∞-Lie theory (higher geometry)
superalgebra and (synthetic ) supergeometry
A super Lie algebra is the analog of a Lie algebra in superalgebra/supergeometry.
See also supersymmetry.
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):
A super Lie algebra is a Lie algebra internal to the symmetric monoidal category $sVect = (Vect^{\mathbb{Z}/2}, \otimes_k, \tau^{super} )$ of super vector spaces. Hence this is
a super vector space $\mathfrak{g}$;
a homomorphism
of super vector spaces (the super Lie bracket)
such that
the bracket is skew-symmetric in that the following diagram commutes
(here $\tau^{super}$ is the braiding natural isomorphism in the category of super vector spaces)
the Jacobi identity holds in that the following diagram commutes
Externally this means the following:
A super Lie algebra according to def. 1 is equivalently
a $\mathbb{Z}/2$-graded vector space $\mathfrak{g}_{even} \oplus \mathfrak{g}_{odd}$;
equipped with a bilinear map (the super Lie bracket)
which is graded skew-symmetric: for $x,y \in \mathfrak{g}$ two elements of homogeneous degree $\sigma_x$, $\sigma_y$, respectively, then
that satisfies the $\mathbb{Z}/2$-graded Jacobi identity in that for any three elements $x,y,z \in \mathfrak{g}$ of homogeneous super-degree $\sigma_x,\sigma_y,\sigma_z\in \mathbb{Z}_2$ then
A homomorphism of super Lie algebras is a homomorphisms of the underlying super vector spaces which preserves the Lie bracket. We write
for the resulting category of super Lie algebras.
For $\mathfrak{g}$ a super Lie algebra of finite dimension, then its Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ is the super-Grassmann algebra on the dual super vector space
equipped with a differential $d_{\mathfrak{g}}$ that on generators is the linear dual of the super Lie bracket
and which is extended to $\wedge^\bullet \mathfrak{g}^\ast$ by the graded Leibniz rule (i.e. as a graded derivation).
$\,$
Here all elements are $(\mathbb{Z} \times \mathbb{Z}/2)$-bigraded, the first being the cohomological grading $n$ in $\wedge^\n \mathfrak{g}^\ast$, the second being the super-grading $\sigma$ (even/odd).
For $\alpha_i \in CE(\mathfrak{g})$ two elements of homogeneous bi-degree $(n_i, \sigma_i)$, respectively, the sign rule is
(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 $CE(\mathfrak{g})$ 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:
Given a super Lie algebra $\mathfrak{g}$, then
an $n$-cocycle on $\mathfrak{g}$ (with coefficients in $\mathbb{R}$) is an element of degree $(n,even)$ in its Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ (def. 2) which is $d_{\mathbb{g}}$ closed.
the cocycle is non-trivial if it is not $d_{\mathfrak{g}}$-exact
hene the super-Lie algebra cohomology of $\mathfrak{g}$ (with coefficients in $\mathbb{R}$) is the cochain cohomology of its Chevalley-Eilenberg algebra
The following says that the Chevalley-Eilenberg algebra is an equivalent incarnation of the super Lie algebra:
The functor
that sends a finite dimensional super Lie algebra $\mathfrak{g}$ to its Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ (def. 2) is a fully faithful functor which hence exibits super Lie algebras as a full subcategory of the opposite category of differential-graded algebras.
Equivalently, a super Lie algebra is a “super-representable” Lie algebra internal to the cohesive (∞,1)-topos Super∞Grpd over the site of super points (Sachse 08).
See the discussion at superalgebra for details on this.
(Kac 77a, Kac 77b) states a classification of super Lie algebras which are
finite dimensional
simple
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.
classical type
Cartan type
(…)
The underlying even-graded Lie algebra for type 2 is as follows
$\mathfrak{g}$ | $\mathfrak{g}_{even}$ | $\mathfrak{g}_{even}$ rep on $\mathfrak{g}_{odd}$ |
---|---|---|
$B(m,n)$ | $B_m \oplus C_n$ | vector $\otimes$ vector |
$D(m,n)$ | $D_m \oplus C_n$ | vector $\otimes$ vector |
$D(2,1,\alpha)$ | $A_1 \oplus A_1 \oplus A_1$ | vector $\otimes$ vector $\otimes$ vector |
$F(4)$ | $B_3\otimes A_1$ | spinor $\otimes$ vector |
$G(3)$ | $G_2\oplus A_1$ | spinor $\otimes$ vector |
$Q(n)$ | $A_n$ | adjoint |
For type 1 the $\mathbb{Z}/2\mathbb{Z}$-grading lifts to an $\mathbb{Z}$-grading with $\mathfrak{g} = \mathfrak{g}_{-1}\oplus \mathfrak{g}_0 \oplus \mathfrak{g}_1$.
$\mathfrak{g}$ | $\mathfrak{g}_{even}$ | $\mathfrak{g}_{even}$ rep on $\mathfrak{g}_{{-1}}$ |
---|---|---|
$A(m,n)$ | $A_m \oplus A_n \oplus C$ | vector $\otimes$ vector $\otimes$ $\mathbb{C}$ |
$A(m,m)$ | $A_m \oplus A_n$ | vector $\otimes$ vector |
$C(n)$ | $\mathbb{C}_{-1} \oplus \mathbb{C}$ | vector $\otimes$ $\mathbb{C}$ |
reviewed e.g. in (Farmer 84, p. 25,26, Minwalla 98, section 4.1).
Some obvious but important classes of examples are the following:
every $\mathbb{Z}/2$-graded vector space $V$ becomes a super Lie algebra (def. 1, prop. 1) by taking the super Lie bracket to be the zero map
These may be called the “abelian” super Lie algebras.
Every ordinary Lie algebras becomes a super Lie algebra (def. 1, prop. 1) concentrated in even degrees. This constitutes a fully faithful functor
which is a coreflective subcategory inclusion in that it has a left adjoint
given on the underlying super vector spaces by restriction to the even graded part
The super Poincare Lie algebra and various of its polyvector extension are super-extension of the ordinary Poincare Lie algebra. These are the supersymmetry algebras in the strict original sense of the word. For more on this see at geometry of physics -- supersymmetry.
higher super Lie algebras
Just as Lie algebras are categorified to L-infinity algebras and L-infinity algebroids, so super Lie algebras categorifie to super L-infinity algebras. A secretly famous example is the
Haag??opusza?ski?Sohnius theorem?
According to Kac77b the definition of super Lie algebra is originally due to
The original references on the classification of super Lie algebras are
Victor Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96.
Victor Kac, A sketch of Lie superalgebra theory, Comm. Math. Phys. Volume 53, Number 1 (1977), 31-64. (EUCLID)
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
Werner Nahm, Supersymmetries and their Representations, Nucl.Phys. B135 (1978) 149 (spire, pdf)
Steven Shnider, The superconformal algebra in higher dimensions, Letters in Mathematical Physics November 1988, Volume 16, Issue 4, pp 377-383
Victor Kac, Classification of supersymmetries, Proceedings of the ICM, Beijing 2002, vol. 1, 319–344 (arXiv:math-ph/0302016)
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