Inmathematics, aLie superalgebra is a generalisation of aLie algebra to include a${\displaystyle \mathbb{Z}/2\mathbb{Z}}$‑grading. Lie superalgebras are important intheoretical physics where they are used to describe the mathematics ofsupersymmetry.

The notion of${\displaystyle \mathbb{Z}/2\mathbb{Z}}$ grading used here is distinct from a second${\displaystyle \mathbb{Z}/2\mathbb{Z}}$ grading having cohomological origins. Agraded Lie algebra (say, graded by${\displaystyle \mathbb{Z}}$ or${\displaystyle \mathbb{N}}$) that is anticommutative and has a gradedJacobi identity also has a${\displaystyle \mathbb{Z}/2\mathbb{Z}}$ 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 apair of${\displaystyle \mathbb{Z}/2\mathbb{Z}}$‑gradations: one of which is supersymmetric, and the other is classical.Pierre Deligne calls the supersymmetric one thesuper gradation, and the classical one thecohomological gradation. These two gradations must be compatible, and there is often disagreement as to how they should be regarded.^[1]

Formally, a Lie superalgebra is a nonassociativeZ₂-graded algebra, orsuperalgebra, over acommutative ring (typicallyR orC) whose product [·, ·], called theLie superbracket orsupercommutator, satisfies the two conditions (analogs of the usualLie algebra axioms, with grading):

Super skew-symmetry:

${\displaystyle [x,y]=-(-1{)}^{|x||y|}[y,x].}$

The super Jacobi identity:^[2]

${\displaystyle (-1{)}^{|x||z|}[x,[y,z]]+(-1{)}^{|y||x|}[y,[z,x]]+(-1{)}^{|z||y|}[z,[x,y]]=0,}$

wherex,y, andz are pure in theZ₂-grading. Here, |x| denotes the degree ofx (either 0 or 1). The degree of [x,y] is the sum of degree of x and y modulo 2.

One also sometimes adds the axioms${\displaystyle [x,x]=0}$ for |x| = 0 (if 2 is invertible this follows automatically) and${\displaystyle [[x,x],x]=0}$ for |x| = 1 (if 3 is invertible this follows automatically). When the ground ring is the integers or the Lie superalgebra is a free module, these conditions are equivalent to the condition that thePoincaré–Birkhoff–Witt theorem holds (and, in general, they are necessary conditions for the theorem to hold).

Just as for Lie algebras, theuniversal enveloping algebra of the Lie superalgebra can be given aHopf algebra structure.

Lie superalgebras show up in physics in several different ways. In conventionalsupersymmetry, theeven elements of the superalgebra correspond tobosons andodd elements tofermions. This corresponds to a bracket that has a grading of zero:

${\displaystyle |[a,b]|=|a|+|b|}$

This is not always the case; for example, inBRST supersymmetry and in theBatalin–Vilkovisky formalism, it is the other way around, which corresponds to the bracket of having a grading of -1:

${\displaystyle |[a,b]|=|a|+|b|-1}$

This distinction becomes particularly relevant when an algebra has not one, but twograded associative products. In addition to the Lie bracket, there may also be an "ordinary" product, thus giving rise to thePoisson superalgebra and theGerstenhaber algebra. Such gradings are also observed indeformation theory.

Let${\displaystyle \mathfrak{g}={\mathfrak{g}}_0\oplus {\mathfrak{g}}_1}$ be a Lie superalgebra. By inspecting the Jacobi identity, one sees that there are eight cases depending on whether arguments are even or odd. These fall into four classes, indexed by the number of odd elements:^[3]

1. No odd elements. The statement is just that${\displaystyle {\mathfrak{g}}_0}$ is an ordinary Lie algebra.
2. One odd element. Then${\displaystyle {\mathfrak{g}}_1}$ is a${\displaystyle {\mathfrak{g}}_0}$-module for the action${\displaystyle {\mathrm{a}\mathrm{d}}_a:b\to [a,b],a\in {\mathfrak{g}}_0,b,[a,b]\in {\mathfrak{g}}_1}$.
3. Two odd elements. The Jacobi identity says that the bracket${\displaystyle {\mathfrak{g}}_1\otimes {\mathfrak{g}}_1\to {\mathfrak{g}}_0}$ is asymmetric${\displaystyle {\mathfrak{g}}_0}$-map.
4. Three odd elements. For all${\displaystyle b\in {\mathfrak{g}}_1}$,${\displaystyle [b,[b,b]]=0}$.

