arXiv subjects
Publications and source records attributed to S. Majid.
Double-bosonisation associates to a braided group in the category of modules of a quantum group, a new quantum group. We announce the semiclassical version of this inductive construction.
We obtain new family of quasitriangular Hopf algebras $C^{0|n}_q\lcross \widetilde{U_q(su_n)}\rcross C^{0|n}_q$ via the author's recent double-bosonisation construction for new quantum groups. They are versions of $U_q(su_{n+1})$ with a fermionic rather than bosonic quantum plane of roots adjoined to $U_q(su_n)$. We give the $n=2$ case in detail. We also consider the anyonic-double of an anyonic ($Z/nZ$-graded) braided group and the double-bosonisation of the free braided group in $n$ variables.
We show that the bicovariant first order differential calculi on a factorisable semisimple quantum group are in 1-1 correspondence with irreducible representations $V$ of the quantum group enveloping algebra. The corresponding calculus is constructed and has dimension $dim V^2$. The differential calculi on a finite group algebra $C G$ are also classified and shown to be in correspondence with pairs consisting of an irreducible representation $V$ and a continuous parameter in $C P^{dim V -1}$. They have dimension $dim V$. For a classical Lie group we obtain an infinite family of non-standard calculi. General constructions for bicovariant calculi and their quantum tangent spaces are also obtained.
We clarify some aspects of quantum group gauge theory and its recent generalisations (by T. Brzezinski and the author) to braided group gauge theory and coalgebra gauge theory. We outline the diagrammatic version of the braided case. We study the bosonisation of any braided group provides as a trivial principal bundle in three ways.
We develop a gauge theory or theory of bundles and connections on them at the level of braids and tangles. Extending recent algebraic work, we provide now a fully diagrammatic treatment of principal bundles, a theory of global gauge transformations, associated braided fiber bundles and covariant derivatives on them. We describe the local structure for a concrete $Z_3$-graded or `anyonic' realization of the theory.
We develop a generalised gauge theory 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.
We show that the action of the special conformal transformations of the usual (undeformed) conformal group is the $q\to 1$ scaling limit of the braided adjoint action or $R$-commutator of $q$-Minkowski space on itself. We also describe the $q$-deformed conformal algebra in $R$-matrix form and its quasi-$*$ structure.
We collect here some less well-known results and formulae about the bosonisation construction which turns braided groups into quantum groups. We clarify the relation with biproduct Hopf algebras (the constructions are not the same), the response to twisting of braided groups and the abstract characterisation via automorphisms of the forgetful functor for the category of (co)modules of a braided group.
We write the fermionic $q$-Fock space representation of $U_q(\hat{sl_n})$ as an infinite extended braided tensor product of finite-dimensional fermionic $U_q(sl_n)$-quantum planes or exterior algebras. Using braided geometrical techniques developed for such quantum exterior algebras, we provide a new approach to the Kashiwara-Miwa-Stern action of the Heisenberg algebra on the $q$-fermionic Fock space, obtaining the action in detail for the lowest nontrivial case $[b_{2},b_{-2}]=2({1-q^{-4n}\over 1-q^{-4}})$. Our R-matrix approach includes other Hecke R-matrices as well.
We show that the affine quantum group $U_q(\hat{sl_2})$ is isomorphic to a bicrossproduct central extension $C Z{}_χ\cross U_q(Lsl_2)$ of the quantum loop group $U_q(Lsl_2)$ by a quantum cocycle $χ$, which we construct. We prove the same result for $U_q(\hat{g})$ in R-matrix form.
We introduce a quasitriangular Hopf algebra or `quantum group' $U(B)$, the {\em double-bosonisation}, associated to every braided group $B$ in the category of $H$-modules over a quasitriangular Hopf algebra $H$, such that $B$ appears as the `positive root space', $H$ as the `Cartan subalgebra' and the dual braided group $B^*$ as the `negative root space' of $U(B)$. The choice $B=f$ recovers Lusztig's construction of $U_q(g)$, where $f$ is Lusztig's algebra associated to a Cartan datum; other choices give more novel quantum groups. As an application, our construction provides a canonical way of building up quantum groups from smaller ones by repeatedly extending their positive and negative root spaces by linear braided groups; we explicitly construct $U_q(sl_3)$ from $U_q(sl_2)$ by this method, extending it by the quantum-braided plane $A_q^2$. We provide a fundamental representation of $U(B)$ in $B$. A projection from the quantum double, a theory of double biproducts and a Tannaka-Krein reconstruction point of view are also provided.
This is an introduction for algebraists to the theory of algebras and Hopf algebras in braided categories. Such objects generalise super-algebras and super-Hopf algebras, aswell as colour-Lie algebras. Basic facts about braided categories C are recalled, the modules and comodules of Hopf algebras in such categories are studied,the notion of `braided -commutative' or `braided-cocommutative' Hopf algebras (braided groups) is reviewed and a fully diagrammatic proof of the reconstruction theorem for a braided group aut(C) is given. The theory has important implications for the theory of quasitriangular Hopf algebras (quantum groups). It also includes important examples such as the degenerate Sklyanin algebra and the quantum plane.
We obtain the universal R-matrix of the non-standard quantum group associated to the Alexander-Conway knot polynomial. We show further that this non-standard quantum group is related to the super-quantum group $U_qgl(1|1)$ by a general process of superization, which we describe. We also study a twisted variant of this non-standard quantum group and obtain, as a result, a twisted version of $U_qgl(1|1)$ as a $q$-supersymmetry of the exterior differential calculus of any quantum plane of Hecke type, acting by mixing the bosonic $x_i$ co-ordinates and the forms $d x_i$.
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.
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.
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.
The braided approach to q-deformation (due to the author and collaborators) gives natural algebras $R_{21}u_1Ru_2=u_2R_{21}u_1R$ and $R_{21}x_1x_2=x_2x_1R$ for q-Minkowski and q-Euclidean spaces respectively. These algebras are covariant under a corresponding background `rotation' quantum group. Semidirect product by this according to the bosonisation procedure (also due to the author) gives the corresponding Poincaré quantum groups. We review the construction and collect the resulting R-matrix formulae for both Euclidean and Minkowski cases in both enveloping algebra and function algebra form, and the duality between them. Axioms for the Poincaré quantum group $*$-structure and the dilaton problem are discussed.
The $q$-Poincaré group of \cite{SWW:inh} is shown to have the structure of a semidirect product and coproduct $B\cocross \widetilde{SO_q(1,3)}$ where $B$ is a braided-quantum group structure on the $q$-Minkowski space of 4-momentum with braided-coproduct $\undΔ\vecp=\vecp\tens 1+1\tens \vecp$. Here the necessary $B$ is not a usual kind of quantum group, but one with braid statistics. Similar braided-vectors and covectors $V(R')$, $V^*(R')$ exist for a general R-matrix. The abstract structure of the $q$-Lorentz group is also studied.