Draft:Lie-isotopic algebras

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 $$A(L)=\{X_1, X_2, ..., X_n; X_iX_j, i, j = 1, ..., n; 1\}$$ 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


 * $$1^* = 1/T*, 1^* \times X_k = X_k \times 1^* = X_k \forall  X_k  in A^*(L^*)$$

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 $$SU^*(2)$$ 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^\dagger,$$



U U^\dagger = 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. which is 'hidden' in the abstract axiom of associativity and allows an exact representation of the Deuteron magnetic moment