Thus the even subalgebra${\displaystyle {\mathfrak{g}}_0}$ of a Lie superalgebra forms a (normal) Lie algebra as all the signs disappear, and the superbracket becomes a normal Lie bracket, while${\displaystyle {\mathfrak{g}}_1}$ is alinear representation of${\displaystyle {\mathfrak{g}}_0}$, and there exists asymmetric${\displaystyle {\mathfrak{g}}_0}$-equivariantlinear map${\displaystyle \{\cdot ,\cdot \}:{\mathfrak{g}}_1\otimes {\mathfrak{g}}_1\to {\mathfrak{g}}_0}$ such that,

${\displaystyle [\left\{x,y\right\},z]+[\left\{y,z\right\},x]+[\left\{z,x\right\},y]=0,x,y,z\in {\mathfrak{g}}_1.}$

Conditions (1)–(3) are linear and can all be understood in terms of ordinary Lie algebras. Condition (4) is nonlinear, and is the most difficult one to verify when constructing a Lie superalgebra starting from an ordinary Lie algebra (${\displaystyle {\mathfrak{g}}_0}$) and a representation (${\displaystyle {\mathfrak{g}}_1}$).

A${\displaystyle \ast }$Lie superalgebra is a complex Lie superalgebra equipped with aninvolutiveantilinear map from itself to itself which respects theZ₂ grading and satisfies[x,y]^* = [y^*,x^*] for allx andy in the Lie superalgebra. (Some authors prefer the convention [x,y]^* = (−1)^|x||y|[y^*,x^*]; changing * to −* switches between the two conventions.) Itsuniversal enveloping algebra would be an ordinary^*-algebra.

Given anyassociative superalgebra${\displaystyle A}$ one can define the supercommutator on homogeneous elements by

${\displaystyle [x,y]=xy-(-1{)}^{|x||y|}yx}$

and then extending by linearity to all elements. The algebra${\displaystyle A}$ together with the supercommutator then becomes a Lie superalgebra. The simplest example of this procedure is perhaps when${\displaystyle A}$ is the space of all linear functions${\displaystyle \mathbf{E}\mathbf{n}\mathbf{d}(V)}$ of a super vector space${\displaystyle V}$ to itself. When${\displaystyle V={\mathbb{K}}^{p|q}}$, this space is denoted by${\displaystyle M^{p|q}}$ or${\displaystyle M(p|q)}$.^[4] With the Lie bracket per above, the space is denoted${\displaystyle \mathfrak{g}\mathfrak{l}(p|q)}$.^[5]

APoisson algebra is an associative algebra together with a Lie bracket. If the algebra is given aZ₂-grading, such that the Lie bracket becomes a Lie superbracket, then one obtains thePoisson superalgebra. If, in addition, the associative product is madesupercommutative, one obtains a supercommutative Poisson superalgebra.

TheWhitehead product on homotopy groups gives many examples of Lie superalgebras over the integers.

Thesuper-Poincaré algebra generates the isometries of flatsuperspace.

The simple complex finite-dimensional Lie superalgebras were classified byVictor Kac.

They are (excluding the Lie algebras):^[6]

Thespecial linear lie superalgebra${\displaystyle \mathfrak{s}\mathfrak{l}(m|n)}$.

The lie superalgebra${\displaystyle \mathfrak{s}\mathfrak{l}(m|n)}$ is the subalgebra of${\displaystyle \mathfrak{g}\mathfrak{l}(m|n)}$ consisting of matrices with super trace zero. It is simple when${\displaystyle m\ne n}$. If${\displaystyle m=n}$, then the identity matrix${\displaystyle I_{2m}}$generates an ideal. Quotienting out this ideal leads to${\displaystyle \mathfrak{s}\mathfrak{l}(m|m)/⟨I_{2m}⟩}$ which is simple for${\displaystyle m\ge 2}$.

Theorthosymplectic Lie superalgebra${\displaystyle \mathfrak{o}\mathfrak{s}\mathfrak{p}(m|2n)}$.

