O. Bershtein, Ye. Kolisnyk, S. Sinel'shchikov
et al.
This paper presents an English version of a chapter of the L.L. Vaksman book `Quantum Bounded Symmetric Domains', see arXiv:0803.3769 [math.QA]. This chapter deals with a quantum analog of a canonical embedding of a bounded symmetric domain.
We introduce the notions of boundary vertex, linear equivalence and effective boundary vertex in the context of Viennot's heaps of pieces. We prove that in the heap of a fully commutative element in a star reducible Coxeter group, every boundary vertex is linearly equivalent to an effective boundary vertex. Using this result, we establish Property W (in the sense of math.QA/0509363) for star reducible Coxeter groups; this corrects a mistake in the latter paper.
Abstract To characterize the four common Qa-1 allelic products, we examined in detail the CTL-defined determinants encoded by Qa-1. In previous studies with anti-Qa-1 CTL and alloantisera, investigators have described antigenic determinants present on Qa-1a and Qa-1b antigens, but they have defined Qa-1c and Qa-1d exclusively by their cross-reactivity with Qa-1a and/or Qa-1b determinants. To delineate further the CTL-defined determinants encoded by Qa-1d, we generated CTL clones with Qa-1d specificity and demonstrated that the Qa-1d molecule expressed determinants that were not detected on Qa-1a, Qa-1b, or Qa-1c target cells. Other CTL clones derived from anti-Qa-1d MLC recognized new antigenic determinants on Qa-1c that cross-reacted with Qa-1d. Each of the four common Qa-1 phenotypes was shown to exhibit unique antigenic determinants. In addition, Qa-1d anti-Qa-1a and Qa-1d anti-Qa-1b CTL confirmed extensive cross-reactivity among these Qa-1 alloantigens. Analysis of CTL from these four immunizations also resulted in the isolation of Qa-1a-specific and Qa-1d-specific CTL clones that cross-reacted with H-2Df and H-2Ks, respectively.
This work presents proofs of the main results of (math.QA/9808015), except those on q-Berezin transform to appear in a subsequent work. The notation and the results of (math.QA/9808037) and (math.QA/9808047) are used.
In our preprint q-alg/9703005 q-analogues of bounded symmetric domains were defined to be homogeneous spaces of the associated quantum groups. The investigation of a simplest among those domains, the quantum matrix ball, was started in math.QA/9803110. This work presents a construction of q-analogues for Hardy-Bergman spaces of 'functions in those balls', together with an explicit form of the Bergman kernel. Besides that, two auxiliary results are also established: a boundedness of matrix balls is proved, and de Rham complexes of differential forms with finite coefficients in those balls are constructed.
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.
Let $S$ be the left bialgebroid $\End {}_BA_B$ over the centralizer $R$ of a right D2 algebra extension $A \| B$, which is to say that its tensor-square is isomorphic as $A$-$B$-bimodules to a direct summand of a finite direct sum of $A$ with itself. We prove that its left endomorphism algebra is a left $S$-Galois extension of $A^{\rm op}$. As a corollary, endomorphism ring theorems for D2 and Galois extensions are derived from the D2 characterization of Galois extension (cf. math.QA/0502188 and math.QA/0409589). We note the converse that a Frobenius extension satisfying a generator condition is D2 if its endomorphism algebra extension is D2.
The papers math.QA/0403527 and math.QA/0409414 v.1 are now merged together. The final version is available at math.QA/0409414 v.2. To avoid duplication of papers, math.QA/0403527 is now removed.
The Dirac reduction technique used previously to obtain solutions of the classical dynamical Yang-Baxter equation on the dual of a Lie algebra is extended to the Poisson-Lie case and is shown to naturally yield certain dynamical r-matrices on the duals of Poisson-Lie groups found by Etingof, Enriquez and Marshall in math.QA/0403283.
This paper of tutorial nature gives some further details of proofs of some theorems related to the quantum dynamical Yang-Baxter equation. This mainly expands proofs given in Lectures on the dynamical Yang-Baxter equation by Etingof and Schiffmann, math.QA/9908064. This concerns the intertwining operator, the fusion matrix, the exchange matrix and the difference operators. The last part expands proofs given in Traces of intertwiners for quantum groups and difference equations, I by Etingof and Varchenko, math.QA/9907181. This concerns the dual Macdonald-Ruijsenaars equations.
This is the continuation of [Y. Li, Affine quivers of type A˜n and canonical bases, math.QA/0501175]. We describe the affine canonical basis elements in the case when the affine quiver has arbitrary orientation. This generalizes the description in [G. Lusztig, Affine quivers and canonical bases, Publ. Math. Inst. Hautes Etudes Sci. 76 (1992) 111–163].
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 is an extended abstract of my talk at the Oberwolfach-Workshop "Representation Theory of Finite-Dimensional Algebras" (February 6 - 12, 2005). It gives self-contained and simplified definitions of quantum cluster algebras introduced and studied in a joint work with A.Berenstein (math.QA/0404446).