Draft:Lie-isotopic algebras

• Comment: Please look at the Lie algebra page. Currently this reads like a textbook extract or set of notes, not something suitable for Wikipedia. In addition, almost everything is from one group, and I have to wonder about a possible COI. Ldm1954 (talk) 00:24, 12 February 2024 (UTC)

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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

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

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 $C_{ij}^{*k}\neq C_{ij}^{k}$. 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)

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

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

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

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