Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · q-alg/9501029

Non-standard quantum (1+1) Poincaré group: a $T$--matrix approach

A. Ballesteros·F. J. Herranz·M. A. del Olmo·C. M. Pereña·M. Santander
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Hopf algebra dual form for the non--standard uniparametric deformation of the (1+1) Poincaré algebra $iso(1,1)$ is deduced. In this framework, the quantum coordinates that generate $Fun_w(ISO(1,1))$ define an infinite dimensional Lie algebra. A change in the basis of the dual form is obtained in order to compare this deformation to the standard one. Finally, a non--standard quantum Heisenberg group acting on a quantum Galilean plane is obtained.

Explore related subjects

q-algmath.QA

Keep this discovery

BibTeXRIS

A. Ballesteros, F. J. Herranz, M. A. del Olmo, C. M. Pereña, M. Santander. 1995-01-27. Non-standard quantum (1+1) Poincaré group: a $T$--matrix approach. https://doi.org/10.1088/0305-4470%2F28%2F24%2F012

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2019

Idempotents of Hecke algebras of type A

We use a skein-theoretic version of the Hecke algebras of type A to present three-dimensional diagrammatic views of Gyoja's idempotent elements, based closely on the corresponding Young diagram. In this context we give straightforward calculations for the eigenvalues of two natural central elements in the Hecke algebras, namely the full curl and the sum of the Murphy operators. We discuss their calculation also in terms of the framing factor associated to the appropriate irreducible representation of the quantum group SU(N,q).

q-alg↗
arXiv2017

Dual Affine Quantum Groups

Let $\hat{\mathfrak{g}}$ be an untwisted affine Kac-Moody algebra, with its Sklyanin-Drinfel'd structure of Lie bialgebra, and let $\hat{\mathfrak{h}}$ be the dual Lie bialgebra. By dualizing the quantum double construction - via formal Hopf algebras - we construct a new quantum group $U_q(\hat{\mathfrak{h}})$, dual of $U_q(\hat{\mathfrak{g}})$. Studying its restricted and unrestricted integer forms and their specializations at roots of 1 (in particular, their classical limits), we prove that $U_q(\hat{\mathfrak{h}})$ yields quantizations of $\hat{\mathfrak{h}}$ and $\hat{G}^\infty$ (the formal group attached to $\hat{\mathfrak{g}}$), and we construct new quantum Frobenius morphisms. The whole picture extends to the untwisted affine case the results known for quantum groups of finite type.

q-alg↗
arXiv2017

A PBW basis for Lusztig's form of untwisted affine quantum groups

Let $ \mathfrak{g} $ be an untwisted affine Kac-Moody algebra over the field $ K \, $, and let $ U_q(\mathfrak{g}) $ be the associated quantum enveloping algebra; let $ \mathfrak{U}_q(g) $ be the Lusztig's integer form of $ U_q(\mathfrak{g}) \, $, generated by $ q $-divided powers of Chevalley generators over a suitable subring $ R $ of $ K(q) \, $. We prove a Poincaré-Birkhoff-Witt like theorem for $ \mathfrak{U}_q(\mathfrak{g}) \, $, yielding a basis over $ R $ made of ordered products of $ q $-divided powers of suitable quantum root vectors.

q-alg↗