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super Lie algebra

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\infty-Lie theory

∞-Lie theory (higher geometry)

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Super-Algebra and Super-Geometry

Contents

Idea

A super Lie algebra is the analog of a Lie algebra in superalgebra/supergeometry.

See also supersymmetry.

Definition

There are various equivalent ways to state the definition of super Lie algebras. Here are a few (for more discussion see at geometry of physics – superalgebra):

As Lie algebras internal to super vector spaces

Definition

A super Lie algebra is a Lie algebra object internal to the symmetric monoidal category sVect=(Vect /2, k,τ super)sVect = (Vect^{\mathbb{Z}/2}, \otimes_k, \tau^{super} ) of super vector spaces (a Lie algebra object in super vector spaces). Hence this is

  1. a super vector space 𝔤\mathfrak{g};

  2. a homomorphism

    [,]:𝔤 k𝔤𝔤 [-,-] \;\colon\; \mathfrak{g} \otimes_k \mathfrak{g} \longrightarrow \mathfrak{g}

    of super vector spaces (the super Lie bracket)

such that

  1. the bracket is skew-symmetric in that the following diagram commutes

    𝔤 k𝔤 τ 𝔤,𝔤 super 𝔤 k𝔤 [,] [,] 𝔤 1 𝔤 \array{ \mathfrak{g} \otimes_k \mathfrak{g} & \overset{\tau^{super}_{\mathfrak{g},\mathfrak{g}}}{\longrightarrow} & \mathfrak{g} \otimes_k \mathfrak{g} \\ {}^{\mathllap{[-,-]}}\downarrow && \downarrow^{\mathrlap{[-,-]}} \\ \mathfrak{g} &\underset{-1}{\longrightarrow}& \mathfrak{g} }

    (here τ super\tau^{super} is the braiding natural isomorphism in the category of super vector spaces)

  2. the Jacobi identity holds in that the following diagram commutes

    𝔤 k𝔤 k𝔤 τ 𝔤,𝔤 super kid 𝔤 k𝔤 k𝔤 [,[,]][[,],] [,[,]] 𝔤. \array{ \mathfrak{g} \otimes_k \mathfrak{g} \otimes_k \mathfrak{g} && \overset{\tau^{super}_{\mathfrak{g}, \mathfrak{g}} \otimes_k id }{\longrightarrow} && \mathfrak{g} \otimes_k \mathfrak{g} \otimes_k \mathfrak{g} \\ & {}_{\mathllap{\left[-,\left[-,-\right]\right]} - \left[\left[-,-\right],-\right] }\searrow && \swarrow_{\mathrlap{\left[-,\left[-,-\right]\right]}} \\ && \mathfrak{g} } \,.

As super-graded Lie algebras

Externally this means the following:

Proposition

A super Lie algebra according to def. is equivalently

  1. a /2\mathbb{Z}/2-graded vector space 𝔤 even𝔤 odd\mathfrak{g}_{even} \oplus \mathfrak{g}_{odd};

  2. equipped with a bilinear map (the super Lie bracket)

    [,]:𝔤 k𝔤𝔤 [-,-] : \mathfrak{g}\otimes_k \mathfrak{g} \to \mathfrak{g}

    which is graded skew-symmetric: for x,y𝔤x,y \in \mathfrak{g} two elements of homogeneous degree σ x\sigma_x, σ y\sigma_y, respectively, then

    [x,y]=(1) σ xσ y[y,x], [x,y] = -(-1)^{\sigma_x \sigma_y} [y,x] \,,
  3. that satisfies the /2\mathbb{Z}/2-graded Jacobi identity in that for any three elements x,y,z𝔤x,y,z \in \mathfrak{g} of homogeneous super-degree σ x,σ y,σ z 2\sigma_x,\sigma_y,\sigma_z\in \mathbb{Z}_2 then

    (1)[x,[y,z]]=[[x,y],z]+(1) σ xσ y[y,[x,z]]. [x, [y, z] ] = [ [x,y],z] + (-1)^{\sigma_x \cdot \sigma_y} [y, [x,z] ] \,.

A homomorphism of super Lie algebras is a homomorphisms of the underlying super vector spaces which preserves the Lie bracket. We write

sLieAlg sLieAlg

for the resulting category of super Lie algebras.

