Quantum Group as Semi-infinite Cohomology

We obtain the quantum group SL _q (2) as semi-infinite cohomology of the Virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges c+[`(c)]=26{c+\bar{c}=26}. Each braided VOA is constructed from the free Fock space realization of the Virasoro algebra with an additional q-deformed harmonic oscillator degree of freedom. The braided VOA structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology. We explicitly provide the four cohomology classes that serve as the generators of SL _q (2) and verify their relations. We also discuss the possible extensions of our construction and its connection to the Liouville model and minimal string theory.     

Interrelations between quantum groups and reflection equation (braided) algebras

We show that the differential complex _B over the braided matrix algebra BM _q (N) represents a covariant comodule with respect to the coaction of the Hopf algebra _A which is a differential extension of GL _q (N). On the other hand, the algebra _A is a covariant braided comodule with respect to the coaction of the braided Hopf algebra _B . Geometrical aspects of these results are discussed.     

GLq(N)-covariant braided differential bialgebras

We study the possibility of defining the (braided) comultiplication for the GL _q (N)-covariant differential complexes on some quantum spaces. We discover suchdifferential bialgebras (and Hopf algebras) on the bosonic and fermionic quantum hyperplanes (with additive coproduct) and on the braided matrix algebra BM _q (N) with both multiplicative and additive coproducts. The latter case is related (forN = 2) to theq-Minkowski space andq-Poincaré algebra.     

On realizations of exterior calculus with d N = 0

We study realizations of the q-exterior calculus with exterior differential d satisfying d ^N = 0, N > 2 on the free associative algebra with one generator and on the generalized Clifford algebras. Analogs of the notions of connection and curvature are discussed in the case of the q-exterior calculus on the generalized Clifford algebra. We show that the q-exterior calculus on the free associative algebra with one generator is related to q-calculus on the braided line.     

The Nakayama Automorphism of the Almost Calabi-Yau Algebras Associated to <Emphasis Type="Italic">SU</Emphasis>(3) Modular Invariants

We determine the Nakayama automorphism of the almost Calabi-Yau algebra A associated to the braided subfactors or nimrep graphs associated to each SU(3) modular invariant. We use this to determine a resolution of A as an A-A bimodule, which will yield a projective resolution of A.     

Quantum orbits of theR-matrix type

Given a simple Lie algebra g, we consider the orbits in g^* which are of theR-matrix type, i.e., which possess a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the so-calledR-matrix bracket. We call an algebra quantizing the latter bracket a quantum orbit of theR-matrix type. We describe some orbits of this type explicitly and we construct a quantization of the whole Poisson pencil on these orbits in a similar way. The notions ofq-deformed Lie brackets, braided coadjoint vector fields, and tangent vector fields are discussed as well.     

Braided differential operators on quantum algebras

We propose a general scheme of constructing braided differential algebras via algebras of “quantum exponentiated vector fields” and those of “quantum functions”. We treat a reflection equation algebra as a quantum analog of the algebra of vector fields. The role of a quantum function algebra is played by a general quantum matrix algebra. As an example we mention the so-called RTT algebra of quantized functions on the linear matrix group GL(m)$GL\left(m\right)$. In this case our construction essentially coincides with the quantum differential algebra introduced by S. Woronowicz. If the role of a quantum function algebra is played by another copy of the reflection equation algebra we get two different braided differential algebras. One of them is defined via a quantum analog of (co)adjoint vector fields, the other algebra is defined via a quantum analog of right-invariant vector fields. We show that the former algebra can be identified with a subalgebra of the latter one. Also, we show that “quantum adjoint vector fields” can be restricted to the so-called “braided orbits” which are counterparts of generic GL(m)$GL\left(m\right)$-orbits in gl^∗(m)$g{l}^{\ast }\left(m\right)$. Such braided orbits endowed with these restricted vector fields constitute a new class of braided differential algebras.     

Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group

Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groupsU _q (g). They have the same FRT generatorsl ^± but a matrix braided-coproductL=LL, whereL=l ⁺ Sl ^–, and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matricesBM _q(2); it is a braided-commutative bialgebra in a braided category. As a second application, we show that the quantum doubleD(U _q (sl ₂)) (also known as the quantum Lorentz group) is the semidirect product as an algebra of two copies ofU _q (sl ₂), and also a semidirect product as a coalgebra if we use braid statistics. We find various results of this type for the doubles of general quantum groups and their semi-classical limits as doubles of the Lie algebras of Poisson Lie groups.     

