Spin Representations of the q-Poincare Algebra

Christian Blohmann

Home/Archive/arXiv

arXiv · math/0110219

Spin Representations of the q-Poincare Algebra

Christian Blohmann

View original source ↗Read source document ↗Export metadata ↓

Abstract

The spin of particles on a non-commutative geometry is investigated within the framework of the representation theory of the q-deformed Poincare algebra. An overview of the q-Lorentz algebra is given, including its representation theory with explicit formulas for the q-Clebsch-Gordan coefficients. The vectorial form of the q-Lorentz algebra (Wess), the quantum double form (Woronowicz), and the dual of the q-Lorentz group (Majid) are shown to be essentially isomorphic. The construction of q-Minkowski space and the q-Poincare algebra is reviewed. The q-Euclidean sub-algebra, generated by rotations and translations, is studied in detail. The results allow for the construction of the q-Pauli-Lubanski vector, which, in turn, is used to determine the q-spin Casimir and the q-little algebras for both the massive and the massless case. Irreducible spin representations of the q-Poincare algebra are constructed in an angular momentum basis, accessible to physical interpretation. It is shown how representations can be constructed, alternatively, by the method of induction. Reducible representations by q-Lorentz spinor wave functions are considered. Wave equations on these spaces are found, demanding that the spaces of solutions reproduce the irreducible representations. As generic examples the q-Dirac equation and the q-Maxwell equations are computed explicitly and their uniqueness is shown.

Explore related subjects

math.QA hep-th

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Christian Blohmann. 2001-10-19. Spin Representations of the q-Poincare Algebra. https://arxiv.org/abs/math/0110219

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

KEEP EXPLORING

Related papers

arXiv2026

Brunnian braids and the inclusion from double shuffle Lie algebra to Kashiwara-Vergne Lie algebra

Schneps \cite{Schneps2012,Schneps2025} and Enriquez-Furusho \cite{EF4} proved that the double shuffle Lie algebra $\mathfrak{dmr}_0$ embeds into the Kashiwara--Vergne Lie algebra $\mathfrak{krv}_2$. We give a Brunnian braid interpretation of a related embedding into the symmetric Kashiwara--Vergne Lie algebra $\mathfrak{krv}_2^{\mathrm{sym}}$. More precisely, the map \[ φ\longmapsto \bigl(φ(-x_0-x_1,x_0),φ(-x_0-x_1,x_1)\bigr) \] defines an injective Lie algebra homomorphism from the subalgebra of $\mathfrak{dmr}_0$ satisfying the condition \[ [x_0,φ(-x_0-x_1,x_0)] +[x_1,φ(-x_0-x_1,x_1)]=0 \] into $\mathfrak{krv}_2^{\mathrm{sym}}$. The proof reformulate the double shuffle and symmetric Kashiwara--Vergne relations through abelianizations of Brunnian Lie algebras associated with the disk and punctured disks. We generalize this inclusion in two directions. First, replacing these abelianizations by higher lower central series quotients yields generalizations of relations and implications among them. Second, we establish explicit identities relating the linear pentagon defect to the stuffle coproduct, the divergence map, and the necklace cobracket.

math.QA↗

arXiv2026

Classification of Strongly Rational Vertex Operator Algebras with Central and Effective Central Charge One

We classify the strongly rational vertex operator algebras whose central charge and effective central charge are both one. Every such algebra is isomorphic to one of $V_L,V_L^+,V_{A_1}^{A_4},V_{A_1}^{S_4},V_{A_1}^{A_5},$ where $L$ ranges over the positive-definite even lattices of rank one and $A_1=\sqrt2\mathbb{Z}$.

math.QA↗

arXiv2026

A $q$-Weyl Freeness Principle for Nichols Algebras and Pointed Hopf Algebras of Square-Free Dimension

Let $H$ be a pointed Hopf algebra of square-free dimension over an algebraically closed field of characteristic $p>0$. We prove that either $H$ is a group algebra or $\dim H/|\G(H)|=p$, and that in the latter case $H$ belongs to exactly one of two explicit families of rank-one pointed Hopf algebras. We develop a truncated $q$-Weyl freeness principle for finite-dimensional Nichols algebras of quandle type. If $V=\bigoplus_{x\in X}\K e_x$, $X'\subsetneq X$ is a nonempty subquandle, $V'=\bigoplus_{x\in X'}\K e_x$, and $s\in X\setminus X'$, then $\mathcal B(V)\simeq\K[e_s]/(e_s^{m_s})\otimes C_{s,X'}\otimes\mathcal B(V')$ for some graded vector space $C_{s,X'}$, where $m_s$ is the nilpotency order of $e_s$; in particular, $(m_s)_z\,\mathcal H_{\mathcal B(V')}(z)\mid\mathcal H_{\mathcal B(V)}(z)$. In the non-group case, this yields a $p^2$-divisibility obstruction that rules out noncentral support for the infinitesimal braiding. Together with a graded-dual argument, the resulting rank-one reduction forces the diagram of $H$ to have dimension $p$.

math.QA↗