arXiv subjects

Shahn Majid

Publications and source records attributed to Shahn Majid.

107 records · Page 6Linked to original sources

arXiv1994

Bicrossproduct structure of $κ$-Poincare group and non-commutative geometry

We show that the $κ$-deformed Poincaré quantum algebra proposed for elementary particle physics has the structure of a Hopf agebra bicrossproduct $U(so(1,3))\cobicross T$. The algebra is a semidirect product of the classical Lorentz group $so(1,3)$ acting in a deformed way on the momentum sector $T$. The novel feature is that the coalgebra is also semidirect, with a backreaction of the momentum sector on the Lorentz rotations. Using this, we show that the $κ$-Poincaré acts covariantly on a $κ$-Minkowski space, which we introduce. It turns out necessarily to be deformed and non-commutative. We also connect this algebra with a previous approach to Planck scale physics.

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arXiv1994

q-Euclidean space and quantum group wick rotation by twisting

We study the quantum matrix algebra $R_{21}x_1x_2=x_2x_1 R$ and for the standard $2\times 2$ case propose it for the co-ordinates of $q$-deformed Euclidean space. The algebra in this simplest case is isomorphic to the usual quantum matrices $M_q(2)$ but in a form which is naturally covariant under the Euclidean rotations $SU_q(2)\otimes SU_q(2)$. We also introduce a quantum Wick rotation that twists this system precisely into the approach to $q$-Minkowski space based on braided-matrices and their associated spinorial $q$-Lorentz group.

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arXiv1993

Braided Matrix Structure of $q$-Minkowski Space and $q$-Poincare Group

We clarify the relation between the approach to $q$-Minkowski space of Carow-Watamura et al. with an approach based on the idea of $2\times 2$ braided Hermitean matrices. The latter are objects like super-matrices but with Bose-Fermi statistics replaced by braid statistics. We also obtain new R-matrix formulae for the $q$-Poincare group in this framework.

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arXiv1993

Solutions Of The Yang-baxter Equations From Braided-Lie Algebras And Braided Groups

We obtain an R-matrix or matrix representation of the Artin braid group acting in a canonical way on the vector space of every (super)-Lie algebra or braided-Lie algebra. The same result applies for every (super)-Hopf algebra or braided-Hopf algebra. We recover some known representations such as those associated to racks. We also obtain new representations such as a non-trivial one on the ring $k[x]$ of polynomials in one variable, regarded as a braided-line. Representations of the extended Artin braid group for braids in the complement of $S^1$ are also obtained by the same method.

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arXiv1993

Lie Algebras and Braided Geometry

We show that every Lie algebra or superLie algebra has a canonical braiding on it, and that in terms of this its enveloping algebra appears as a flat space with braided-commuting coordinate functions. This also gives a new point of view about $q$-Minkowski space which arises in a similar way as the enveloping algebra of the braided Lie algebra $gl_{2,q}$. Our point of view fixes the signature of the metric on $q$-Minkowski space and hence also of ordinary Minkowski space at $q=1$. We also describe an abstract construction for left-invariant integration on any braided group.

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arXiv1993

Cross Product Quantisation, Nonabelian Cohomology And Twisting Of Hopf Algebras

This is an introduction to work on the generalisation to quantum groups of Mackey's approach to quantisation on homogeneous spaces. We recall the bicrossproduct models of the author, which generalise the quantum double. We describe the general extension theory of Hopf algebras and the nonAbelian cohomology spaces $\CH^2(H,A)$ which classify them. They form a new kind of topological quantum number in physics which is visible only in the quantum world. These same cross product quantisations can also be viewed as trivial quantum principal bundles in quantum group gauge theory. We also relate this nonAbelian cohomology $\CH^2(H,\C )$ to Drinfeld's theory of twisting.

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arXiv1993

On the Addition of Quantum Matrices

We introduce an addition law for the usual quantum matrices $A(R)$ by means of a coaddition $\underlineΔ t=t\otimes 1+1\otimes t$. It supplements the usual comultiplication $Δt=t\otimes t$ and together they obey a codistributivity condition. The coaddition does not form a usual Hopf algebra but a braided one. The same remarks apply for rectangular $m\times n$ quantum matrices. As an application, we construct left-invariant vector fields on $A(R)$ and other quantum spaces. They close in the form of a braided Lie algebra. As another application, the wave-functions in the lattice approximation of Kac-Moody algebras and other lattice fields can be added and functionally differentiated.

