I am looking for the earliest reference to the fact that any associative algebra becomes a Lie algebra with bracket $AXB-BXA$, where $X$ is a fixed element of the algebra. This is observed in the following paper:

Yanovski, A. B. "Linear bundles of Lie algebras and their applications." Journal of Mathematical Physics 41.11 (2000): 7869-7882.

But surely this is a much older result, at least for the case of matrix algebras.

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    $\begingroup$ Isn't it just because $(A,B)\mapsto AXB$ is associative? $\endgroup$ – YCor May 24 '15 at 13:51
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    $\begingroup$ Making a new semigroup or of an old one by fixing an element x and mapping (a,b) to axb is called a variant and has been around since probably the 50s or sixties. $\endgroup$ – Benjamin Steinberg May 24 '15 at 15:57
  • $\begingroup$ I know that the proof is trivial; I am interested in who first observed the fact. $\endgroup$ – Arnold Neumaier May 24 '15 at 16:25
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    $\begingroup$ My point is that it's the combination of two distinct facts and I think that in this case, the first appearance of this combination deserves less consideration than the appearance of these two facts separately, unless we speak of some particular interesting feature of this very combination. $\endgroup$ – YCor May 24 '15 at 18:20
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    $\begingroup$ Question: For two different $x$ and $y$ (say $n \times n$-matrices) is it known when the Lie algebras associated are isomorphic? $\endgroup$ – user74329 May 29 '15 at 13:22

I think it has first been considered by A.A. Albert in $1948$, in connection with so-called Lie-admissible algebras. An algebra $(A,\cdot)$ is called Lie-admissible, if $[a,b]=a\cdot b-b\cdot a$ defines a Lie bracket on the vector space of $A$. The bracket $[a,b]=axb-bxa$ has been considered later (around $1967$ for the bracket $[a,b]=\lambda ab-\mu ba$ for scalars $\lambda,\mu$, and $1978$ for the bracket $[a,b]=apb-bqa$ for fixed $p,q$ in an associative algebra) in physics (Lie-Santilli bracket). For references see (for example) here, page $8$.

  • $\begingroup$ I saw Albert's definition on p.575, but it is a different context, and he apparently doesn't state that the product $a\cdot b=axb$ constructed from an associative product is Lie admissible. $\endgroup$ – Arnold Neumaier May 26 '15 at 12:39
  • $\begingroup$ Right. I have seen many papers attributing this to Santilli (under different names, isotopy Lie algebras, Lie-Santilli bracket etc.). $\endgroup$ – Dietrich Burde May 26 '15 at 12:44
  • $\begingroup$ The first paper by Santilli containing the construction (called by him isotopic lifting) seems to be from 1983: Lie-isotopic lifting of the special relativity for extended deformable particles (r-m-santilli-foundation.com/docs/Santilli-50.pdf ) I didn't find it in several of his 1967/8 papers. $\endgroup$ – Arnold Neumaier May 26 '15 at 13:32
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    $\begingroup$ According to Makhlouf's paper on page $8$, Santilli introduced the bracket $[A,B]=APB-BQA$ for fixed $P,Q$ in $1978$, see the references given there. $\endgroup$ – Dietrich Burde May 26 '15 at 13:56
  • $\begingroup$ Thanks! The main paper from 1978 is available at rmsfoundation.org/docs/Santilli-58.pdf . The isotopic version of the product appears on p.287, and the Lie algebra version on p.333. $\endgroup$ – Arnold Neumaier May 26 '15 at 14:24

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