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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Lie superalgebra is a generalisation of a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
to include a Z2 grading. Lie superalgebras are important in
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
where they are used to describe the mathematics of
supersymmetry In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
. In most of these theories, the ''even'' elements of the superalgebra correspond to
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
s and ''odd'' elements to
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s (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 In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the se ...
, or ''
superalgebra In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. T ...
'', over a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
(typically R or C) whose product ·, Â· called the Lie superbracket or supercommutator, satisfies the two conditions (analogs of the usual
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
axioms, with grading): Super skew-symmetry: : ,y-(-1)^ ,x\ The super Jacobi identity: :(-1)^ ,_[y,_z_+_(-1)^[y,_[z,_x.html"_;"title=",_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^[y,_[z,_x">,_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^ ,_[y,_z_+_(-1)^[y,_[z,_x.html"_;"title=",_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^[y,_[z,_x">,_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^[y,_[z,_x_+_(-1)^[z,_[x,_y">,_z">,_[y,_z_+_(-1)^[y,_[z,_x.html"_;"title=",_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^[y,_[z,_x">,_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^[y,_[z,_x_+_(-1)^[z,_[x,_y_=_0,_ where_''x'',_''y'',_and_''z''_are_pure_in_the_Z2-grading._Here,_.html" ;"title=",_[z,_x_+_(-1)^ ,_[y,_z_+_(-1)^[y,_[z,_x.html"_;"title=",_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^[y,_[z,_x">,_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^[y,_[z,_x_+_(-1)^[z,_[x,_y">,_z">,_[y,_z_+_(-1)^[y,_[z,_x.html"_;"title=",_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^[y,_[z,_x">,_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^[y,_[z,_x_+_(-1)^[z,_[x,_y_=_0,_ where_''x'',_''y'',_and_''z''_are_pure_in_the_Z2-grading._Here,_">''x''.html" ;"title=",_[x,_y.html" ;"title=",_z">,_[y,_z_+_(-1)^[y,_[z,_x.html" ;"title=",_z.html" ;"title=", [y, z">, ,_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^[y,_[z,_x_+_(-1)^[z,_[x,_y">,_z">,_[y,_z_+_(-1)^[y,_[z,_x.html"_;"title=",_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^[y,_[z,_x">,_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^[y,_[z,_x_+_(-1)^[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,yis_the_sum_of_degree_of_x_and_y_modulo_2. One_also_sometimes_adds_the_axioms_[x,x0_for_.html" ;"title=", z + (-1)^[y, [z, x">,_z.html" ;"title=", ,_[y,_z_+_(-1)^[y,_[z,_x_+_(-1)^[z,_[x,_y">,_z">,_[y,_z_+_(-1)^[y,_[z,_x.html"_;"title=",_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^[y,_[z,_x">,_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^[y,_[z,_x_+_(-1)^[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,yis_the_sum_of_degree_of_x_and_y_modulo_2. One_also_sometimes_adds_the_axioms_[x,x0_for_">''x''.html" ;"title=", z">, [y, z + (-1)^[y, [z, x + (-1)^[z, [x, y">,_z">,_[y,_z_+_(-1)^[y,_[z,_x.html" ;"title=",_z.html" ;"title=", [y, z">, [y, z + (-1)^[y, [z, x">,_z.html" ;"title=", [y, z">, [y, z + (-1)^[y, [z, x + (-1)^[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,yis the sum of degree of x and y modulo 2. One also sometimes adds the axioms [x,x0 for ">''x''"> = 0 (if 2 is invertible this follows automatically) and x,xx0 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 In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the represent ...
of the Lie superalgebra can be given a Hopf algebra structure. A
graded Lie algebra In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket oper ...
(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.


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: # No odd elements. The statement is just that \mathfrak g_0 is an ordinary Lie algebra. # One odd element. Then \mathfrak g_1 is a \mathfrak g_0-module for the action \mathrm_a: b \rightarrow , b \quad a \in \mathfrak g_0, \quad b, , b\in \mathfrak g_1. # 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. # Three odd elements. For all b \in \mathfrak g_1, ,[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_representation_of_a_Lie_algebra.html" "title=",b.html" ;"title=",[b,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 representation of a Lie algebra">linear representation Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...
of \mathfrak g_0, and there exists a symmetric \mathfrak g_0-equivariant linear map \:\mathfrak g_1\otimes \mathfrak g_1\rightarrow \mathfrak g_0 such that, :[\left\,z]+[\left\,x]+[\left\,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 involutive
antilinear In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if \begin f(x + y) &= f(x) + f(y) && \qquad \text \\ f(s x) &= \overline f(x) && \qquad \text \\ \end hold for all vectors x, y ...
map from itself to itself which respects the Z2 grading and satisfies 'x'',''y''sup>* =  'y''*,''x''*for all ''x'' and ''y'' in the Lie superalgebra. (Some authors prefer the convention 'x'',''y''sup>* = (−1), ''x'', , ''y'', 'y''*,''x''* changing * to −* switches between the two conventions.) Its
universal enveloping algebra In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the represent ...
would be an ordinary *-algebra.


