Gelfand–Naimark–Segal construction

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).

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):

Super skew-symmetry:

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

The super Jacobi identity:

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

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 $[[x,x],x]=0$ 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 Poincaré-Birkhoff-Witt theorem holds (and, in general, they are necessary conditions for the theorem to hold).

Just as for Lie algebras, the universal enveloping algebra of the Lie superalgebra can be given a Hopf algebra structure.

Distinction from graded Lie algebra

A 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 note at graded Lie algebra for discussion.

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-equivariant linear map $\{\cdot ,\cdot \}:L_{1}\otimes L_{1}\rightarrow L_{0}$ such that for all x,y and z in L₁,

$\left\{x,y\right\}[z]+\left\{y,z\right\}[x]+\left\{z,x\right\}[y]=0.$

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.

Examples

Given any associative superalgebra 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 Whitehead product on homotopy groups gives many examples of Lie superalgebras over the integers.

Classification

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

The basic classical compact Lie superalgebras (that are not Lie algebras) are: [1]

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-commuative 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 Orthosymplectic groups. They have invariants given by:

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

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

D(2/1;$\alpha$) This is a set of superalgebras paramaterised 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)xO(3)xO(3). So the invariants are:

$A_{\mu }A_{\mu }+B_{\mu }B_{\mu }+C_{\mu }C_{\mu }+\psi ^{\alpha \beta \gamma }\psi ^{\alpha '\beta '\gamma '}\varepsilon _{\alpha \alpha '}\varepsilon _{\beta \beta '}\varepsilon _{\gamma \gamma '}$

$A_{\{1}A_{2}A_{3\}}+B_{\{1}B_{2}B_{3\}}+C_{\{1}C_{2}C_{3\}}+A_{\mu }\Gamma _{\mu }^{\alpha \alpha '}\psi \psi +B_{\mu }\Gamma _{\mu }^{\beta \beta '}\psi \psi +C_{\mu }\Gamma _{\mu }^{\gamma \gamma '}\psi \psi$

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:

$B_{\mu \nu }+B_{\nu \mu }=0$

$A_{\mu }A_{\mu }+B_{\mu \nu }B_{\mu \nu }+\psi _{\{1}^{\alpha }\psi _{2\}}^{\alpha }$

$A_{\{1}A_{2}A_{3\}}+B_{\{\mu \nu }B_{\nu \tau }B_{\tau \mu \}}+B_{\mu \nu }\sigma _{\mu \nu }^{\alpha \beta }\psi _{k}^{\alpha }\psi _{k}^{\beta }+A_{\mu }\Gamma _{\mu }^{\alpha \beta }\psi _{\alpha }^{k}\psi _{\beta }^{k}+(sym.)$

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)xG2. The invariants are similar to the above (it being a subalgebra of the F(4)?) so the first invariant is:

$A_{\mu }A_{\mu }+C_{\alpha }^{\mu }C_{\alpha }^{\mu }+\psi _{\{1}^{\mu }\psi _{2\}}^{\nu }$

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 5 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)xSU(2)xU(1) as their zero level algebra. Infinite dimensional (affine) Lie superalgebras are important symmetries in superstring theory.

Category-theoretic definition

In category theory, a Lie superalgebra can be defined as a nonassociative superalgebra whose product satisfies

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

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

See also

• Anyonic Lie algebra
• Grassmann algebra
• Representation of a Lie superalgebra
• Superspace
• Supergroup

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
• Pavel Grozman, Dimitry Leites and Irina Shchepochkina. "LIE SUPERALGEBRAS OF STRING THEORIES"

External links

• Irving Kaplansky + Lie Superalgebras