Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1995-06-22
J.Phys.A29:1703-1722,1996
Physics
High Energy Physics
High Energy Physics - Theory
22 pages, latex, version to appear in J. Phys. A. see http://www.mth.kcl.ac.uk/~delius/q-lie.html for calculations and further
Scientific paper
10.1088/0305-4470/29/8/018
As a natural generalization of ordinary Lie algebras we introduce the concept of quantum Lie algebras ${\cal L}_q(g)$. We define these in terms of certain adjoint submodules of quantized enveloping algebras $U_q(g)$ endowed with a quantum Lie bracket given by the quantum adjoint action. The structure constants of these algebras depend on the quantum deformation parameter $q$ and they go over into the usual Lie algebras when $q=1$. The notions of q-conjugation and q-linearity are introduced. q-linear analogues of the classical antipode and Cartan involution are defined and a generalised Killing form, q-linear in the first entry and linear in the second, is obtained. These structures allow the derivation of symmetries between the structure constants of quantum Lie algebras. The explicitly worked out examples of $g=sl_3$ and $so_5$ illustrate the results.
Delius Gustav W.
Hueffmann Andreas
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