Consider an even, non-degenerate, supersymmetric bilinear form${\displaystyle ⟨\cdot ,\cdot ⟩}$ on${\displaystyle {\mathbb{C}}^{m|2n}}$. Then the orthosymplectic Lie superalgebra is the subalgebra of${\displaystyle \mathfrak{g}\mathfrak{l}(m|2n)}$ consisting of matrices that leave this form invariant:$${\displaystyle \mathfrak{o}\mathfrak{s}\mathfrak{p}(m|2n)=\{X\in \mathfrak{g}\mathfrak{l}(m|2n)\mid ⟨Xu,v⟩+(-1{)}^{|X||u|}⟨u,Xv⟩=0\text{ for all }u,v\in {\mathbb{C}}^{m|2n}\}.}$$ Its even part is given by${\displaystyle \mathfrak{s}\mathfrak{o}(m)\oplus \mathfrak{s}\mathfrak{p}(2n)}$.

Theexceptional Lie superalgebra${\displaystyle D(2,1;\alpha )}$.

There is a family of (9∣8)-dimensional Lie superalgebras depending on a parameter${\displaystyle \alpha }$. These are deformations of${\displaystyle D(2,1)=\mathfrak{o}\mathfrak{s}\mathfrak{p}(4|2)}$. If${\displaystyle \alpha \ne 0}$ and${\displaystyle \alpha \ne -1}$, then D(2,1,α) is simple. Moreover${\displaystyle D(2,1;\alpha )\cong D(2,1;\beta )}$ if${\displaystyle \alpha }$ and${\displaystyle \beta }$ are under the same orbit under the maps${\displaystyle \alpha \mapsto {\alpha }^{-1}}$ and${\displaystyle \alpha \mapsto -1-\alpha }$.

Theexceptional Lie superalgebra${\displaystyle F(4)}$.

It has dimension (24|16). Its even part is given by${\displaystyle \mathfrak{s}\mathfrak{l}(2)\oplus \mathfrak{s}\mathfrak{o}(7)}$.

Theexceptional Lie superalgebra${\displaystyle G(3)}$.

It has dimension (17|14). Its even part is given by${\displaystyle \mathfrak{s}\mathfrak{l}(2)\oplus G_2}$.

There are also two so-calledstrange series called${\displaystyle \mathfrak{p}\mathfrak{e}(n)}$ and${\displaystyle \mathfrak{q}(n)}$.

TheCartan types. They can be divided in four families:${\displaystyle W(n)}$,${\displaystyle S(n)}$,${\displaystyle \tilde{S}(2n)}$ and${\displaystyle H(n)}$. For the Cartan type of simple Lie superalgebras, the odd part is no longer completely reducible under the action of the even part.

The classification consists of the 10 seriesW(m,n),S(m,n) ((m, n) ≠ (1, 1)),H(2m, n),K(2m + 1,n),HO(m, m) (m ≥ 2),SHO(m,m) (m ≥ 3),KO(m,m + 1),SKO(m, m + 1; β) (m ≥ 2),SHO ~ (2m, 2m),SKO ~ (2m + 1, 2m + 3) and the five exceptional algebras:

E(1, 6),E(5, 10),E(4, 4),E(3, 6),E(3, 8)

The last two are particularly interesting (according to Kac) because they have the standard model gauge groupSU(3)×SU(2)×U(1) as their zero level algebra. Infinite-dimensional (affine) Lie superalgebras are important symmetries insuperstring theory. Specifically, the Virasoro algebras with${\displaystyle \mathcal{N}}$ supersymmetries are${\displaystyle K(1,\mathcal{N})}$ which only have central extensions up to${\displaystyle \mathcal{N}=4}$.^[7]

Incategory theory, aLie superalgebra can be defined as a nonassociativesuperalgebra whose product satisfies

• ${\displaystyle [\cdot ,\cdot ]\circ (\mathrm{id}+{\tau }_{A,A})=0}$
• ${\displaystyle [\cdot ,\cdot ]\circ ([\cdot ,\cdot ]\otimes \mathrm{id}\circ (\mathrm{id}+\sigma +{\sigma }^2)=0}$

where σ is the cyclic permutation braiding${\displaystyle (\mathrm{id}\otimes {\tau }_{A,A})\circ ({\tau }_{A,A}\otimes \mathrm{id})}$. In diagrammatic form:

• Gerstenhaber algebra
• Anyonic Lie algebra
• Grassmann algebra
• Representation of a Lie superalgebra
• Superspace
• Supergroup
• Universal enveloping algebra

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• Irving Kaplansky + Lie Superalgebras

• Supersymmetry
• Lie algebras

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