As formal duals of a Chevalley-Eilenberg super-algebras

Definition

For 𝔤\mathfrak{g} a super Lie algebra of finite dimension, then its Chevalley-Eilenberg algebra CE(𝔤)CE(\mathfrak{g}) is the super-Grassmann algebra on the dual super vector space

𝔤 * \wedge^\bullet \mathfrak{g}^\ast

equipped with a differential d 𝔤d_{\mathfrak{g}} that on generators is the linear dual of the super Lie bracket

d 𝔤[,] *:𝔤 *𝔤 *𝔤 * d_{\mathfrak{g}} \;\coloneqq\; [-,-]^\ast \;\colon\; \mathfrak{g}^\ast \to \mathfrak{g}^\ast \wedge \mathfrak{g}^\ast

and which is extended to 𝔤 *\wedge^\bullet \mathfrak{g}^\ast by the graded Leibniz rule (i.e. as a graded derivation).

\,

Here all elements are (×/2)(\mathbb{Z} \times \mathbb{Z}/2)-bigraded, the first being the cohomological grading nn in n𝔤 *\wedge^\n \mathfrak{g}^\ast, the second being the super-grading σ\sigma (even/odd).

For α iCE(𝔤)\alpha_i \in CE(\mathfrak{g}) two elements of homogeneous bi-degree (n i,σ i)(n_i, \sigma_i), respectively, the sign rule is

α 1α 2=(1) n 1n 2(1) σ 1σ 2α 2α 1. \alpha_1 \wedge \alpha_2 = (-1)^{n_1 n_2} (-1)^{\sigma_1 \sigma_2}\; \alpha_2 \wedge \alpha_1 \,.

(See at signs in supergeometry for discussion of this sign rule and of an alternative sign rule that is also in use. )

We may think of CE(𝔤)CE(\mathfrak{g}) equivalently as the dg-algebra of left-invariant super differential forms on the corresponding simply connected super Lie group .

The concept of Chevalley-Eilenberg algebras is traditionally introduced as a means to define Lie algebra cohomology:

Definition

Given a super Lie algebra 𝔤\mathfrak{g}, then

  1. an nn-cocycle on 𝔤\mathfrak{g} (with coefficients in \mathbb{R}) is an element of degree (n,even)(n,even) in its Chevalley-Eilenberg algebra CE(𝔤)CE(\mathfrak{g}) (def. ) which is d 𝕘d_{\mathbb{g}} closed.

  2. the cocycle is non-trivial if it is not d 𝔤d_{\mathfrak{g}}-exact

  3. hene the super-Lie algebra cohomology of 𝔤\mathfrak{g} (with coefficients in \mathbb{R}) is the cochain cohomology of its Chevalley-Eilenberg algebra

    H (𝔤,)=H (CE(𝔤)). H^\bullet(\mathfrak{g}, \mathbb{R}) = H^\bullet(CE(\mathfrak{g})) \,.

The following says that the Chevalley-Eilenberg algebra is an equivalent incarnation of the super Lie algebra:

Proposition

The functor

CE:sLieAlg findgAlg op CE \;\colon\; sLieAlg^{fin} \hookrightarrow dgAlg^{op}

that sends a finite dimensional super Lie algebra 𝔤\mathfrak{g} to its Chevalley-Eilenberg algebra CE(𝔤)CE(\mathfrak{g}) (def. ) is a fully faithful functor which hence exibits super Lie algebras as a full subcategory of the opposite category of differential-graded algebras.

As super-representable Lie algebras in the topos over superpoints

Equivalently, a super Lie algebra is a “super-representable” Lie algebra object internal to the cohesive (∞,1)-topos Super∞Grpd over the site of super points (Sachse 08, Section 3.2, towards cor. 3.3).

See the discussion at superalgebra for details on this.

Properties

Classification

(Kac 77a, Kac 77b) states a classification of super Lie algebras which are

  1. finite dimensional

  2. simple

  3. over a field of characteristic zero.

Such an algebra is called of classical type if the action of its even-degree part on the odd-degree part is completely reducible. Those simple finite dimensional algebras not of classical type are of Cartan type.

