Abstract
We introduce most of the concepts for q-Lie algebras in a way independent of the base field K. Again it turns out that we can keep the same Lie algebra with a small modification. We use very similar definitions for all quantities, which means that the proofs are similar. In particular, the quantities solvable, nilpotent, semisimple q-Lie algebra, Weyl group and Weyl chamber are identical with the ordinary case q = 1. The computations of sample q-roots for certain well-known q-Lie groups contain an extra q-addition, and consequently, for most of the quantities which are q-deformed, we add a prefix q in the respective name. Important examples are the q-Cartan subalgebra and the q-Cartan Killing form. We introduce the concept q-homogeneous spaces in a formal way exemplified by the examples
S
U
q
(
1
,
1
)
S
O
q
(
2
)
{{S{U_q}\left( {1,1} \right)} \over {S{O_q}\left( 2 \right)}}
and
S
O
q
(
3
)
S
O
q
(
2
)
{{S{O_q}\left( 3 \right)} \over {S{O_q}\left( 2 \right)}}
with corresponding q-Lie groups and q-geodesics. By introducing a q-deformed semidirect product, we can define exact sequences of q-Lie groups and some other interesting q-homogeneous spaces. We give an example of the corresponding q-Iwasawa decomposition for SLq(2).
On the Killing form of Lie Algebras in Symmetric Ribbon Categories
