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arXiv1995

Quasi-{$*$} Structure on {$q$}-Poincare algebras

We use braided groups to introduce a theory of $*$-structures on general inhomogeneous quantum groups, which we formulate as {\em quasi-$*$} Hopf algebras. This allows the construction of the tensor product of unitary representations up to a quantum cocycle isomorphism, which is a novel feature of the inhomogeneous case. Examples include $q$-Poincaré quantum group enveloping algebras in $R$-matrix form appropriate to the previous $q$-Euclidean and $q$-Minkowski spacetime algebras $R_{21}x_1x_2=x_2x_1R$ and $R_{21}u_1Ru_2=u_2R_{21}u_1R$. We obtain unitarity of the fundamental differential representations. We show further that the Euclidean and Minkowski Poincaré quantum groups are twisting equivalent by a another quantum cocycle.

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arXiv1995

Finite Group Factorisations and Braiding

We compute the quantum double, braiding and other canonical Hopf algebra constructions for the bicrossproduct Hopf algebra $H$ associated to the factorization of a finite group into two subgroups. The representations of the quantum double are described by a notion of bicrossed bimodules, generalising the cross modules of Whitehead. We also show that self-duality structures for the bicrossproduct Hopf algebras are in one-one correspondence with factor-reversing group isomorphisms. The example $Z_6Z_6$ is given in detail. We show further that the quantum double $D(H)$ is the twisting of $D(X)$ by a non-trivial quantum cocycle, where $X$ is the associated double cross product group.

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arXiv1995

Representations of Quantum Affine Algebras

In this paper we define a quantum version of the ``fusion'' tensor product of two representations of an affine Kac-Moody algebra.It is replaced by what we call fusion action of the category of finite-dimensional representations of quantum affine algebra on its highest weight representations. We construct a quantum version of the associativity constraint. We give categorical treatment of the subject and related questions ( like quantum Knizhnik-Zamolodchikov equations).

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arXiv1995

Comments on Bosonisation and Biproducts

We clarify the relation between the `bosonisation' construction (due to the author) which can be used to turn a Hopf algebra $B$ in the category of $H$-modules or $H$-comodules into an equivalent ordinary Hopf algebra, and a version of Radfords theorem (also due in this form to the author) which does the same for $B$ in the category of crossed $H$-modules. We also comment on reconstruction from the category of $B$-comodules.

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arXiv1995

Linear representations of groups with translation invariant defining relationships. Some new series of braid group representations and new invariants of links and knots

In this paper we indicate one method of construction of linear representations of groups and algebras with translation invariant (except, maybe , finite number) defining relationships. As an illustration of this method, we give one approach to the construction of linear representations of braid group and derive some series of such representations. Some invariants of oriented knots and links are constructed. The author is grateful to Yuri Drozd, Sergey Ovsienko and other members of algebraic seminar at Kiev University for the creative atmosphere without which this work could hardly appear.

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arXiv1995

Cohomology of Drinfel'd algebras: A General Nonsense Approach

In our paper [Markl, Shnider: Drinfel'd Algebra Deformations and the Associahedra, IMRN 1994, no. 4, 169-176] we announced a construction of a cohomology controlling deformations of quasi-coassociative (or Drinfel'd) bialgebras. The full version of the paper will appear as [Markl, Shnider: Drinfel'd Algebra Deformations, Homotopy Comodules and the Associahedra] in Trans. Amer. Math. Soc. The construction in the paper was based on very explicit arguments using deep combinatorial properties of the associahedra. The present paper gives an alternative, general nonsense approach to the construction. So, we just prove the existence of such a cohomology without explicitly constructing it. This should be compared with the two approaches to the cohomology of associative algebras: we either describe explicitly the Hochschild complex and say "Behold! this is the cohomology" or we prove the existence of a projective resolution and define the cohomology as the derived functor.

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arXiv1995

Cohomology of Quantum Groups

Lecture notes. Introduction to the cohomology of algebras, Lie algebras, Lie bialgebras and quantum groups. Contains a new derivation of the classification of classical r-matrices in terms of deformation cohomology, and a calculation of the esoteric forms of quantum gl(N) by deformation theory.