  1. classical type

    1. four infinite series

      1. A(m,n)A(m,n)

      2. B(m,n)=B(m,n) = osp(2m+1,2n)(2m+1,2n) m0m\geq 0, n>0n \gt 0

      3. C(n)C(n)

      4. D(m,n)=D(m,n) = osp(2m,2n)(2m,2n) m2m \geq 2, n>0n \gt 0

    2. two exceptional ones

      1. F(4)F(4)

      2. G(3)G(3)

    3. a family D(2,1;α)D(2,1;\alpha) of deformations of D(2,1)D(2,1)

    4. two “strange” series

      1. P(n)P(n)

      2. Q(n)Q(n)

  2. Cartan type

    (…)

The underlying even-graded Lie algebra for type 2 is as follows

𝔤\mathfrak{g}𝔤 even\mathfrak{g}_{even}𝔤 even\mathfrak{g}_{even} rep on 𝔤 odd\mathfrak{g}_{odd}
B(m,n)B(m,n)B mC nB_m \oplus C_nvector \otimes vector
D(m,n)D(m,n)D mC nD_m \oplus C_nvector \otimes vector
D(2,1,α)D(2,1,\alpha)A 1A 1A 1A_1 \oplus A_1 \oplus A_1vector \otimes vector \otimes vector
F(4)F(4)B 3A 1B_3\otimes A_1spinor \otimes vector
G(3)G(3)G 2A 1G_2\oplus A_1spinor \otimes vector
Q(n)Q(n)A nA_nadjoint

For type 1 the /2\mathbb{Z}/2\mathbb{Z}-grading lifts to an \mathbb{Z}-grading with 𝔤=𝔤 1𝔤 0𝔤 1\mathfrak{g} = \mathfrak{g}_{-1}\oplus \mathfrak{g}_0 \oplus \mathfrak{g}_1.

𝔤\mathfrak{g}𝔤 even\mathfrak{g}_{even}𝔤 even\mathfrak{g}_{even} rep on 𝔤 1\mathfrak{g}_{{-1}}
A(m,n)A(m,n)A mA nCA_m \oplus A_n \oplus Cvector \otimes vector \otimes \mathbb{C}
A(m,m)A(m,m)A mA nA_m \oplus A_nvector \otimes vector
C(n)C(n) 1\mathbb{C}_{-1} \oplus \mathbb{C}vector \otimes \mathbb{C}

reviewed e.g. in (Farmer 84, p. 25,26, Minwalla 98, section 4.1).

Relation to dg-Lie algebras

A dg-Lie algebra (𝔤,,[,])(\mathfrak{g}, \partial, [-,-]) may be understood equivalently as a super Lie algebra

(𝔤=𝔤 even𝔤 odd,[,,]) (\mathfrak{g} = \mathfrak{g}_{even} \oplus \mathfrak{g}_{odd}, [-,-,])

equipped with

  1. a lift of the 2\mathbb{Z}_2-grading of the underlying vector space to a \mathbb{Z}-graded vector space through the projection /2= 2\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} = \mathbb{Z}_2, hence

    𝔤 evenn𝔤 2n,AAA𝔤 oddn𝔤 2n+1 \mathfrak{g}_{even} \;\simeq\; \underset{n }{\oplus} \mathfrak{g}_{2n} \,, \phantom{AAA} \mathfrak{g}_{odd} \;\simeq\; \underset{n }{\oplus} \mathfrak{g}_{2n+1}

    such that

    [,]:𝔤 n 1𝔤 n 2𝔤 n 1+n 2 [-,-] \;\colon\; \mathfrak{g}_{n_1} \otimes \mathfrak{g}_{n_2} \longrightarrow \mathfrak{g}_{n_1 + n_2}
  2. an element Q𝔤 1Q \in \mathfrak{g}_{-1}

    such that

    (2)[Q,Q]=0 [Q,Q] = 0

(See also at “NQ-supermanifold”.)

Given this, define the differential to be the adjoint action by QQ:

[Q,]. \partial \;\coloneqq\; [Q,-] \,.

