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My Lie algebra with commutation relation

$[e_2, e_3] = e_3,\;[e_2, e_4] = -e_4,\;[e_3, e_4] = -e_1$

is isomorphic to Lie algebra $A_{4.7}^{-1}$ through transformations

$e_1\mapsto e_1,\;e_2\mapsto - e_4,\;e_3\mapsto e_3,\;e_4\mapsto e_2$

I tried to find classification of algebra $A_{4.7}^{-1}$ in Patera and Winternitz but there I find only classification of $A_{4.7}$.

Can anybody please suggest me article where classification of $A_{4.7}^{-1}$ is given ?

IgotiT
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  • How is $A_{4.7}^{-1}$ defined ? – Dietrich Burde Jun 19 '16 at 11:49
  • Roman Popovych suggested me this algebra, I don't know this algebra either. – IgotiT Jun 19 '16 at 12:04
  • @DietrichBurde: Actually I was reading paper Realizations of Real Low-Dimensional Lie Algebras by Roman O. Popovych and I asked him query about realization of algebra $[e_2, e_3] = e_3,;[e_2, e_4] = -e_4,;[e_3, e_4] = -e_1$ and he responded with algebra $A_{4.7}^{-1}$, unfortunately I don't know how this algebra is different from $A_{4.7}$. – IgotiT Jun 19 '16 at 13:35

1 Answers1

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Your Lie algebra is exactly $A_{4,8}^{-1}$ in Roman Popovych's classification, page $17$, Table $5$, after the suggested base change. The Lie brackets are given by $$ [x_2,x_3]=x_1,\;[x_2,x_4]=x_2,\;[x_3,x_4]=-x_3. $$ So Roman Popovych is right. There only was a typo in the index, it seems.

Dietrich Burde
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