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arXiv1995

Elliptic Algebras and Equivariant Elliptic Cohomology

In this paper we explain the parallelism in the classification of three different kinds of mathematical objects: (i) Classical r-matrices. (ii) Generalized cohomology theories that have Chern classes for complex vector bundles. (iii) 1-dimensional formal groups. The main point of the paper is a construction of the elliptic algebra associated to Belavin's classical elliptic r-matrix in terms of Equivariant elliptic cohomology of the Steinberg varieties associated to some partial flag manifolds.

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arXiv1995

Universal T-matrix for Twisted Quantum gl(N)

The Universal T-matrix is the capstone of the structure that consists of a quantum group and its dual, and the central object from which spring the T-matrices (monodromies) of all the associated integrable models. A closed expression is obtained for the case of multiparameter (twisted) quantum $gl(N)$. The factorized nature of standard quantum groups, that allows the explicit expression for $U\hskip-1mm T $ to be obtained with relative ease, extends to some nonstandard quantum groups, such as those based on $A_n^{(2)}$, and perhaps to all. The paper is mostly concerned with parameters in general position, but the extension to roots of unity is also explored, in the case of $g\ell(N)$. The structure of the dual is now radically different, and an interesting generalization of the $q$-exponential appears in the formulas for the Universal T- and R-matrices. The projection to quantum $sl(N)$ is simple and direct; this allows, in particular, to apply recent results concerning deformations of twisted $gl(N)$ to the semisimple quotient.

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arXiv1995

A theory of tensor products for module categories for a vertex operator algebra, IV

This is the fourth part of a series of papers developing a tensor product theory of modules for a vertex operator algebra. In this paper, We establish the associativity of $P(z)$-tensor products for nonzero complex numbers $z$ constructed in Part III of the present series under suitable conditions. The associativity isomorphisms constructed in this paper are analogous to associativity isomorphisms for vector space tensor products in the sense that it relates the tensor products of three elements in three modules taken in different ways. The main new feature is that they are controlled by the decompositions of certain spheres with four punctures into spheres with three punctures using a sewing operation. We also show that under certain conditions, the existence of the associativity isomorphisms is equivalent to the associativity (or (nonmeromorphic) operator product expansion in the language of physicists) for the intertwining operators (or chiral vertex operators). Thus the associativity of tensor products provides a means to establish the (nonmeromorphic) operator product expansion.

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arXiv1995

On Hopf algebras and the elimination theorem for free Lie algebras

The elimination theorem for free Lie algebras, a general principle which describes the structure of a free Lie algebra in terms of free Lie subalgebras, has been recently used by E. Jurisich to prove that R. Borcherds' ``Monster Lie algebra'' has certain large free Lie subalgebras, illuminating part of Borcherds' proof that the moonshine module vertex operator algebra obeys the Conway-Norton conjectures. In the present expository note, we explain how the elimination theorem has a very simple and natural generalization to, and formulation in terms of, Hopf algebras. This fact already follows from general results contained in unpublished 1972 work, unknown to us when we wrote this note, of R. Block and P. Leroux.

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arXiv1995

On Vassiliev knot invariants induced from finite type 3-manifold invariants

We prove that the knot invariant induced by a $\Bbb Z$-homology 3-sphere invariant of order $\leq k$ in Ohtsuki's sense, where $k\geq 4$, is of order $\leq k-2$. The method developed in our computation shows that there is no $\Bbb Z$-homology 3-sphere invariant of order 5. This result agrees with a conjecture of Rozansky based on physical predictions about the asymptotic behavior of Witten's Chern-Simons path integral.

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arXiv1995

On Deformations of Commutation Relation Algebras

This paper is on $C$ - symmetric creation and annihilation operators, which are constructed on Wick's algebras which fulfil consistency conditions. The essential assumption is that every algebraic action must be constant on equivalence classes. All consistency conditions follow from the above assumption. In this way we obtain well defined quotient algebras with some additional relations.

