Some Remarks on Producing Hopf Algebras

Details Some Remarks on Producing Hopf Algebras Some Remarks on Producing Hopf Algebras

1995-07-17

arxiv.org/abs/q-alg/9507012v1

Mathematics

Quantum Algebra

6 pages, LATEX, no figures, Contribution to the Proceedings of the 4th Colloquium "Quantum Groups and Integrable Systems" (Pra

Scientific paper

10.1007/BF01688822

We report some observations concerning two well-known approaches to construction of quantum groups. Thus, starting from a bialgebra of inhomogeneous type and imposing quadratic, cubic or quartic commutation relations on a subset of its generators we come, in each case, to a q-deformed universal enveloping algebra of a certain simple Lie algebra. An interesting correlation between the order of initial commutation relations and the Cartan matrix of the resulting algebra is observed. Another example demonstrates that the bialgebra structure of sl_q(2) can be completely determined by requiring the q-oscillator algebra to be its covariant comodule, in analogy with Manin's approach to define SL_q(2) as a symmetry algebra of the bosonic and fermionic quantum planes.

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