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arXiv1993

Glueing operation for r-matrices, quantum groups and link-invariants of Hecke type

We introduce an associative glueing operation $\oplus_q$ on the space of solutions of the Quantum Yang-Baxter Equations of Hecke type. The corresponding glueing operations for the associated quantum groups and quantum vector spaces are also found. The former involves $2\times 2$ quantum matrices whose entries are themselves square or rectangular quantum matrices. The corresponding glueing operation for link-invariants is introduced and involves a state-sum model with Boltzmann weights determined by the link invariants to be glued. The standard $su(n)$ solution, its associated quantum matrix group, quantum space and link-invariant arise at once by repeated glueing of the one-dimensional case.

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arXiv1993

The Quantum Double as Quantum Mechanics

We introduce $*$-structures on braided groups and braided matrices. Using this, we show that the quantum double $D(U_q(su_2))$ can be viewed as the quantum algebra of observables of a quantum particle moving on a hyperboloid in q-Minkowski space (a three-sphere in the Lorentz metric), and with the role of angular momentum played by $U_q(su_2)$. This provides a new example of a quantum system whose algebra of observables is a Hopf algebra. Furthermore, its dual Hopf algebra can also be viewed as a quantum algebra of observables, of another quantum system. This time the position space is a q-deformation of $SL(2,\R)$ and the momentum group is $U_q(su_2^*)$ where $su_2^*$ is the Drinfeld dual Lie algebra of $su_2$. Similar results hold for the quantum double and its dual of a general quantum group.

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arXiv1993

Quantum group gauge theory on quantum spaces

We construct quantum group-valued canonical connections on quantum homogeneous spaces, including a q-deformed Dirac monopole on the quantum sphere of Podles quantum differential coming from the 3-D calculus of Woronowicz on $SU_q(2)$ . The construction is presented within the setting of a general theory of quantum principal bundles with quantum group (Hopf algebra) fiber, associated quantum vector bundles and connection one-forms. Both the base space (spacetime) and the total space are non-commutative algebras (quantum spaces).

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arXiv1993

Quantum and Braided Lie Algebras

We introduce the notion of a braided Lie algebra consisting of a finite-dimensional vector space $\CL$ equipped with a bracket $[\ ,\ ]:\CL\tens\CL\to \CL$ and a Yang-Baxter operator $Ψ:\CL\tens\CL\to \CL\tens\CL$ obeying some axioms. We show that such an object has an enveloping braided-bialgebra $U(\CL)$. We show that every generic $R$-matrix leads to such a braided Lie algebra with $[\ ,\ ]$ given by structure constants $c^{IJ}{}_K$ determined from $R$. In this case $U(\CL)=B(R)$ the braided matrices introduced previously. We also introduce the basic theory of these braided Lie algebras, including the natural right-regular action of a braided-Lie algebra $\CL$ by braided vector fields, the braided-Killing form and the quadratic Casimir associated to $\CL$. These constructions recover the relevant notions for usual, colour and super-Lie algebras as special cases. In addition, the standard quantum deformations $U_q(g)$ are understood as the enveloping algebras of such underlying braided Lie algebras with $[\ ,\ ]$ on $\CL\subset U_q(g)$ given by the quantum adjoint action.

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arXiv1993

Free Braided Differential Calculus, Braided Binomial Theorem and the Braided Exponential Map

Braided differential operators $\del^i$ are obtained by differentiating the addition law on the braided covector spaces introduced previously (such as the braided addition law on the quantum plane). These are affiliated to a Yang-Baxter matrix $R$. The quantum eigenfunctions $\exp_R(\vecx|\vecv)$ of the $\del^i$ (braided-plane waves) are introduced in the free case where the position components $x_i$ are totally non-commuting. We prove a braided $R$-binomial theorem and a braided-Taylors theorem $\exp_R(\veca|\del)f(\vecx)=f(\veca+\vecx)$. These various results precisely generalise to a generic $R$-matrix (and hence to $n$-dimensions) the well-known properties of the usual 1-dimensional $q$-differential and $q$-exponential. As a related application, we show that the q-Heisenberg algebra $px-qxp=1$ is a braided semidirect product $\C[x]\cocross \C[p]$ of the braided line acting on itself (a braided Weyl algebra). Similarly for its generalization to an arbitrary $R$-matrix.