Braided m-Lie Algebras

Braided m-Lie algebras induced by multiplication are introduced, which generalize Lie algebras, Lie color algebras and quantum Lie algebras. The necessary and sufficient conditions for the braided m-Lie algebras to be strict Jacobi braided Lie algebras are given. Two classes of braided m-Lie algebras are given, which are generalized matrix braided m-Lie algebras and braided m-Lie subalgebras of End _F M, where M is a Yetter–Drinfeld module over B with dimB < . In particular, generalized classical braided m-Lie algebras sl _{q, f} (GM _G (A), F) and osp _{q, t} (GM _G (A), M, F) of generalized matrix algebra GM _G (A) are constructed and their connection with special generalized matrix Lie superalgebra sl _{s, f} (GM _{Z_2}(A ^s ), F) and orthosymplectic generalized matrix Lie super algebra osp _{s, t} (GM _{Z_2}(A ^s ), M ^s , F) are established. The relationship between representations of braided m-Lie algebras and their associated algebras are established.This revised version was published online in March 2005 with corrections to the cover date.     

Braided Covariance of the Braided Differential Bialgebras Under Quantized Braided Groups

The braided differential bialgebras on braided matrix algebras (with bothmultiplicative and additive coproducts) and on quantum hyperplanes (withadditive coproduct) are proven to be covariant under the braided coactions ofthe quantized braided groups, which contain the usual quantum group-covarianceas a special case. This means that the above braided differential bialgebras havemore and richer symmetries. It is also shown that the braided matrix algebraitself and the related braided differential algebra constitute two braided rings withthe two above-mentioned coproducts.     

Braided Hopf Algebras Obtained from Coquasitriangular Hopf Algebras

In this paper, we study the generalized quantum double construction for paired Hopf algebras with particular attention to the case when the generalized quantum double is a Hopf algebra with projection. Applying our theory to a coquasitriangular Hopf algebra (H, σ), we see that H has an associated structure of braided Hopf algebra in the category of Yetter-Drinfeld modules over , where H _σ is a subHopf algebra of H ⁰, the finite dual of H. Specializing to the quantum group H = SL _q (N), we find that H _σ is , so that the duality between these quantum groups is just the evaluation map. Furthermore, we obtain explicit formulas for the braided Hopf algebra structure of SL _q (N) in the category of left Yetter-Drinfeld modules over . The second author held a postdoctoral fellowship at Mount Allison University from 2005 to 2007 and would like to thank Mount Allison for their warm hospitality. Support for the first author’s research and partial support for the postdoctoral position of the second author came from an NSERC Discovery Grant. The second author now holds research support from Grant 434/1.10.2007 of CNCSIS.     

Braided Tensor Categories and Extensions of Vertex Operator Algebras

Let V be a vertex operator algebra satisfying suitable conditions such that in particular its module category has a natural vertex tensor category structure, and consequently, a natural braided tensor category structure. We prove that the notions of extension (i.e., enlargement) of V and of commutative associative algebra, with uniqueness of unit and with trivial twist, in the braided tensor category of V-modules are equivalent.     

Decoupling braided tensor factors

We briefly report on our result that the braided tensor product algebra of two module algebras A ₁, A ₂ of a quasitriangular Hopf algebra H is equal to the ordinary tensor product algebra of A ₁ with a subalgebra isomorphic to A ₂ and commuting with A ₁, provided there exists a realization of H within A ₁. As applications of the theorem, we consider the braided tensor product algebras of two or more quantum group covariant quantum spaces or deformed Heisenberg algebras.     

Open-String Vertex Algebras,Tensor Categories and Operads

We introduce notions of open-string vertex algebra, conformal open-string vertex algebra and variants of these notions. These are open-string-theoretic, noncommutative generalizations of the notions of vertex algebra and of conformal vertex algebra. Given an open-string vertex algebra, we show that there exists a vertex algebra, which we call the meromorphic center, inside the original algebra such that the original algebra yields a module and also an intertwining operator for the meromorphic center. This result gives us a general method for constructing open-string vertex algebras. Besides obvious examples obtained from associative algebras and vertex (super)algebras, we give a nontrivial example constructed from the minimal model of central charge We establish an equivalence between the associative algebras in the braided tensor category of modules for a suitable vertex operator algebra and the grading-restricted conformal open-string vertex algebras containing a vertex operator algebra isomorphic to the given vertex operator algebra. We also give a geometric and operadic formulation of the notion of grading-restricted conformal open-string vertex algebra, we prove two isomorphism theorems, and in particular, we show that such an algebra gives a projective algebra over what we call the Swiss-cheese partial operad.Acknowledgement. We would like to thank Jürgen Fuchs and Christoph Schweigert for helpful discussions and comments. We are also grateful to Jim Lepowsky for comments. The research of Y.-Z. H. is supported in part by NSF grant DMS-0070800.     

