In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z₂-grading. 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). Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field (or commutative ring) with an extra piece of structure, known as a gradation (or grading). ... Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. ... This article or section is in need of attention from an expert on the subject. ... In particle physics, bosons, named after Satyendra Nath Bose, are particles having integer spin. ... In particle physics, fermions are particles with half-integer spin, such as protons and electrons. ... Possible alternative meanings of BRST are: BRST formalism Big Red Switch Time (or Big Red Switch Treatment): computer jargon for switching your computer off, when all other options for a more elegant shutdown have been exhausted. ...

Definition

Formally, a Lie superalgebra is a (nonassociative) Z₂-graded algebra, or superalgebra, over a commutative ring (typically R or C) whose product [·, ·], called the Lie superbracket or supercommutator, satisfies the two conditions (analogs of the usual Lie algebra axioms, with grading): In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field (or commutative ring) with an extra piece of structure, known as a gradation (or grading). ... In mathematics and theoretical physics, a superalgebra over a field K generally refers to a Z2-graded algebra over K (here Z2 is the cyclic group of order 2). ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...

Super skew-symmetry:

[x,y] = − ( − 1) ^{| x | | y |} [y,x]

The super Jacobi identity:

( − 1) ^{| z | | x |} [x,[y,z]] + ( − 1) ^{| x | | y |} [y,[z,x]] + ( − 1) ^{| y | | z |} [z,[x,y]] = 0

where x, y, and z are pure in the Z₂-grading. Here, |x| denotes the degree of x (either 0 or 1).

One also sometimes adds the axioms [x,x] = 0 for |x|=0 (if 2 is invertible this follows automatically) and [[x,x],x] = 0 for |x|=1 (if 3 is invertible this follows automatically).

Just as for Lie algebras, the universal enveloping algebra of the Lie superalgebra can be given a Hopf algebra structure. In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U(L). ... In mathematics, a Hopf algebra, named after Heinz Hopf, is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. ...

Distinction from graded Lie algebra

A graded Lie algebra (say, graded by Z or N) that is commutative and Jacobi in the graded sense also has a Z₂ grading (which is called "rolling up" the algebra into odd and even parts), but is not referred to as "super". See note at graded Lie algebra for discussion. In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. ... In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. ... In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. ...

Even and odd parts

Note that the even subalgebra of a Lie superalgebra forms a (normal) Lie algebra as all the signs disappear, and the superbracket becomes a normal Lie bracket.

One way of thinking about a Lie superalgebra is to consider its even and odd parts, L₀ and L₁ separately. Then, L₀ is a Lie algebra, L₁ is a linear representation of L₀, and there exists a symmetric L₀-intertwiner such that for all x,y and z in L₁, In mathematics, if φ: G→H is a homomorphism of Lie groups, and g and h are the Lie algebras of G and H respectively, then the induced map φ* on tangent spaces is a homomorphism of Lie algebras, i. ... Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. ...

Involution

A ^* Lie superalgebra is a complex Lie superalgebra equipped with an involutive antilinear map from itself to itself which respects the Z₂ grading and satisfies [x,y]^*=[y^*,x^*] for all x and y in the Lie superalgebra. (Some authors prefer the convention [x,y]^*=(−1)^|x||y|[y^*,x^*]; changing * to −* switches between the two conventions.) Its universal enveloping algebra would be an ordinary ^*-algebra. This page is about involution in mathematics; for involution in philosophy and integral theory, see Involution (philosophy). ... In mathematics, a real linear transformation f from a complex vector space V to another is said to be antilinear (or conjugate-linear or semilinear) if for all c, d in C and all x, y in V. See also: complex conjugate, sesquilinear form ... In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U(L). ... In mathematics, a *-algebra is an associative algebra over the field of complex numbers with an antilinear antiautomorphism * : A->A which is an involution. ...

Examples

Given any associative superalgebra A one can define the supercommutator on homogeneous elements by In mathematics and theoretical physics, a superalgebra over a field K is another name for a Z2-graded algebra over K. Specifically, a superalgebra is a super vector space A = A0 ⊕ A1 over K together with a bilinear multiplication which is an even morphism of super vector spaces. ...

[x,y] = xy − ( − 1) ^{| x | | y |} yx

and then extending by linearity to all elements. The algebra A together with the supercommutator then becomes a Lie superalgebra.

The Whitehead product on homotopy groups gives many examples of Lie superalgebras over the integers. The Whitehead product is a super quasi-Lie algebra structure on the homotopy groups of a space. ...

Classification

The simple complex finite dimensional Lie superalgebras were classified by Victor Kac. Victor Kac (born Dec 19, 1943 in Russia) is an American mathematician at M.I.T., known for his work in representation theory. ...

Category-theoretic definition

In category theory, a Lie superalgebra can be defined as a nonassociative superalgebra whose product satisfies In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics and theoretical physics, a superalgebra over a field K generally refers to a Z2-graded algebra over K (here Z2 is the cyclic group of order 2). ...

where σ is the cyclic permutation braiding . In diagrammatic form:

Image File history File links latex compiled File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

See also

• Algebra representation of a Lie superalgebra
• Anyonic Lie algebra
• Graded Lie algebra
• Grassmann algebra
• Representation of a Lie superalgebra
• Superspace
• Supergroup
• Quantum group
• Universal enveloping algebra

If we have a Lie superalgebra L, then, a (not necessarily associative) Z2 gradedalgebra A is an algebra representation of L if as a Z2graded vector space, A is a vector space rep of L and in addition, the elements of L acts as derivations/antiderivations. ... An anyonic Lie algebra is a U(1) graded vector space L over C equipped with a bilinear operator [.,.] and linear maps ε:L->C and Δ:L->L⊗L satisfying ε([X,Y])=ε(X)ε(Y) for pure graded elements X, Y and Z. See also Lie... In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... In the theory of Lie superalgebras, a representation of a Lie superalgebra L is the action of L upon a Z2-graded vector space V such that if A and B are any two pure elements of L (remember that L is Z2-graded) and X and Y are any... Superspace has had two meanings in physics. ... The concept of supergroup is a generalization of a that of group. ... In abstract algebra, a Hopf algebra is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes . (Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. ... In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U(L). ...

References

• Kac, V. G. Lie superalgebras. Advances in Math. 26 (1977), no. 1, 8--96.
• Manin, Yuri I. Gauge field theory and complex geometry. Grundlehren der Mathematischen Wissenschaften, 289. Springer-Verlag, Berlin, 1997. ISBN 3-540-61378-1