Hasil untuk "math.QA"

V. Danilov, A. Karzanov

In ArXiv:1604.00338[math.QA] we gave a complete combinatorial characterization of homogeneous quadratic identities for minors of quantum matrices. It was obtained as a consequence of results on minors of matrices of a special sort, the so-called path matrices $Path_G$ generated by paths in special planar directed graphs $G$. In this paper we prove two assertions that were stated but left unproved in ArXiv:1604.00338[math.QA]. The first one says that any minor of $Path_G$ is determined by a system of disjoint paths, called a flow, in $G$ (generalizing a similar result of Lindstr\"om's type for the path matrices of Cauchon graphs by Casteels). The second, more sophisticated, assertion concerns certain transformations of pairs of flows in $G$.

A. Lauda

A 2-category was introduced in math.QA/0803.3652 that categorifies Lusztig’s integral version of quantum sl(2). Here we construct for each positive integer N a representation of this 2-category using the equivariant cohomology of iterated flag varieties. This representation categorifies the irreducible (N + 1)-dimensional representation of quantum sl(2).

C. Torossian

This is a Bourbaki's seminar text. We introduce the combinatorial Kashiwara-Vergne conjecture on the Baker-Campbell-Hausdorff serie. After recalling previous results and consequences, we explain the Alekseev-Meinrenken's proof [math.QA/0506499]

V. Tolstoy

We discussed twisted quantum deformations of D = 4 Lorentz and Poincare alge- bras. In the case of Poincare algebra it is shown that almost all classical r-matrices of S. Zakrzewski classification can be presented as a sum of subordinated r-matrices of Abelian and Jordanian types. Corresponding twists describing quantum deforma- tions are obtained in explicit form. This work is an extended version of the paper arXiv:0704.0081v1(math.QA).

O. A. Bershtein, Ye. K. Kolisnyk, S. D. 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.

B. Shoikhet

We prove that the Kontsevich integrals (in the sense of the formality theorem) of all even wheels are equal to zero. These integrals appear in the approach to the Duflo formula via the formality theorem. The result means that for any finite-dimensional Lie algebra g, and for invariant polynomials f, g ∈ [S·(g)]g one has f · g = f * g, where * is the Kontsevich star product, corresponding to the Kirillov–Poisson structure on g*. We deduce this theorem form the result contained in math.QA/0010321 on the deformation quantization with traces.

D. Tamarkin

We prove that the chain operad of small squares is formal. This fact clarifies situation with the proof of M. Kontsevich formality theorem in the paper of the author math.QA/9803025, revised Sept 24. The formality of the operad follows quite easily from the existence of an associator.

P. Etingof, Shlomo Gelaki

It is shown in math.QA/0301027 that a finite dimensional quasi-Hopf algebra with radical of codimension 1 is semisimple and 1-dimensional. On the other hand, there exist quasi-Hopf (in fact, Hopf) algebras, whose radical has codimension 2. Namely, it is known that these are exactly the Nichols Hopf algebras H_{2^n} of dimension 2^n, n\ge 1 (one for each value of n). The main result of this paper is that if H is a finite dimensional quasi-Hopf algebra over C with radical of codimension 2, then H is twist equivalent to a Nichols Hopf algebra H_{2^n}, n\ge 1, or to a lifting of one of the four special quasi-Hopf algebras H(2), H_+(8), H_-(8), H(32) of dimensions 2, 8, 8, and 32, defined in Section 3. As a corollary we obtain that any finite tensor category which has two invertible objects and no other simple object is equivalent to \Rep(H_{2^n}) for a unique n\ge 1, or to a deformation of the representation category of H(2), H_+(8), H_-(8), or H(32). As another corollary we prove that any nonsemisimple quasi-Hopf algebra of dimension 4 is twist equivalent to H_4.

