Lie superalgebra

In, a Lie superalgebra is a generalisation of a to include a Z2-. Lie superalgebras are important in where they are used to describe the mathematics of. In most of these theories, the even elements of the superalgebra correspond to s and odd elements to s (but this is not always true; for example, the is the other way around).

Definition
Formally, a Lie superalgebra is a nonassociative Z2-, or ', over a (typically R or C) whose product [·, ·], called the Lie superbracket or supercommutator''', satisfies the two conditions (analogs of the usual  axioms, with grading):

Super skew-symmetry:


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

The super Jacobi identity:


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

where x, y, and z are pure in the Z2-grading. Here, |x| denotes the degree of x (either 0 or 1). The degree of [x,y] is the sum of degree of x and y modulo 2.

One also sometimes adds the axioms $$[x,x]=0$$ for |x| = 0 (if 2 is invertible this follows automatically) and $${{Wikipedia link|x,x],x]=0$$ 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 the {{Wikipedia link|Poincaré–Birkhoff–Witt theorem}} holds (and, in general, they are necessary conditions for the theorem to hold).

Just as for Lie algebras, the {{Wikipedia link|universal enveloping algebra}} of the Lie superalgebra can be given a {{Wikipedia link|Hopf algebra}} structure.

A {{Wikipedia link|graded Lie algebra}} (say, graded by Z or N) that is anticommutative and Jacobi in the graded sense also has a $$Z_2$$ grading (which is called "rolling up" the algebra into odd and even parts), but is not referred to as "super". See {{Wikipedia link|Graded Lie algebra#note-0|note at graded Lie algebra}} for discussion.

Properties
Let $$\mathfrak g = \mathfrak g_0 \oplus \mathfrak g_1$$ 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 $$\mathfrak g_0$$ is an ordinary Lie algebra.
 * 2) One odd element. Then $$\mathfrak g_1$$ is a $$\mathfrak g_0$$-module for the action $$\mathrm{ad}_a: b \rightarrow [a, b], \quad a \in \mathfrak g_0, \quad b, [a, b] \in \mathfrak g_1$$.
 * 3) Two odd elements. The Jacobi identity says that the bracket $$\mathfrak g_1 \otimes \mathfrak g_1 \rightarrow \mathfrak g_0$$ is a symmetric $$\mathfrak g_1$$-map.
 * 4) Three odd elements. For all $$b \in \mathfrak g_1$$, $$[b,[b,b}} = 0$$.

Thus the even subalgebra $$\mathfrak g_0$$ of a Lie superalgebra forms a (normal) Lie algebra as all the signs disappear, and the superbracket becomes a normal Lie bracket, while $$\mathfrak g_1$$ is a of $$\mathfrak g_0$$, and there exists a  $$\mathfrak g_0$$-  $$\{\cdot,\cdot\}:\mathfrak g_1\otimes \mathfrak g_1\rightarrow \mathfrak g_0$$ such that,


 * $$[\left\{x, y\right\},z]+[\left\{y, z\right\},x]+[\left\{z, x\right\},y]=0, \quad x,y, z \in \mathfrak g_1.$$

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 ($$\mathfrak g_0$$) and a representation ($$\mathfrak g_1$$).

Involution
A ∗ Lie superalgebra is a complex Lie superalgebra equipped with an  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)undefined[y*,x*]; changing * to −* switches between the two conventions.) Its would be an ordinary.

Examples
Given any $$A$$ one can define the supercommutator on homogeneous elements by


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

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

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

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

The basic classical compact Lie superalgebras (that are not Lie algebras) are:

SU(m/n) These are the superunitary Lie algebras which have invariants:


 * $$ z.\overline{z}+iw.\overline{w}$$

This gives two orthosymplectic (see below) invariants if we take the m z variables and n w variables to be non-commutative and we take the real and imaginary parts. Therefore, we have


 * $$SU(m/n)=OSp(2m/2n)\cap OSp(2n/2m)$$

SU(n/n)/U(1) A special case of the superunitary Lie algebras where we remove one U(1) generator to make the algebra simple.

OSp(m/2n) These are the s. They have invariants given by:


 * $$x.x+y.z-z.y$$

for m commutative variables (x) and n pairs of anti-commutative variables (y,z). They are important symmetries in theories.

D(2/1;$$\alpha$$) This is a set of superalgebras parameterised by the variable $$\alpha$$. It has dimension 17 and is a sub-algebra of OSp(9|8). The even part of the group is O(3)×O(3)×O(3). So the invariants are:


 * $$A_\mu A_\mu+B_\mu B_\mu+C_\mu C_\mu +\psi^{\alpha \beta \gamma}\psi^{\alpha' \beta' \gamma'}\varepsilon_{\alpha \alpha'}\varepsilon_{\beta \beta'}\varepsilon_{\gamma \gamma'}$$


 * $$ A_{\{1} A_2 A_{3\}} + B_{\{1} B_2 B_{3\}} + C_{\{1} C_2 C_{3\}} + A_\mu \Gamma^{\alpha \alpha'}_\mu \psi\psi + B_\mu \Gamma^{\beta \beta'}_\mu \psi\psi + C_\mu \Gamma^{\gamma \gamma'}_\mu \psi\psi$$

for particular constants $$\gamma$$.

F(4) This exceptional Lie superalgebra has dimension 40 and is a sub-algebra of OSp(24|16). The even part of the group is O(3)xSO(7) so three invariants are:


 * $$B_{\mu \nu} + B_{\nu \mu} = 0 $$
 * $$A_\mu A_\mu + B_{\mu \nu}B_{\mu \nu} + \psi_{\{1}^\alpha \psi_{2\}}^\alpha$$
 * $$A_{\{1} A_2 A_{3\}} + B_{\{\mu \nu} B_{\nu \tau} B_{\tau \mu\}} + B_{\mu \nu} \sigma_{\mu \nu}^{\alpha \beta} \psi^\alpha_k \psi^\beta_k + A_\mu \Gamma_\mu^{\alpha \beta} \psi^k_\alpha \psi^k_\beta + (\text{sym.})$$

This group is related to the octonions by considering the 16 component spinors as two component octonion spinors and the gamma matrices acting on the upper indices as unit octonions. We then have $$f^{\mu \nu \tau}\sigma_{\nu \tau} \equiv \gamma_\mu$$ where f is the structure constants of octonion multiplication.

G(3) This exceptional Lie superalgebra has dimension 31 and is a sub-algebra of OSp(17|14). The even part of the group is O(3)×G2. The invariants are similar to the above (it being a subalgebra of the F(4)?) so the first invariant is:


 * $$A_\mu A_\mu + C^\mu_\alpha C^\mu_\alpha + \psi_{\{1}^\mu \psi_{2\}}^\nu$$

There are also two so-called strange series called p(n) and q(n).

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.

Category-theoretic definition
In, a Lie superalgebra can be defined as a nonassociative  whose product satisfies

where σ is the cyclic permutation braiding $$({\operatorname{id}} \otimes\tau_{A,A}) \circ (\tau_{A,A}\otimes {\operatorname{id}})$$. In diagrammatic form:
 * $$[\cdot,\cdot]\circ ({\operatorname{id}}+\tau_{A,A})=0$$
 * $$[\cdot,\cdot]\circ ([\cdot,\cdot]\otimes {\operatorname{id}} \circ({\operatorname{id}}+\sigma+\sigma^2)=0$$