Lie-isotopic algebra

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Recall that a finite-dimensional Lie algebra $L$ with generators $X_1,X_2,...,X_n$ and commutation rules

$[X_i{X}_j]=X_i{X}_j-X_j{X}_i=C_{ij}^k{X}_k,$

can be defined (particularly in physics) as the totally anti-symmetric algebra $A(L{)}^{-}$ attached to the universal enveloping associative algebra $A(L)=\{X_1,X_2,...,X_n;X_i{X}_j,i,j=1,...,n;1\}$ equipped with the associative product $X_i\times X_j$ over a numeric field $F$ with multiplicative unit $1$.

Consider now the axiom-preserving lifting of $A(L)$ into the form $A^{*}(L^{*})=\{X_1,X_2,...,X_n;X_i\times X_j,i,j=1,...,n;1^{*}\}$, called universal enveloping isoassociative algebra, with isoproduct

$X_i\times X_j=X_i{T}^{*}{X}_j,$

verifying the isoassociative law

$X_i\times (X_j\times X_k)=X_i\times (X_j\times X_k)$

and multiplicative isounit

$1^{*}=1/T*,1^{*}\times X_k=X_k\times 1^{*}=X_k\mathrm{\forall }{X}_{k}in{A}^{*}(L^{*})$

where $T^{*}$, called the isotopic element, is not necessarily an element of $A(L)$ which is solely restricted by the condition of being positive-definite, $T^{*}>0$ , but otherwise having any desired dependence on local variables, and the products $X_i{T}^{*},T^{*}{X}_j,etc.$ are conventional associative products in $A(L)$.

Then a Lie-isotopic algebra $L^{*}$ can be defined as the totally antisymmetric algebra attached to the enveloping isoassociative algebra. $L^{*}=A^{*}(L^{*}{)}^{-}$ with isocommutation rules

$[X_i,X_j{]}^{*}=X_i\times X_j-X_j\times X_i=X_i{T}^{*}{X}_j-X_j{T}^{*}{X}_i=C_{ij}^{*k}{X}_k.$

It is evident that: 1) The isoproduct and the isounit coincide at the abstract level with the conventional product and; 2) The isocommutators $[X_i,X_j{]}^{*}$ verify Lie's axioms; 3) In view of the infinitely possible isotopic elements $T^{*}$ (as numbers, functions, matrices, operators, etc.), any given Lie algebra $L$ admits an infinite class of isotopes; 4) Lie-isotopic algebras are called regular whenever $C_{ij}^{*k}=C_{ij}^k$, and irregular whenever $C_{ij}^{*k}\ne C_{ij}^k$. 5) All regular Lie-isotope $L^{*}$ are evidently isomorphic to $L$. However, the relationship between irregular isotopes $L^{*}$ and $L$ does not appear to have been studied to date (Jan. 20, 2024).

An illustration of the applications cf Lie-isotopic algebras in physics is given by the isotopes $S{U}^{*}(2)$ of the $SU(2)$-spin symmetry whose fundamental representation on a Hilbert space $H$ over the field of complex numbers $C$ can be obtained via the nonunitary transformation of the fundamental reopreserntation of $SU(2)$ (Pauli matrices)

${\sigma }_k^{*}=U{\sigma }_k{U}^{†},$

$U{U}^{†}=I^{*}=Diag.({\lambda }^{-1},\lambda ),Det{1}^{*}=1,$

${\sigma }_1^{*}=\left(\begin{array}{cc}0& \lambda \\ {\lambda }^{-1}& 0\end{array}\right),{\sigma }_2^{*}=\left(\begin{array}{cc}0& -i\lambda \\ i{\lambda }^{-1}& 0\end{array}\right),{\sigma }_3^{*}=\left(\begin{array}{cc}{\lambda }^{-1}& 0\\ 0& -\lambda \end{array}\right),$

providing an explicit and concrete realization of Bohm's hidden variables math>\lambdca</math>. which is 'hidden' in the abstract axiom of associativity and allows an exact representation of the Deuteron magnetic moment

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