That this differential squares to 0 follows by the super-Jacobi identity (1) and by the nilpotency (2):

[Q,[Q,]]=[[Q,Q]=0,][Q,[Q,]]AAAA[Q,[Q,]]=0 [Q,[Q,-] ] \;=\; [ \underset{= 0}{\underbrace{[Q,Q]}}, - ] - [ Q, [Q, -] ] \phantom{AA} \Rightarrow \phantom{AA} [ Q,[Q,-] ] = 0

and the derivation-property of the differential over the bracket follows again with the super Jacobi identity (1):

[Q,[x,y]]=[[Q,x],y]+(1) deg(x)[x,[Q,y]]. [Q,[x,y] ] \;=\; [ [Q,x],y] + (-1)^{deg(x)} [x, [Q,y] ] \,.

Examples

Basic examples

Some obvious but important classes of examples are the following:

Example

every /2\mathbb{Z}/2-graded vector space VV becomes a super Lie algebra (def. , prop. ) by taking the super Lie bracket to be the zero map

[,]=0. [-,-] = 0 \,.

These may be called the “abelian” super Lie algebras.

Example

Every ordinary Lie algebras becomes a super Lie algebra (def. , prop. ) concentrated in even degrees. This constitutes a fully faithful functor

LieAlgsLieAlg. LieAlg \hookrightarrow sLieAlg \,.

which is a coreflective subcategory inclusion in that it has a left adjoint

LieAlg()sLieAlg LieAlg \underoverset {\underset{ \overset{ \rightsquigarrow}{(-)} }{\longleftarrow}} {\hookrightarrow} {\bot} sLieAlg

given on the underlying super vector spaces by restriction to the even graded part

𝔰=𝔰 even. \overset{\rightsquigarrow}{\mathfrak{s}} = \mathfrak{s}_{even} \,.

Super-Poincaré super Lie algebras (supersymmetry)

Embedding tensors and tensor hierarchy

The following example is highlighted in Palmkvist 13, 3.1 (reviewed more clearly in Lavau-Palmkvist 19, 2.4) where it is attributed to I. L. Kantor (1970).

Definition

Let VV be a finite-dimensional vector space over some ground field kk.

Define a \mathbb{Z}-graded vector space

V^Vect k , \widehat V \;\in \; Vect_k^{\mathbb{Z}} \,,

concentrated in degrees 1\leq 1, recursively as follows:

For n=1n =1 we set

V^ 1V. \widehat V_{1} \;\coloneqq\; V \,.

For n0n \leq 0 \in \mathbb{Z}, the component space in degree n1n-1 is taken to be the vector space of linear maps from VV to the component space in degree nn:

V^ n1Hom k(V,V^ n). \widehat V_{n-1} \;\coloneqq\; Hom_k( V, \widehat V_n ) \,.

Hence:

(3)V^ 1 =V V^ 0 =Hom k(V,V)=𝔤𝔩(V) V^ 1 =Hom k(V,Hom k(V,V))Hom k(VV,V) V^ 2 =Hom k(V,Hom k(V,Hom k(V,V)))Hom k(VVV,V) \begin{aligned} \widehat V_1 & = V \\ \widehat V_0 & = Hom_k(V,V) = \mathfrak{gl}(V) \\ \widehat V_{-1} & = Hom_k(V, Hom_k(V,V)) \simeq Hom_k(V \otimes V, V) \\ \widehat V_{-2} & = Hom_k(V, Hom_k(V, Hom_k(V,V))) \simeq Hom_k(V \otimes V \otimes V, V) \\ \vdots \end{aligned}

Consider then the direct sum of these component spaces as a super vector space with the even number/odd number-degrees being in super-even/super-odd degree, respectively.

On this super vector space consider a super Lie bracket defined recusively as follows:

For all v 1,v 2V^ 1=Vv_1, v_2 \in \widehat V_1 = V we set

[v 1,v 2]=0. [v_1, v_2] \;=\; 0 \,.