Examples

Given any associative superalgebra A one can define the supercommutator on homogeneous elements by : ,y= xy - (-1)^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 (V) of a super vector space V to itself. When V = \mathbb K^, this space is denoted by M^ or M(p, q). With the Lie bracket per above, the space is denoted \mathfrak (p, q). The
Whitehead product In mathematics, the Whitehead product is a graded quasi-Lie algebra structure on the homotopy groups of a space. It was defined by J. H. C. Whitehead in . The relevant MSC code is: 55Q15, Whitehead products and generalizations. Definition ...
on homotopy groups gives many examples of Lie superalgebras over the integers. The
super-Poincaré algebra In theoretical physics, a super-Poincaré algebra is an extension of the Poincaré algebra to incorporate supersymmetry, a relation between bosons and fermions. They are examples of supersymmetry algebras (without central charges or internal symme ...
generates the isometries of flat
superspace Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions ''x'', ''y'', ''z'', ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann numb ...
.


Classification

The simple complex finite-dimensional Lie superalgebras were classified by
Victor Kac Victor Gershevich (Grigorievich) Kac (russian: link=no, Виктор Гершевич (Григорьевич) Кац; born 19 December 1943) is a Soviet and American mathematician at MIT, known for his work in representation theory. He co-disco ...
. 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+iw.\overline 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''/2''n'') 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-commutative variables (''y'',''z''). They are important symmetries in
supergravity In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
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^\psi^\varepsilon_\varepsilon_\varepsilon_ : A_ + B_ + C_ + A_\mu \Gamma^_\mu \psi\psi + B_\mu \Gamma^_\mu \psi\psi + C_\mu \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_ + B_ = 0 :A_\mu A_\mu + B_B_ + \psi_^\alpha :A_ + B_ + B_ \sigma_^ \psi^\alpha_k \psi^\beta_k + A_\mu \Gamma_\mu^ \psi^k_\alpha \psi^k_\beta + (\text) 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^\sigma_ \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_^\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(2''m'' + 1, ''n''), HO(m, m) (''m'' ≥ 2), SHO(''m'', ''m'') (''m'' ≥ 3), KO(''m'', ''m'' + 1), SKO(m, m + 1; β) (''m'' ≥ 2), SHO âˆ¼ (2''m'', 2''m''), SKO âˆ¼ (2''m'' + 1, 2''m'' + 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 Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. 'Superstring theory' is a shorthand for supersymmetric string t ...
. Specifically, the Virasoro algebras with \mathcal supersymmetries are K(1, \mathcal) which only have central extensions up to \mathcal = 4.


Category-theoretic definition

In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, a Lie superalgebra can be defined as a nonassociative
superalgebra In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. T ...
whose product satisfies * cdot,\cdotcirc (+\tau_)=0 * cdot,\cdotcirc ( cdot,\cdototimes \circ(+\sigma+\sigma^2)=0 where σ is the cyclic permutation braiding ( \otimes\tau_) \circ (\tau_\otimes ). In diagrammatic form: :


See also

*
Gerstenhaber algebra In mathematics and theoretical physics, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an algebraic structure discovered by Murray Gerstenhaber (1963) that combines the structures of a supercommutative ring ...
* Anyonic Lie algebra *
Grassmann algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
*
Representation of a Lie superalgebra In the mathematical field of representation theory, a representation of a Lie superalgebra is an action of Lie superalgebra ''L'' on a Z2-graded vector space ''V'', such that if ''A'' and ''B'' are any two pure elements of ''L'' and ''X'' and ''Y' ...
*
Superspace Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions ''x'', ''y'', ''z'', ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann numb ...
* Supergroup *
Universal enveloping algebra In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the represent ...


Notes


References

* * * * * * * *


Historical

*. * * *


External links


Irving Kaplansky + Lie Superalgebras
{{Authority control Supersymmetry Lie algebras