M. Jimbo, H. Konno, S. Odake et al.

P. Etingof, Shlomo Gelaki

Let p be a prime, and denote the class of radically graded finite dimensional quasi-Hopf algebras over C, whose radical has codimension p, by RG(p). The purpose of this paper is to continue the structure theory of finite dimensional quasi-Hopf algebras started in math.QA/0310253 (p=2) and math.QA/0402159 (p>2). More specifically, we completely describe the class RG(p) for p>2. Namely, we show that if H\in RG(p) has a nontrivial associator, then the rank of H[1] over H[0] is \le 1. This yields the following classification of H\in RG(p), p>2, up to twist equivalence: (a) Duals of pointed Hopf algebras with p grouplike elements, classified in math.QA/9806074. (b) Group algebra of Z_p with associator defined by a 3-cocycle. (c) The algebras A(q), introduced in math.QA/0402159. This result implies, in particular, that if p>2 is a prime then any finite tensor category over C with exactly p simple objects which are all invertible must have Frobenius-Perron dimension p^N, N=1,2,3,4,5 or 7. In the second half of the paper we construct new examples of finite dimensional quasi-Hopf algebras H, which are not twist equivalent to a Hopf algebra. They are radically graded, and H/Rad(H)=C[Z_n^m], with a nontrivial associator. For instance, to every finite dimensional simple Lie algebra g and an odd integer n, coprime to 3 if g=G_2, we attach a quasi-Hopf algebra of dimension n^{dim(g)}.

P. Etingof, Shlomo Gelaki

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.

Magnus Jacobsson

Haisheng Li

Inspired by the Borcherds' work on ``$G$-vertex algebras,'' we formulate and study an axiomatic counterpart of Borcherds' notion of $G$-vertex algebra for the simplest nontrivial elementary vertex group, which we denote by $G_{1}$. Specifically, we formulate a notion of axiomatic $G_{1}$-vertex algebra, prove certain basic properties and give certain examples, where the notion of axiomatic $G_{1}$-vertex algebra is a nonlocal generalization of the notion of vertex algebra. We also show how to construct axiomatic $G_{1}$-vertex algebras from a set of compatible $G_{1}$-vertex operators. The results of this paper were reported in June 2001, at the International Conference on Lie Algebras in the Morningside center, Beijing, China, and were reported on November 30, 2001, in the Quantum Mathematics Seminar, at Rutgers-New Brunswick. We noticed that a paper of Bakalov and Kac appeared today (math.QA/0204282) on noncommutative generalizations of vertex algebras, which has certain overlaps with the current paper. On the other hand, most of their results are orthogonal to the results of this paper.

N. Hu, Xiuling Wang

Abstract We quantize the generalized-Witt algebra in characteristic 0 with its Lie bialgebra structures discovered by Song–Su [G. Song, Y. Su, Lie bialgebras of generalized-Witt type, arXiv: math.QA/0504168 , Sci. China Ser. A 49 (4) (2006) 533–544]. Via a modulo p reduction and a modulo “p-restrictedness” reduction process, we get 2 n − 1 families of truncated p-polynomial noncocommutative deformations of the restricted universal enveloping algebra of the Jacobson–Witt algebra W ( n ; 1 ) (for the Cartan type simple modular restricted Lie algebra of W type). They are new families of noncommutative and noncocommutative Hopf algebras of dimension p 1 + n p n in characteristic p. Our results generalize a work of Grunspan [C. Grunspan, Quantizations of the Witt algebra and of simple Lie algebras in characteristic p, J. Algebra 280 (2004) 145–161] in rank n = 1 case in characteristic 0. In the modular case, the argument for a refined version follows from the modular reduction approach (different from [C. Grunspan, Quantizations of the Witt algebra and of simple Lie algebras in characteristic p, J. Algebra 280 (2004) 145–161]) with some techniques from the modular Lie algebra theory.

R. Taillefer

Abstract We prove that the category of Hopf bimodules over any Hopf algebra has enough injectives, which enables us to extend some results on the unification of Hopf bimodule cohomologies of [R. Taillefer, PhD thesis, 2001; arXiv preprint math.QA/0005019 ] to the infinite dimensional case. We also prove that the cup-product defined on these cohomologies is graded-commutative. Unlike the algebra case (see [S. Schwede, J. Reine Angew. Math. 498 (1998) 153–172]), these methods do not give a non-trivial Gerstenhaber algebra structure on the cohomology we consider. We also comment that the other approach to finding such a structure that we know of (see [M. Farinati, A. Solotar, arXiv preprint math.KT/0207243 ]) also gives a trivial Gerstenhaber algebra structure.