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arXiv1995

Poisson-Lie Structures on Infinite-Dimensional Jet Groups and Quantum Groups Related to Them

We study the problem of classifying all Poisson-Lie structures on the group $G_{\infty}$ of formal diffeomorphisms of the real line $\zR^{1}$ which leave the origin fixed, as well as the extended group of diffeomorphisms $G_{0\infty}\supset G_{\infty}$ whose action on $\zR^{1}$ does not necessarily fix the origin. A complete local classification of all Poisson-Lie structures on the groups $G_{\infty}$ and $G_{0\infty}$ is given. This includes a classification of all Lie-bialgebra structures on the Lie algebra $\Cal G_{\infty}$ of $G_{\infty}$, which we prove to be all of coboundary type, and a classification of all Lie-bialgebra strucutures on the Lie algebra $\Cal G_{0\infty}$ (the Witt algebra) of $G_{0\infty}$ which also turned out to be all of coboundary type. A large class of Poisson structures on the space $V_λ$ of $λ$-densities on the real line is found such that $V_λ$ becomes a homogeneous Poisson space under the action of the Poisson-Lie group $G_{\infty}$. We construct a series of quantum semigroups whose quasiclassical limits are finite-dimensional Poisson-Lie factor groups of $G_{\infty}$ and $G_{0\infty}$.

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arXiv1995

On Finite Type 3-Manifold Invariants II

This paper continues the study of finite-type invariants of homology spheres studied by Ohtsuki and Garoufalidis. We apply the surgery classification of links to give a diagrammatic description, using ideas of Ohtsuki. This uses a computation of the surgery equivalence classes of pure braids. We show that the order of any invariant, in Ohtsukis sense, is a multiple of 3. We also study the relation between the order of an invariant and that of the knot invariant it defines.

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arXiv1995

On the Cremmer-Gervais quantizations of SL(n)

Non-standard quantum groups $C_R [GL(n)]$ and $C_R [SL(n)]$ are constructed for a two parameter version of the Cremmer-Gervais $R$-matrix. An epimorphism is constructed from $C_R [GL(n)]$ onto the restricted dual $U_{\bar{R}}(\frak{gl}(n-1))$ associated to a related smaller $R$-matrix of the same form. A related result is proved concerning factorizable Lie bialgebras. For any such Lie bialgebra, the dual Lie bialgebra has a canonical homomorphic image which is again factorizable.

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arXiv1995

Decomposition of $q$-deformed Fock spaces

A decomposition of the level-one $q$-deformed Fock representations of $\uqn$ is given. It is found that the action of $\upqn$ on these Fock spaces is centralized by a Heisenberg algebra, which arises from the center of the affine Hecke algebra $\widehat{H}_N$ in the limit $N \rightarrow \infty$. The $q$-deformed Fock space is shown to be isomorphic as a $\upqn$-Heisenberg-bimodule to the tensor product of a level-one irreducible highest weight representation of $\upqn$ and the Fock representation of the Heisenberg algebra. The isomorphism is used to decompose the $q$-wedging operators, which are intertwiners between the $q$-deformed Fock spaces, into constituents coming from $\upqn$ and from the Heisenberg algebra.

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arXiv1995

On inner product in modular tensor categories. I

In this paper we study modular tensor categories (braided rigid balanced tensor categories with additional finiteness and non-degeneracy conditions), in particular, representations of quantum groups at roots of unity. We show that the action of modular group on certain spaces of morphisms in MTC is unitary with respect to the natural inner product on these spaces. In a special case of category based on representations of the quantum group U_q sl_n at roots of unity we show that in some of these spaces of morphisms (for U_q sl_2, in all of them) the action of modular group can be written in terms of values of Macdonald's polynomials of type A at roots of unity. This gives identities for these special values, both known before (symmetry identity) and new ones. The paper contains a detailed exposition of the theory of modular categories as well as construction of modular categories from representation of quantum groups at roots of unity

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