Generalization of a Lie algebra

In mathematics, a Lie-isotopic algebra is a generalization of Lie algebras proposed by physicist R. M. Santilli in 1978.

Definition

Recall that a finite-dimensional Lie algebra^[1] ${\displaystyle L}$ with generators ${\displaystyle X_{1},X_{2},...,X_{n}}$ and commutation rules

${\displaystyle [X_{i}X_{j}]=X_{i}X_{j}-X_{j}X_{i}=C_{ij}^{k}X_{k},}$

can be defined (particularly in physics) as the totally anti-symmetric algebra ${\displaystyle A(L)^{-}}$ attached to the universal enveloping associative algebra ${\displaystyle A(L)=\{X_{1},X_{2},...,X_{n};X_{i}X_{j},i,j=1,...,n;1\}}$ equipped with the associative product ${\displaystyle X_{i}\times X_{j}}$ over a numeric field ${\displaystyle F}$ with multiplicative unit ${\displaystyle 1}$.

Consider now the axiom-preserving lifting of ${\displaystyle A(L)}$ into the form ${\displaystyle A^{*}(L^{*})=\{X_{1},X_{2},...,X_{n};X_{i}\times X_{j},i,j=1,...,n;1^{*}\}}$, called universal enveloping isoassociative algebra,^[2] with isoproduct

${\displaystyle X_{i}\times X_{j}=X_{i}T^{*}X_{j},}$

verifying the isoassociative law

${\displaystyle X_{i}\times (X_{j}\times X_{k})=X_{i}\times (X_{j}\times X_{k})}$

and multiplicative isounit

${\displaystyle 1^{*}=1/T*,1^{*}\times X_{k}=X_{k}\times 1^{*}=X_{k}\forall X_{k}inA^{*}(L^{*})}$

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

Then a Lie-isotopic algebra^[3] ${\displaystyle L^{*}}$ can be defined as the totally antisymmetric algebra attached to the enveloping isoassociative algebra. ${\displaystyle L^{*}=A^{*}(L^{*})^{-}}$ with isocommutation rules

${\displaystyle [X_{i},X_{j}]^{*}=X_{i}\times X_{j}-X_{j}\times X_{i}=X_{i}T^{*}X_{j}-X_{j}T^{*}X_{i}=C_{ij}^{*k}X_{k}.}$

It is evident that:^[4][5] 1) The isoproduct and the isounit coincide at the abstract level with the conventional product and; 2) The isocommutators ${\displaystyle [X_{i},X_{j}]^{*}}$ verify Lie's axioms; 3) In view of the infinitely possible isotopic elements ${\displaystyle T^{*}}$ (as numbers, functions, matrices, operators, etc.), any given Lie algebra ${\displaystyle L}$ admits an infinite class of isotopes; 4) Lie-isotopic algebras are called^[6] regular whenever ${\displaystyle C_{ij}^{*k}=C_{ij}^{k}}$, and irregular whenever ${\displaystyle C_{ij}^{*k}\neq C_{ij}^{k}}$. 5) All regular Lie-isotope ${\displaystyle L^{*}}$ are evidently isomorphic to ${\displaystyle L}$. However, the relationship between irregular isotopes ${\displaystyle L^{*}}$ and ${\displaystyle L}$ 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 ${\displaystyle SU^{*}(2)}$ of the ${\displaystyle SU(2)}$-spin symmetry ^[7] whose fundamental representation on a Hilbert space ${\displaystyle H}$ over the field of complex numbers ${\displaystyle C}$ can be obtained via the nonunitary transformation of the fundamental reopreserntation of ${\displaystyle SU(2)}$ (Pauli matrices)

${\displaystyle \sigma _{k}^{*}=U\sigma _{k}U^{\dagger },}$

${\displaystyle UU^{\dagger }=I^{*}=Diag.(\lambda ^{-1},\lambda ),Det1^{*}=1,}$

${\displaystyle \sigma _{1}^{*}=\left(\!{\begin{array}{cc}0&\lambda \\\lambda ^{-1}&0\end{array}}\!\right),\sigma _{2}^{*}=\left(\!{\begin{array}{cc}0&-i\!\lambda \\i\!\lambda ^{-1}&0\end{array}}\!\right),\sigma _{3}^{*}=\left(\!{\begin{array}{cc}\lambda ^{-1}&0\\0&-\lambda \end{array}}\!\right),}$

providing an explicit and concrete realization of Bohm's hidden variables ${\displaystyle \lambda }$,^[8] which is 'hidden' in the abstract axiom of associativity and allows an exact representation of the Deuteron magnetic moment.^[9]