In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a ‑grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry.
The notion of grading used here is distinct from a second grading having cohomological origins. A graded Lie algebra (say, graded by or ) that is anticommutative and has a graded Jacobi identity also has a grading; this is the "rolling up" of the algebra into odd and even parts. This rolling-up is not normally referred to as "super". Thus, supergraded Lie superalgebras carry a pair of ‑gradations: one of which is supersymmetric, and the other is classical. Pierre Deligne calls the supersymmetric one the super gradation, and the classical one the cohomological gradation. These two gradations must be compatible, and there is often disagreement as to how they should be regarded.[1]
a Liesuperalgebra is a generalisation of a Lie algebra to include a Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } ‑grading. Liesuperalgebras are...
field of representation theory, a representation of a Liesuperalgebra is an action of Liesuperalgebra L on a Z2-graded vector space V, such that if A and...
universal enveloping algebra of a Liesuperalgebra which is a unital, associative superalgebra. Let A be a superalgebra over a commutative ring K. The submodule...
semisimple Lie algebra with the structure of a graded Lie algebra. Any parabolic Lie algebra is also a graded Lie algebra. A graded Liesuperalgebra extends...
product, a Lie superbracket [ ⋅ , ⋅ ] : A ⊗ A → A {\displaystyle [\cdot ,\cdot ]:A\otimes A\to A} such that (A, [·,·]) is a Liesuperalgebra and the operator...
supersymmetric quantum mechanics, an application of the supersymmetry superalgebra to quantum mechanics as opposed to quantum field theory. It was hoped...
of a Liesuperalgebra that are analogous to representations of a Lie algebra. Each Lie algebra has an associated Lie group and a Liesuperalgebra can sometimes...
representation theory of affine Kac–Moody algebras. Liesuperalgebras are generalizations of Lie algebras in which the underlying vector space has a Z2-grading...
therefore may be written as a triple (t, Θ, Θ*). The coordinates form a Liesuperalgebra, in which the gradation degree of t is even and that of Θ and Θ* is...
is called a Liesuperalgebra. Just as one can have representations of a Lie algebra, one can also have representations of a Liesuperalgebra, called supermultiplets...
spaces. This leads to a treatment of "superobjects" such as superalgebras, Liesuperalgebras, supergroups, etc. that is completely analogous to their ungraded...
The analogous observation for Liesuperalgebras gives the notion of a Poisson superalgebra. Representation of a Lie group Weight (representation theory)...
In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements x, y we...
Wigner's classification Gauge theory Grand unification theory Supergroup Liesuperalgebra Twistor theory Anyon Witt algebra Virasoro algebra Erlangen programme...
theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two...
then V is called a vertex operator superalgebra. One of the simplest examples is the vertex operator superalgebra generated by a single free fermion ψ...
that combines the structures of a supercommutative ring and a graded Liesuperalgebra. It is used in the Batalin–Vilkovisky formalism. It appears also in...
(The name heterotic SO(32) is slightly inaccurate since among the SO(32) Lie groups, string theory singles out a quotient Spin(32)/Z2 that is not equivalent...
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