talk in Berlin. It gives some background to Goddard's talk (math.QA/9808136) about the moonshine conjectures.

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