Braided algebras and the <Emphasis Type="Italic">κ</Emphasis>-deformed oscillators

Recently there were presented several proposals how to formulate the binary relations describing κ-deformed oscillator algebras. In this paper we shall consider multilinear products of κ-deformed oscillators consistent with the axioms of braided algebras. In general case the braided triple products are quasi-associative and satisfy the hexagon condition depending on the coassociator \({\Phi \in A\otimes A\otimes A}\) . We shall consider only the products of κ-oscillators consistent with co-associative braided algebra, with \({\Phi =1}\) . We shall consider three explicit examples of binary κ-deformed oscillator algebra relations and describe briefly their multilinear coassociative extensions satisfying the postulates of braided algebras. The third example, describing κ-deformed oscillators in group manifold approach to κ-deformed fourmomenta, is a new result.     

Geometrical Meaning of R-Matrix Action for Quantum Groups at Roots of 1

The present work splits in two parts: first, we perform a straightforward generalization of results from [Re], proving that quantum groups and their unrestricted specializations at roots of 1, in particular the function algebra F[H] of the Poisson group H dual of G, are braided; second, as a main contribution, we prove the convergence of the (specialized) R-matrix action to a birational automorphism of a -fold ramified covering of when is a primitive -th root of 1, and of a 2-fold ramified covering of H, thus giving a geometric content to the notion of braiding for quantum groups at roots of 1. Received: 23 April 1996/Accepted: 12 August 1996     

Non-trivially associated tensor categories and left coset representatives

In this talk I will motivate and describe a generalisation of the group doublecross product construction, involving a set of coset representatives for the left action of a subgroup on a group. From this data a non-trivially associated tensor category can be made. I shall briefly mention the corresponding double construction, which gives a non-trivially associated braided tensor category, containing a braided Hopf algebra. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Coalgebra Bundles

We develop a generalised theory of bundles and connections on them in which the role of gauge group is played by a coalgebra and the role of principal bundle by an algebra. The theory provides a unifying point of view which includes quantum group gauge theory, embeddable quantum homogeneous spaces and braided group gauge theory, the latter being introduced now by these means. Examples include ones in which the gauge groups are the braided line and the quantum plane. Received: 22 February 1996 / Accepted: 29 May 1997     

On α-Induction, Chiral Generators and¶Modular Invariants for Subfactors

We consider a type III subfactor N⊂N of finite index with a finite system of braided N-N morphisms which includes the irreducible constituents of the dual canonical endomorphism. We apply α-induction and, developing further some ideas of Ocneanu, we define chiral generators for the double triangle algebra. Using a new concept of intertwining braiding fusion relations, we show that the chiral generators can be naturally identified with the α-induced sectors. A matrix Z is defined and shown to commute with the S- and T-matrices arising from the braiding. If the braiding is non-degenerate, then Z is a “modular invariant mass matrix” in the usual sense of conformal field theory. We show that in that case the fusion rule algebra of the dual system of M-M morphisms is generated by the images of both kinds of α-induction, and that the structural information about its irreducible representations is encoded in the mass matrix Z. Our analysis sheds further light on the connection between (the classifications of) modular invariants and subfactors, and we will construct and analyze modular invariants from SU(n) _k loop group subfactors in a forthcoming publication, including the treatment of all SU(2) _k modular invariants. Received: 13 April 1999 / Accepted: 13 July 1999     

Fine Gradings on Non-simple Lie Algebras: Example of o(4,C)

On any Lie algebra L, it is of significant convenience to have at one's disposal all the possible fine gradings of L, since they reflect the basic structural properties of the Lie algebra. They also provide useful bases of the representations of the algebra -- namely such bases that are preserved by the commutator.We list all the six fine gradings on the non-simple Lie algebra o(4,C) and we explain their relation to the fine gradings of the Lie algebra sl(2,C) where relevant. The existence of such relation is not surprising, since o(4,C) is in fact a product of two specimen of sl(2,C). The example of o(4,C) is especially important due to the fact that one of its fine gradings is not generated by any MAD-group. This proves that, unlike in the case of classical simple Lie algebras over C, on the non-simple classical Lie algebras over C there can exist a fine grading that is not generated by any MAD-group on the Lie algebra.