Narrow Results

In this paper a finite dimensional unital associative algebra is presented, and its group of algebra automorphisms is detailed. The studied algebra can physically be understood as the creation operator algebra in a formal quantum field theory at fixed momentum for a spin 1/2 particle along with its antiparticle. It is shown that the essential part of the corresponding automorphism group can naturally be related to the conformal Lorentz group. In addition, the non-semisimple part of the automorphism group can be understood as “dressing” of the pure one-particle states. The studied mathematical structure may help in constructing quantum field theories in a non-perturbative manner. In addition, it provides a simple example of circumventing Coleman–Mandula theorem using non-semisimple groups, without SUSY.

We study the Leibniz $n$-algebra ${\mathbf{U}}_n(\mathfrak{L})$, whose multiplication is defined via the bracket of a Leibniz algebra $\mathfrak{L}$ as $[x_1,\dots ,x_{\mathrm{n}}]=[x_1,[\dots ,[x_{n-2},[x_{n-1},x_n]]\dots ]]$. We show that ${\mathbf{U}}_n(\mathfrak{L})$ is simple if and only if $\mathfrak{L}$ is a simple Lie algebra. An analog of Levi's theorem for Leibniz algebras in ${\mathbf{U}}_n(\mathbf{Lb})$ is established and it is proven that the Leibniz $n$-kernel of ${\mathbf{U}}_n(\mathfrak{L})$ for any semisimple Leibniz algebra $\mathfrak{L}$ is the $n$-algebra ${\mathbf{U}}_n(\mathfrak{L})$.

In this paper a finite dimensional unital associative algebra is presented, and its group of algebra automorphisms is detailed. The studied algebra can physically be understood as the creation operator algebra in a formal quantum field theory at fixed momentum for a spin 1/2 particle along with its antiparticle. It is shown that the essential part of the corresponding automorphism group can naturally be related to the conformal Lorentz group. In addition, the nonsemisimple part of the automorphism group can be understood as “dressing” of the pure one-particle states. The studied mathematical structure may help in constructing quantum field theories in a non-perturbative manner. In addition, it provides a simple example of circumventing Coleman–Mandula theorem using non-semisimple groups, without SUSY.