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arXiv1993

Beyond Supersymmetry and Quantum Symmetry (an introduction to braided groups and braided matrices)

This is a systematic introduction for physicists to the theory of algebras and groups with braid statistics, as developed over the last three years by the author. There are braided lines, braided planes, braided matrices and braided groups all in analogy with superlines, superplanes etc. The main idea is that the bose-fermi $\pm1$ statistics between Grassmannn coordinates is now replaced by a general braid statistics $Ψ$, typically given by a Yang-Baxter matrix $R$. Most of the algebraic proofs are best done by drawing knot and tangle diagrams, yet most constructions in supersymmetry appear to generalise well. Particles of braid statistics exist and can be expected to be described in this way. At the same time, we find many applications to ordinary quantum group theory: how to make quantum-group covariant (braided) tensor products and spin chains, action-angle variables for quantum groups, vector addition on $q$-Minkowski space and a semidirect product q-Poincaré group are among the main applications so far. Every quantum group can be viewed as a braided group, so the theory contains quantum group theory as well as supersymmetry. There also appears to be a rich theory of braided geometry, more general than super-geometry and including aspects of quantum geometry. Braided-derivations obey a braided-Leibniz rule and recover the usual Jackson $q$-derivative as the 1-dimensional case.

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arXiv1992

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 groups $U_q(g)$. They have the same FRT generators $l^\pm$ but a matrix braided-coproduct $\undΔL=L\und\tens L$ where $L=l^+Sl^-$, and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matrices $BM_q(2)$; it is a braided-commutative bialgebra in a braided category. As a second application, we show that the quantum double $D(\usl)$ (also known as the `quantum Lorentz group') is the semidirect product as an algebra of two copies of $\usl$, 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.

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arXiv1992

A Class of Bicovariant Differential Calculi on Hopf Algebras

We introduce a large class of bicovariant differential calculi on any quantum group $A$, associated to $Ad$-invariant elements. For example, the deformed trace element on $SL_q(2)$ recovers Woronowicz' $4D_\pm$ calculus. More generally, we obtain a sequence of differential calculi on each quantum group $A(R)$, based on the theory of the corresponding braided groups $B(R)$. Here $R$ is any regular solution of the QYBE.

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arXiv1992

Quantum and Braided Linear Algebra

Quantum matrices $A(R)$ are known for every $R$ matrix obeying the Quantum Yang-Baxter Equations. It is also known that these act on `vectors' given by the corresponding Zamalodchikov algebra. We develop this interpretation in detail, distinguishing between two forms of this algebra, $V(R)$ (vectors) and $V^*(R)$ (covectors). $A(R)\to V(R_{21})\tens V^*(R)$ is an algebra homomorphism (i.e. quantum matrices are realized by the tensor product of a quantum vector with a quantum covector), while the inner product of a quantum covector with a quantum vector transforms as a scaler. We show that if $V(R)$ and $V^*(R)$ are endowed with the necessary braid statistics $Ψ$ then their braided tensor-product $V(R)\und\tens V^*(R)$ is a realization of the braided matrices $B(R)$ introduced previously, while their inner product leads to an invariant quantum trace. Introducing braid statistics in this way leads to a fully covariant quantum (braided) linear algebra. The braided groups obtained from $B(R)$ act on themselves by conjugation in a way impossible for the quantum groups obtained from $A(R)$.

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arXiv1992

Quantum and super-quantum group related to the Alexander-Conway polynomial

We describe the quasitriangular structure (universal $R$-matrix) on the non-standard quantum group $U_q(H_1,H_2,X^\pm)$ associated to the Alexander-Conway matrix solution of the Yang-Baxter equation. We show that this Hopf algebra is connected with the super-Hopf algebra $U_qgl(1|1)$ by a general process of superization.

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