Lie superalgebra

Comments

Lie superalgebras show up in physics in several different ways. In conventional supersymmetry, the even elements of the superalgebra correspond to bosons and odd elements to fermions. This corresponds to a bracket that has a grading of zero:

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

This is not always the case; for example, in BRST supersymmetry and in the Batalin–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 two graded associative products. In addition to the Lie bracket, there may also be an "ordinary" product, thus giving rise to the Poisson superalgebra and the Gerstenhaber algebra. Such gradings are also observed in deformation theory.

Properties

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:

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 a symmetric ${\displaystyle {\mathfrak{g}}_0}$-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 a linear representation of ${\displaystyle {\mathfrak{g}}_0}$, and there exists a symmetric ${\displaystyle {\mathfrak{g}}_0}$-equivariant linear 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 (${\displaystyle {\mathfrak{g}}_1}$).

Involution

A ${\displaystyle \ast }$ 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.

Examples

Given any associative 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 ${\displaystyle M(p|q)}$. With the Lie bracket per above, the space is denoted ${\displaystyle \mathfrak{g}\mathfrak{l}(p|q)}$.

A Poisson algebra is an associative algebra together with a Lie bracket. If the algebra is given a Z₂-grading, such that the Lie bracket becomes a Lie superbracket, then one obtains the Poisson superalgebra. If, in addition, the associative product is made supercommutative, one obtains a supercommutative Poisson superalgebra.

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

The super-Poincaré algebra generates the isometries of flat superspace.

Classification

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

They are (excluding the Lie algebras):

The special 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 ${\displaystyle m\ge 2}$.

The orthosymplectic 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 ${\displaystyle \mathfrak{s}\mathfrak{o}(m)\oplus \mathfrak{s}\mathfrak{p}(2n)}$.

The exceptional 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 ${\displaystyle \alpha \mapsto -1-\alpha }$.

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

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

The exceptional 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-called strange series called ${\displaystyle \mathfrak{p}\mathfrak{e}(n)}$ and ${\displaystyle \mathfrak{q}(n)}$.

The Cartan 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.

Classification of infinite-dimensional simple linearly compact Lie superalgebras

The classification consists of the 10 series W(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 group SU(3)×SU(2)×U(1) as their zero level algebra. Infinite-dimensional (affine) Lie superalgebras are important symmetries in superstring theory. Specifically, the Virasoro algebras with ${\displaystyle \mathcal{N}}$ supersymmetries are ${\displaystyle K(1,\mathcal{N})}$ which only have central extensions up to ${\displaystyle \mathcal{N}=4}$.

Category-theoretic definition

In category theory, a Lie superalgebra can be defined as a nonassociative superalgebra 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: