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| In [[mathematics]], a '''Lie superalgebra''' is a generalisation of a [[Lie algebra]] to include a '''Z'''<sub>2</sub>-[[graded algebra|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 [[boson]]s and ''odd'' elements to [[fermion]]s (but this is not always true; for example, the [[BRST supersymmetry]] is the other way around).
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| ==Definition==
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| Formally, a '''Lie superalgebra''' is a (nonassociative) '''Z'''<sub>2</sub>-[[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):
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| | | <li>[http://www.btalk.tk/activity/p/6175/ http://www.btalk.tk/activity/p/6175/]</li> |
| Super skew-symmetry:
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| | | <li>[http://www.oisex.cn/forum.php?mod=viewthread&tid=664643 http://www.oisex.cn/forum.php?mod=viewthread&tid=664643]</li> |
| :<math>[x,y]=-(-1)^{|x| |y|}[y,x].\ </math>
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| | | <li>[http://www.histoirepassion.eu/spip.php?article1761/ http://www.histoirepassion.eu/spip.php?article1761/]</li> |
| The super Jacobi identity:
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| | | <li>[http://bbs.whxx.mhedu.sh.cn/forum.php?mod=viewthread&tid=1396555 http://bbs.whxx.mhedu.sh.cn/forum.php?mod=viewthread&tid=1396555]</li> |
| :<math>[x,[y,z]]=[[x,y],z]+(-1)^{|x| |y|}[y,[x,z]]\ </math>
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| where ''x'', ''y'', and ''z'' are pure in the '''Z'''<sub>2</sub>-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.
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| One also sometimes adds the axioms <math>[x,x]=0</math> for |''x''|=0 (if 2 is invertible this follows automatically) and <math>[[x,x],x]=0</math> 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).
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| Just as for Lie algebras, the [[universal enveloping algebra]] of the Lie superalgebra can be given a [[Hopf algebra]] structure.
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| ===Distinction from graded Lie algebra===
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| A [[graded Lie algebra]] (say, graded by '''Z''' or '''N''') that is anticommutative and Jacobi in the graded sense also has a <math>Z_2</math> grading (which is called "rolling up" the algebra into odd and even parts), but is not referred to as "super". See [[Graded_Lie_algebra#_note-0|note at graded Lie algebra]] for discussion.
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| ===Even and odd parts===
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| 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.
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| One way of thinking about a Lie superalgebra is to consider its even and odd parts, L<sub>0</sub> and L<sub>1</sub> separately. Then, L<sub>0</sub> is a Lie algebra, L<sub>1</sub> is a [[representation of a Lie algebra|linear representation]] of L<sub>0</sub>, and there exists a [[symmetric]] L<sub>0</sub>-[[equivariant]] [[linear map]] <math>\{\cdot,\cdot\}:L_1\otimes L_1\rightarrow L_0</math> such that for all x,y and z in L<sub>1</sub>,
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| :<math>\left\{x, y\right\}[z]+\left\{y, z\right\}[x]+\left\{z, x\right\}[y]=0.</math>
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| ===Involution=== | |
| A '''<sup>*</sup> Lie superalgebra''' is a complex Lie superalgebra equipped with an [[Involution (mathematics)|involutive]] [[antilinear]] map from itself to itself which respects the '''Z'''<sub>2</sub> grading and satisfies
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| [x,y]<sup>*</sup>=[y<sup>*</sup>,x<sup>*</sup>] for all x and y in the Lie superalgebra. (Some authors prefer the convention [x,y]<sup>*</sup>=(−1)<sup>|x||y|</sup>[y<sup>*</sup>,x<sup>*</sup>]; changing * to −* switches between the two conventions.) Its [[universal enveloping algebra]] would be an ordinary [[star-algebra|<sup>*</sup>-algebra]].
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| ==Examples==
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| Given any [[associative superalgebra]] ''A'' one can define the supercommutator on homogeneous elements by
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| :<math>[x,y] = xy - (-1)^{|x||y|}yx\ </math>
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| and then extending by linearity to all elements. The algebra ''A'' together with the supercommutator then becomes a Lie superalgebra. | |
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| The [[Whitehead product]] on homotopy groups gives many examples of Lie superalgebras over the integers.
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| ==Classification==
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| The simple complex finite dimensional Lie superalgebras were classified by [[Victor Kac]].
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| The basic classical compact Lie superalgebras (that are not Lie algebras) are: [http://www.springerlink.com/content/f380116p6858n46n/]
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| '''SU(m/n)''' These are the superunitary Lie algebras which have invariants:
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| :<math> z.\overline{z}+iw.\overline{w}</math>
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| 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
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| :<math>SU(m/n)=OSp(2m/2n)\cap OSp(2n/2m)</math>
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| '''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.
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| '''OSp(m/2n)''' These are the Orthosymplectic groups. They have invariants given by:
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| :<math>x.x+y.z-z.y</math>
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| for m commutative variables (x) and n pairs of anti-commuative variables (y,z). They are important symmetries in [[supergravity]] theories.
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| '''D(2/1;<math>\alpha</math>)''' This is a set of superalgebras paramaterised by the variable '''<math>\alpha</math>'''. 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:
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| :<math>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'}</math>
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| :<math> 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
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| + B_\mu \Gamma^{\beta \beta'}_\mu \psi\psi + C_\mu \Gamma^{\gamma \gamma'}_\mu \psi\psi</math>
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| for particular constants <math>\gamma</math>.
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| '''F(4)'''
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| 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:
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| :<math>B_{\mu \nu} + B_{\nu \mu} = 0 </math> | |
| :<math>A_\mu A_\mu + B_{\mu \nu}B_{\mu \nu} + \psi_{\{1}^\alpha \psi_{2\}}^\alpha</math>
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| :<math>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 + (sym.)</math>
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| 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 <math>f^{\mu \nu \tau}\sigma_{\nu \tau} \equiv \gamma_{\mu}</math> where f is the structure constants of octonion multiplication.
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| '''G(3)'''
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| 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:
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| :<math>A_\mu A_\mu + C^{\mu}_\alpha C^{\mu}_\alpha + \psi_{\{1}^\mu \psi_{2\}}^\nu</math>
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| There are also two so-called '''strange''' series called '''p(n)''' and '''q(n)'''.
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| ==Classification of Infinite Dimensional Simple Linearly Compact Lie Superalgebras==
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| 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)'''
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| (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:
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| ::'''E(1,6)''', '''E(5,10)''', '''E(4,4)''', '''E(3,6)''', '''E(3,8)'''
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| 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]].
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| ==Category-theoretic definition==
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| In [[category theory]], a '''Lie superalgebra''' can be defined as a nonassociative [[superalgebra]] whose product satisfies
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| *<math>[\cdot,\cdot]\circ (id+\tau_{A,A})=0</math>
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| *<math>[\cdot,\cdot]\circ ([\cdot,\cdot]\otimes id)\circ(id+\sigma+\sigma^2)=0</math>
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| where σ is the cyclic permutation braiding <math>(id\otimes\tau_{A,A})\circ(\tau_{A,A}\otimes id)</math>. In diagrammatic form:
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| :<center>[[Image:Liealgebra.png|center]]</center>
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| == See also ==
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| * [[Anyonic Lie algebra]]
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| * [[Grassmann algebra]]
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| * [[Representation of a Lie superalgebra]]
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| * [[Superspace]]
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| * [[Supergroup (physics)|Supergroup]]
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| ==References==
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| *Kac, V. G. ''Lie superalgebras.'' Advances in Math. 26 (1977), no. 1, 8--96.
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| * Manin, Yuri I. ''Gauge field theory and complex geometry.'' Grundlehren der Mathematischen Wissenschaften, 289. Springer-Verlag, Berlin, 1997. ISBN 3-540-61378-1
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| *Pavel Grozman, Dimitry Leites and Irina Shchepochkina. [http://arxiv.org/abs/hep-th/9702120v1 "LIE SUPERALGEBRAS OF STRING THEORIES"]
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| ==External links== | |
| *[http://www.justpasha.org/math/links/subj/lie/kaplansky/ Irving Kaplansky + Lie Superalgebras]
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| [[Category:Supersymmetry]]
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| [[Category:Lie algebras]]
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