Gustav W Delius and Andreas Hüffmann 1996 J. Phys. A: Math. Gen. 29 1703 doi:10.1088/0305-4470/29/8/018
On quantum Lie algebras and quantum root systems
Gustav W Delius and Andreas Hüffmann
Show affiliationsAs a natural generalization of ordinary Lie algebras we introduce the concept of quantum Lie algebras 
. We define these in terms of certain adjoint submodules of quantized enveloping algebras 
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 generalized 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 
and 
illustrate the results.
- PACS
-
02.20.Qs General properties, structure, and representation of Lie groups
- MSC
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16S30 Universal enveloping algebras of Lie algebras (See mainly 17B35)
81R50 Quantum groups and related algebraic methods (See also 16W35, 17B37)
22E60 Lie algebras of Lie groups (For the algebraic theory of Lie algebras, see 17Bxx)
- Subjects
- Dates
-
Issue 8 (21 April 1996)
Received 7 November 1995, in final form 15 January 1996
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On quantum Lie algebras and quantum root systems
Gustav W Delius and Andreas Hüffmann 1996 J. Phys. A: Math. Gen. 29 1703
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