Supergroup (physics)The concept of supergroup is a generalization of that of group. In other words, every supergroup carries a natural group structure, but there may be more than one way to structure a given group as a supergroup. A supergroup is like a Lie group in that there is a well defined notion of smooth function defined on them. However the functions may have even and odd parts. Moreover, a supergroup has a super Lie algebra which plays a role similar to that of a Lie algebra for Lie groups in that they determine most of the representation theory and which is the starting point for classification.
Grassmann numberIn mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra over the complex numbers. The special case of a 1-dimensional algebra is known as a dual number. Grassmann numbers saw an early use in physics to express a path integral representation for fermionic fields, although they are now widely used as a foundation for superspace, on which supersymmetry is constructed.
Extended supersymmetryIn theoretical physics, extended supersymmetry is supersymmetry whose infinitesimal generators carry not only a spinor index , but also an additional index where is integer (such as 2 or 4). Extended supersymmetry is also called , supersymmetry, for example. Extended supersymmetry is very important for analysis of mathematical properties of quantum field theory and superstring theory. The more extended supersymmetry is, the more it constrains physical observables and parameters.
Supersymmetry algebraIn theoretical physics, a supersymmetry algebra (or SUSY algebra) is a mathematical formalism for describing the relation between bosons and fermions. The supersymmetry algebra contains not only the Poincaré algebra and a compact subalgebra of internal symmetries, but also contains some fermionic supercharges, transforming as a sum of N real spinor representations of the Poincaré group. Such symmetries are allowed by the Haag–Łopuszański–Sohnius theorem. When N>1 the algebra is said to have extended supersymmetry.
Lie superalgebraIn mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the even elements of the superalgebra correspond to bosons and odd elements to fermions (but this is not always true; for example, the BRST supersymmetry is the other way around).