For fV^ n0f \in \widehat V_{n \leq 0} and vV^ 1=Vv \in \widehat V_1 = V we set

(4)[f,v]f(v) [f, v] \;\coloneqq\; f(v)

Finally, for fV^ deg(f)0f\in \widehat V_{ deg(f) \leq 0 } and gV^ deg(g)0g\in \widehat V_{deg(g) \leq 0} we set

(5)[f,g] :v[f,g(v)](1) deg(f)deg(g)[g,f(v)] \begin{aligned} [f, g] & \colon\; v \;\mapsto\; [f, g(v)] - (-1)^{ deg(f) deg(g) } [ g, f(v) ] \\ \end{aligned}
Remark

By (4) the definition (5) is equivalent to

[[f,g],v]=[f,[g,v]](1) deg(f)deg(g)[g,[f,v]] [ [f,g],v ] \;=\; [f, [g,v] ] - (-1)^{ deg(f) deg(g) } [ g, [f,v] ]

Hence (5) is already implied by (4) if the bracket is to satisfy the super Jacobi identity (1).

It remains to show that:

Proposition

Def. indeed gives a super Lie algebra in that the bracket (5) satisfies the super Jacobi identity (1).

Proof

We proceed by induction:

By Remark we have that the super Jacobi identity holds for for all triples f 1,f 2,f 3V^f_1, f_2, f_3 \in \widehat{V} with deg(f 3)0deg(f_3) \geq 0.

Now assume that the super Jacobi identity has been shown for triples (f 1,f 2,f 3(v))(f_1, f_2, f_3(v)) and (f 1,f 3,f 2(v))(f_1, f_3, f_2(v)), for any vVv \in V. The following computation shows that then it holds for (f 1,f 2,f 3)(f_1, f_2, f_3):

[f 1,[f 2,f 3]](v) =[f 1,[f 2,f 3](v)](1) deg(f 1)(deg(f 2)+deg(f 3))[[f 2,f 3],f 1(v)] =[f 1,[f 2,f 3(v)]] =(1) deg(f 2)deg(f 3)[f 1,[f 3,f 2(v)]] =(1) deg(f 1)(deg(f 2)+deg(f 3))[f 2,[f 3,f 1(v)]] =+(1) deg(f 1)(deg(f 2)+deg(f 3))+deg(f 2)deg(f 3)[f 3,[f 2,f 1(v)]] =[f 1,[f 2,f 3(v)]](1) deg(f 1)deg(f 2)[f 2,[f 1,f 3(v)]] =(1) deg(f 2)deg(f 3)([f 1,[f 3,f 2(v)]](1) deg(f 1)deg(f 3)[f 3,[f 1,f 2(v)]]) =(1) deg(f 1)(deg(f 2)+deg(f 3))([f 2,[f 3,f 1(v)]](1) deg(f 1)deg(f 3)[f 2,[f 1,f 3(v)]]) =+(1) deg(f 1)deg(f 2)+deg(f 1)deg(f 3)+deg(f 2)deg(f 3)([f 3,[f 2,f 1(v)]](1) deg(f 1)deg(f 2)[f 3,[f 1,f 2(c)]]) =+(1) deg(f 1)deg(f 2)(+[f 2,[f 1,f 3(v)]][f 2,[f 1,f 3(v)]]=0) =+(1) deg(f 1)deg(f 3)+deg(f 2)deg(f 3)([f 3,[f 1,f 2(v)]][f 3,[f 1,f 2(v)]]=0) =[[f 1,f 2],f 3(v)] =(1) deg(f 2)deg(f 3)[[f 1,f 3],f 2(c)] =+(1) deg(f 1)deg(f 2)[f 2,[f 1,f 3](v)] =(1) deg(f 3)(deg(f 1)+deg(f 2))[f 3,[f 1,f 2](v)] =[[f 1,f 2],f 3](v)+(1) deg(f 1)deg(f 2))[f 2,[f 1,f 3]](v) \begin{aligned} [f_1, [f_2, f_3] ] (v) & = [ f_1, [f_2, f_3](v) ] - (-1)^{deg(f_1)(deg(f_2) + deg(f_3))} [ [ f_2, f_3 ], f_1(v) ] \\ & = [ f_1, [ f_2, f_3(v) ] ] \\ & \phantom{=} - (-1)^{deg(f_2)deg(f_3)} [ f_1, [ f_3, f_2(v) ] ] \\ & \phantom{=} - (-1)^{deg(f_1)(deg(f_2) + deg(f_3))} [ f_2, [ f_3, f_1(v) ] ] \\ & \phantom{=} + (-1)^{deg(f_1)(deg(f_2) + deg(f_3)) + deg(f_2)deg(f_3)} [ f_3, [ f_2, f_1(v) ] ] \\ & = [ f_1, [ f_2, f_3(v) ] ] - (-1)^{deg(f_1) deg(f_2)} { \color{green} [ f_2, [ f_1, f_3(v) ] ] } \\ & \phantom{=} - (-1)^{deg(f_2) deg(f_3)} \big( [ f_1, [ f_3, f_2(v) ] ] - (-1)^{deg(f_1) deg(f_3)} { \color{orange} [ f_3, [ f_1, f_2(v) ] ] } \big) \\ & \phantom{=} - (-1)^{deg(f_1)(deg(f_2) + deg(f_3))} \big( [ f_2, [ f_3, f_1(v) ] ] - (-1)^{deg(f_1)deg(f_3)} { \color{blue} [ f_2, [ f_1, f_3(v) ] ] } \big) \\ & \phantom{=} + (-1)^{deg(f_1) deg(f_2 ) + deg(f_1) deg(f_3) + deg(f_2) deg(f_3)} \big( [ f_3, [ f_2, f_1(v) ] ] - (-1)^{deg(f_1) deg(f_2)} { \color{cyan} [ f_3, [ f_1, f_2(c) ] ] } \big) \\ & \phantom{=} + (-1)^{deg(f_1) deg(f_2)} \big( \underset{ = 0 }{ \underbrace{ + { \color{green} [ f_2, [ f_1, f_3(v) ] ] } - { \color{blue} [ f_2, [ f_1, f_3(v) ] ] } } } \big) \\ & \phantom{=} + (-1)^{deg(f_1) deg(f_3) + deg(f_2) deg(f_3)} \big( \underset{ = 0 }{ \underbrace{ { \color{orange} [ f_3, [ f_1, f_2(v) ] ] } - { \color{cyan} [ f_3, [ f_1, f_2(v) ] ] } } } \big) \\ & = \big[ [f_1, f_2], f_3(v) \big] \\ & \phantom{=} - (-1)^{ deg(f_2) deg(f_3) } \big[ [f_1, f_3], f_2(c) \big] \\ & \phantom{=} + (-1)^{ deg(f_1) deg(f_2) } \big[ f_2, [f_1, f_3](v) \big] \\ & \phantom{=} - (-1)^{ deg(f_3)( deg(f_1) + deg(f_2) ) } \big[ f_3, [f_1, f_2](v) \big] \\ & = \big[ [f_1, f_2], f_3 \big](v) + (-1)^{deg(f_1)deg(f_2))} \big[ f_2, [f_1, f_3] \big](v) \end{aligned}

