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Quantum deformations of simple Lie algebras

  Published:1997-06-01
 Printed: Jun 1997
  • Murray Bremner
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Abstract

It is shown that every simple complex Lie algebra $\fg$ admits a 1-parameter family $\fg_q$ of deformations outside the category of Lie algebras. These deformations are derived from a tensor product decomposition for $U_q(\fg)$-modules; here $U_q(\fg)$ is the quantized enveloping algebra of $\fg$. From this it follows that the multiplication on $\fg_q$ is $U_q(\fg)$-invariant. In the special case $\fg = {\ss}(2)$, the structure constants for the deformation ${\ss}(2)_q$ are obtained from the quantum Clebsch-Gordan formula applied to $V(2)_q \otimes V(2)_q$; here $V(2)_q$ is the simple 3-dimensional $U_q\bigl({\ss}(2)\bigr)$-module of highest weight $q^2$.
MSC Classifications: 17B37, 17A01 show english descriptions Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23]
General theory
17B37 - Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23]
17A01 - General theory
 

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