The Problem of Differential Calculus on Quantum Groups

The bicovariant differential calculi on quantum groups of Woronowicz have the drawback that their dimensions do not agree with that of the corresponding classical calculus. In this paper we discuss the first-order differential calculus which arises from a simple quantum Lie algebra. This calculus has the correct dimension and is shown to be bicovariant and complete. But it does not satisfy the Leibniz rule. For sl_n this approach leads to a differential calculus which satisfies a simple generalization of the Leibniz rule.Comment: Contribution to the proceedings of the Colloquium on Quantum Groups and Integrable Systems Prague, June 1996. amslatex, 9 pages. For related information see http://www.mth.kcl.ac.uk/~delius/q-lie.htm

From Refuge to resistance: Botsabelo, Mafolofolo and Johannes Dinkwanyane: Missionaries and converts under the authority of the Zuid-Afrikaansche Republiek, 1860-1876

African Studies Seminar series. Paper presented 23 February, 1981In the 1870s the Transvaal witnessed an intensification of struggles over land and labour. This development was particularly marked in its eastern districts and was partly stimulated by the impact on the local and regional political economy of the discovery and exploitation of diamonds and gold. Also important was the changing nature of Z.A.R. control over, and intervention in, the countryside and the growing power of the Pedi polity. The latter had by the 1870s emerged as an alternative focus of power and authority to both the Z.A.R. and the Swazi kingdom. These factors shaped the disputes which culminated in the war between the Pedi and the Z.A.R. in 1876. This conflict in turn provided one of the pretexts for the British annexation of the Transvaal in early 1877

Sebatakgomo: Migrant organization, the ANC and the Sekhukhuneland Revolt

African Studies Seminar series. Paper presented. No dates given in the paper. No citations in copy. Marked 'Additional Seminar paper' and on the 1st page of text as 'informal'.In the 1940s and 1950s in reserve and trust area from the Zoutpansberg to the Ciskei bitter battles were fought against first Betterment Schemes and then Bantu Authorities. Communities believed - with good reason - that these state initiatives posed a mortal threat to their residual, but cherished, economic and political autonomy. These episodes are usually treated under the rubric of rural or peasant resistance but the centrality of migrant labour to the South African political economy has always undermined simple divisions between town and countryside. A closer examination shows that in virtually every instance of resistance urban-based migrant organizations played vital roles. Yet this is difficult to explain for groups like the Zoutpansberg Cultural Association, the Bahurutshe Association or the Mpondo Association step almost entirely unheralded onto the stage. We have the barest idea of the long history of migrant organization which preceded their part in these events. It has also become commonplace in the literature on 'rural resistance' to suggest that the ANC, while not entirely insensitive to rural issues in the 194Os and 1950s, nonetheless failed to establish effective rural organization and played at best a marginal role in the various revolts. This conclusion is partly based on the sparseness of Congress branches in the countryside. But it has been arrived at without any systematic attempt to examine a crucial question. Did migrants and their organizations provide a partly unseen but effective bridge between the ANC, the SACP and rural politics? These gaps in our understanding of 'rural resistance' will not easily be filled . This article, however, attempts to provide some illumination of these issues by means of a study of the role of migrants in the Sekhukhuneland Revolt of 1957 — 1961. To give some indication of the destination of the argument, the evidence suggests that a movement established in 1954 from within the ANC and the SACP - Sebatakqqmg - won widespread migrant support and played a key role in organizing and sustaining the resistance in the eastern Transvaal. The journey to this conclusion will, however, be long and prone to detour - for in order to be able to explain the interaction between migrants, the ANC, and rural conflict in the 1950s it is necessary to trace the changing patterns of Pedi employment and association from at least the 1930s

The structure of quantum Lie algebras for the classical series B_l, C_l and D_l

The structure constants of quantum Lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a Lie algebra at $q=1$. We explain the relationship between the structure constants of quantum Lie algebras and quantum Clebsch-Gordan coefficients for adjoint x adjoint ---> adjoint. We present a practical method for the determination of these quantum Clebsch-Gordan coefficients and are thus able to give explicit expressions for the structure constants of the quantum Lie algebras associated to the classical Lie algebras B_l, C_l and D_l. In the quantum case also the structure constants of the Cartan subalgebra are non-zero and we observe that they are determined in terms of the simple quantum roots. We introduce an invariant Killing form on the quantum Lie algebras and find that it takes values which are simple q-deformations of the classical ones.Comment: 25 pages, amslatex, eepic. Final version for publication in J. Phys. A. Minor misprints in eqs. 5.11 and 5.12 correcte

Quantum Lie algebras associated to Uq(gln)U_q(gl_n) and Uq(sln)U_q(sl_n)

Quantum Lie algebras \qlie{g} are non-associative algebras which are embedded into the quantized enveloping algebras $U_q(g)$ of Drinfeld and Jimbo in the same way as ordinary Lie algebras are embedded into their enveloping algebras. The quantum Lie product on \qlie{g} is induced by the quantum adjoint action of $U_q(g)$. We construct the quantum Lie algebras associated to $U_q(gl_n)$ and $U_q(sl_n)$. We determine the structure constants and the quantum root systems, which are now functions of the quantum parameter $q$. They exhibit an interesting duality symmetry under $q\leftrightarrow 1/q$.Comment: Latex 9 page

Representations of the Generalized Lie Algebra sl(2)_q

We construct finite-dimensional irreducible representations of two quantum algebras related to the generalized Lie algebra \ssll (2)_q introduced by Lyubashenko and the second named author. We consider separately the cases of $q$ generic and $q$ at roots of unity. Some of the representations have no classical analog even for generic $q$. Some of the representations have no analog to the finite-dimensional representations of the quantised enveloping algebra $U_q(sl(2))$, while in those that do there are different matrix elements.Comment: 14 pages, plain-TEX file using input files harvmac.tex, amssym.de

Boundary breathers in the sinh-Gordon model

We present an investigation of the boundary breather states of the sinh-Gordon model restricted to a half-line. The classical boundary breathers are presented for a two parameter family of integrable boundary conditions. Restricting to the case of boundary conditions which preserve the \phi --> -\phi symmetry of the bulk theory, the energy spectrum of the boundary states is computed in two ways: firstly, by using the bootstrap technique and subsequently, by using a WKB approximation. Requiring that the two descriptions of the spectrum agree with each other allows a determination of the relationship between the boundary parameter, the bulk coupling constant, and the parameter appearing in the reflection factor derived by Ghoshal to describe the scattering of the sinh-Gordon particle from the boundary.Comment: 16 pages amslate

Solutions to the Quantum Yang-Baxter Equation with Extra Non-Additive Parameters

We present a systematic technique to construct solutions to the Yang-Baxter equation which depend not only on a spectral parameter but in addition on further continuous parameters. These extra parameters enter the Yang-Baxter equation in a similar way to the spectral parameter but in a non-additive form. We exploit the fact that quantum non-compact algebras such as $U_q(su(1,1))$ and type-I quantum superalgebras such as $U_q(gl(1|1))$ and $U_q(gl(2|1))$ are known to admit non-trivial one-parameter families of infinite-dimensional and finite dimensional irreps, respectively, even for generic $q$. We develop a technique for constructing the corresponding spectral-dependent R-matrices. As examples we work out the the $R$-matrices for the three quantum algebras mentioned above in certain representations.Comment: 13 page

Quantum affine Toda solitons

We review some of the progress in affine Toda field theories in recent years, explain why known dualities cannot easily be extended, and make some suggestions for what should be sought instead.Comment: 16pp, LaTeX. Minor revision