Liang Kong

Abstract We study the operadic and categorical formulations of (conformal) full field algebras. In particular, we show that a grading-restricted R × R -graded full field algebra is equivalent to an algebra over a partial operad constructed from spheres with punctures and local coordinates. This result is generalized to conformal full field algebras over V L ⊗ V R , where V L and V R are two vertex operator algebras satisfying certain finiteness and reductivity conditions. We also study the geometry interpretation of conformal full field algebras over V L ⊗ V R equipped with a nondegenerate invariant bilinear form. By assuming slightly stronger conditions on V L and V R , we show that a conformal full field algebra over V L ⊗ V R equipped with a nondegenerate invariant bilinear form exactly corresponds to a commutative Frobenius algebra with a trivial twist in the category of V L ⊗ V R -modules. The so-called diagonal constructions [Y.-Z. Huang, L. Kong, Full field algebras, arXiv: math.QA/0511328 ] of conformal full field algebras are given in tensor-categorical language.

P. Etingof, Shlomo Gelaki

In this paper we contribute to the classification of Hopf algebras of dimension pq, where p,q are distinct prime numbers. More precisely, we prove that if p and q are odd primes with p<q<2p+3, then any complex Hopf algebra of dimension pq is semisimple and hence isomorphic to either a group algebra or to the dual of a group algebra by the previous works math.QA/9801129 and math.QA/9801128.

Shlomo Gelaki

It is shown in math.QA/0310253 that a finite dimensional quasi-Hopf algebra over the complex numbers with radical of codimension 2 is twist equivalent to a Nichols Hopf algebra, or to a lifting of one of four special quasi-Hopf algebras of dimensions 2, 8, 8, and 32. The purpose of this paper is to construct new finite dimensional basic quasi-Hopf algebras A(q) of dimension n^3, n>2, parametrized by primitive roots of unity q of order n^2, with radical of codimension n, which generalize the construction of the basic quasi-Hopf algebras of dimension 8 given in math.QA/0310253. These quasi-Hopf algebras are not twist equivalent to a Hopf algebra, and may be regarded as quasi-Hopf analogs of Taft Hopf algebras. By math.QA/0301027, our construction is equivalent to the construction of new finite tensor categories whose simple objects form a cyclic group of order n, and which are not tensor equivalent to a representation category of a Hopf algebra. In a later publication we plan to use our construction to classify finite tensor categories whose simple objects form a cyclic group of prime order n. We also prove that if H is a finite dimensional radically graded quasi-Hopf algebra with H[0]=(C[\Z/n\Z],\Phi), where n is prime and \Phi is a nontrivial associator, such that H[1] is a free left module over H[0] of rank 1 (it is always free) then H is isomorphic to A(q).

K. Costello

This is the sequel to my preprint "TCFTs and Calabi-Yau categories", math.QA/0412149. Here we extend the results of that paper to construct, for certain Calabi-Yau A-infinity categories, something playing the role of the Gromov-Witten potential. This is a state in the Fock space associated to periodic cyclic homology, which is a symplectic vector space. Applying this to a suitable A-infinity version of the derived category of sheaves on a Calabi-Yau yields the B model potential, at all genera. The construction doesn't go via the Deligne-Mumford spaces, but instead uses the Batalin-Vilkovisky algebra constructed from the uncompactified moduli spaces of curves by Sen and Zwiebach. The fundamental class of Deligne-Mumford space is replaced here by a certain solution of the quantum master equation, essentially the "string vertices" of Zwiebach. On the field theory side, the BV operator has an interpretation as the quantised differential on the Fock space for periodic cyclic chains. Passing to homology, something satisfying the master equation yields an element of the Fock space.