We introduce two classes of novel color superalgebras of $ \mathbb{Z}_2 \times \mathbb{Z}_2 $ grading. This is done by realizing members of each in the universal enveloping algebra of the ${\cal N}=1$ supersymmetric extension of the conformal Galilei algebra. This allows us to upgrade any representation of the super conformal Galilei algebras to a representation of the $ \mathbb{Z}_2 \times \mathbb{Z}_2 $ graded algebra. As an example, boson-fermion Fock space representation of one class is given. We also provide a vector field realization of members of the other class by using a generalization of the Grassmann calculus to $ \mathbb{Z}_2 \times \mathbb{Z}_2 $ graded setting.
N=4 l-conformal Galilei superalgebras inspired by D(2,1;a) supermultiplets
通过在一维D（2，1;a）上适当地扩张最一般的N=4超共形群所对应的李超代数，构造了l-共形Galilei代数的N=4超对称扩张.如果超代数中的加速度生成元形成D（2，1;a）的不可约（1，4，3）-，（2，4，2）-，（3，4，1）-和（4，4，0）-超多重态的类似物，则参数a被Jacobi恒等式约束。相反，如果加速度发生器的塔类似于一般实超场的分量分解，这是D（2，1;a）的可约表示，a仍然是任意的。一个N=4的l-共形Galilei超代数最近在[Phys. Lett. B 771（2017）401]被证明是这项工作中更一般的构造的一个特定实例。
In the recent paper, Bruce and Duplij introduced a double-graded version of supersymmetric quantum mechanics (SQM). It is an extension of Lie superalgebraic nature of ${\cal N}=1$ SQM to a $\mathbb{Z}_2^2$-graded superalgebra. In this work, we propose an extension of Bruce-Duplij model to higher values of $\cal N.$ Furthermore, it is shown that our construction of double-graded SQM is a special case of the method which converts a given Lie superalgebra to a $\mathbb{Z}_2^2$-graded superalgebra. By employing this method one may convert a model of superconformal mechanics to its double-graded ve…
We construct superconformal invariants in superspace which are used to build 3-point correlators of spinning operators in general $\cal{N}=2$ superconformal field theories in three dimensions. Our systematic analysis includes various relations between these invariants and provides a minimal set of parity-even and parity-odd invariants which is further used to construct general 3-point functions in any 3d $\cal{N}=2$ SCFT. For conserved (super)currents, we explicitly compute various 3-point functions using Wick contractions in the free field case, and express them in terms of the constructed pa…
The noncommutative spacetimes associated to the $κ$-Poincaré relativistic symmetries and their "non-relativistic" (Galilei) and "ultra-relativistic" (Carroll) limits are indistinguishable, since their coordinates satisfy the same algebra. In this work, we show that the three quantum kinematical models can be differentiated when looking at the associated spaces of time-like worldlines. Specifically, we construct the noncommutative spaces of time-like geodesics with $κ$-Galilei and $κ$-Carroll symmetries as contractions of the corresponding $κ$-Poincaré space and we show that these three spaces …