(Fine, but is this sufficient to induct over the full range of all three degrees?)

Example

For f,gV^ 0=Hom k(V,V)f,g \in \widehat V_0 = Hom_k(V,V) (3) we have that the bracket on V^\widehat V in Def. restricts to

[f,g](v)=[f,g(v)][g,f(v)]=f(g(v))g(f(v)) [f,g](v) \;=\; [f,g(v)] - [g,f(v)] \;=\; f(g(v)) - g(f(v))

(by combining (5) with (4)).

This is the Lie bracket of the general linear Lie algebra 𝔤𝔩(V)\mathfrak{gl}(V), as indicated on the right in (3).

Proposition

(embedding tensors are square-0 elements in V^\widehat{V})

Let kk be a ground field of characteristic zero.

An element in degree -1 of the super Lie algebra V^\widehat V from Def. ,

ΘV^ 1Hom k(V,𝔤𝔩(V)), \Theta \in \widehat V_{-1} \simeq Hom_{k}(V, \mathfrak{gl}(V)) \,,

which by Example is identified with a linear map

Θ:V𝔤𝔤𝔩(V) \Theta \;\colon\; V \longrightarrow \mathfrak{g} \coloneqq \mathfrak{gl}(V)

from VV to the general linear Lie algebra on VV, is square-0 (2) precisely if it is an embedding tensor, in that:

[Θ,Θ]=0AAAAAA[Θ(v 1),Θ(v 2)]=Θ(ρ Θ(v)1)(v 2)). [\Theta, \Theta] \;=\; 0 \phantom{AAA} \Leftrightarrow \phantom{AAA} [\Theta(v_1), \Theta(v_2) ] \;=\; \Theta( \rho_{\Theta(v)1)}(v_2) ) \,.

Here on the right, [,][-,-] denotes the Lie bracket in 𝔤𝔩(V)\mathfrak{gl}(V), while ρ\rho denotes the canonical Lie algebra action of 𝔤𝔩(V)\mathfrak{gl}(V) on VV.

Proof

By unwinding of the definition (4) and (5) and using again Example we compute as follows:

(12[Q,Q](v 1))(v 2) =[Q,Q(v 1)](v 2) =[Q,(Q(v 1))(v 2)=ρ Θ(v 1)(v 2)][Q(v 1),Q(v 2)] =Θ(ρ Θ(v 1)(v 2))[Θ(v 1),Θ(v)] \begin{aligned} \big( \tfrac{1}{2} [Q,Q](v_1) \big)(v_2) & = [Q, Q(v_1)](v_2) \\ & = [Q, \underset{ \mathclap{ = \rho_{\Theta(v_1)}(v_2) } } { \underbrace{ (Q(v_1))(v_2) } } ] - [Q(v_1), Q(v_2)] \\ & = \Theta( \rho_{\Theta(v_1)}(v_2) ) - [ \Theta(v_1), \Theta(v) ] \end{aligned}
Remark

(embedding tensors induce tensor hierarchies)

In view of the relation between super Lie algebras and dg-Lie algebras (above), Prop. says that every choice of an embedding tensor for a faithful representation on a vector space VV induces a dg-Lie algebra (V^,[,],[Θ,])(\widehat V, [-,-], \partial \coloneqq [\Theta, -]).

According to Palmkvist 13, 3.1, Lavau-Palmkvist 19, 2.4 this dg-Lie algebra (or some extension of some sub-algebra of it) is the tensor hierarchy associated with the embedding tensor.

References

According to Kac77b the definition of super Lie algebra is originally due to

  • Felix Berezin, G. I. Kac, Math. Sbornik 82, 343—351 (1970) (Russian)

See also

  • Isaiah Kantor, Graded Lie algebras, Trudy Sem. Vektor. Tenzor. Anal 15 (1970): 227-266.

The original references on the classification of super Lie algebras are

See also

  • Werner Nahm, V. Rittenberg, Manfred Scheunert, The classification of graded Lie algebras , Physics Letters B Volume 61, Issue 4, 12 April 1976, Pages 383–384 (publisher)

  • M. Parker, Classification Of Real Simple Lie Superalgebras Of Classical Type, J.Math.Phys. 21 (1980) 689-697 (spire)

Further discussion of classification related specifically to classification of supersymmetry is due to

Introductions and surveys include

  • Richard Joseph Farmer, Orthosymplectic superalgebras in mathematics and science, PhD Thesis (1984) (web, pdf)

  • L. Frappat, A. Sciarrino, P. Sorba, Dictionary on Lie Superalgebras (arXiv:hep-th/9607161)

  • Groeger, Super Lie groups and super Lie algebras, lecture notes 2011 (pdf)

  • L. Frappat, A. Sciarrino, P. Sorba, Dictionary on Lie Superalgebras (arXiv:hep-th/9607161)

  • D. Leites, Lie superalgebras, J. Soviet Math. 30 (1985), 2481–2512 (web)

  • Manfred Scheunert, The theory of Lie superalgebras. An introduction, Lect. Notes Math. 716 (1979)

  • D. Westra, Superrings and supergroups (pdf)

  • Shiraz Minwalla, Restrictions imposed by superconformal invariance on quan tum field theories Adv. Theor. Math. Phys. 2, 781 (1998)

    (arXiv:hep-th/9712074).

Discussion in the topos over superpoints is in

Discussion of Lie algebra extensions for super Lie algebras includes

On Lie algebra weight systems arising from super Lie algebras:

On Lie algebra cohomology of super Lie algebras (see also the brane scan) in relation to integrable forms of coset supermanifolds:

Last revised on December 11, 2020 at 03:10:27. See the history of this page for a list of all contributions to it.