∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
superalgebra and (synthetic ) supergeometry
A super Lie algebra is the analog of a Lie algebra in superalgebra/supergeometry.
See also at 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 object internal to the symmetric monoidal category of super vector spaces (a Lie algebra object in super vector spaces). Hence this is
a homomorphism
of super vector spaces (the super Lie bracket)
such that
the bracket is skew-symmetric in that the following diagram commutes
(here 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. is equivalently
a -graded vector space ;
equipped with a bilinear map (the super Lie bracket)
which is graded skew-symmetric: for two elements of homogeneous degree , , respectively, then
that satisfies the -graded Jacobi identity in that for any three elements of homogeneous super-degree 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 a super Lie algebra of finite dimension, then its Chevalley-Eilenberg algebra is the super-Grassmann algebra on the dual super vector space
equipped with a differential that on generators is the linear dual of the super Lie bracket
and which is extended to by the graded Leibniz rule (i.e. as a graded derivation).
Here all elements are -bigraded, the first being the cohomological grading in , the second being the super-grading (even/odd).
For two elements of homogeneous bi-degree , 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 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 , then
an -cocycle on (with coefficients in ) is an element of degree in its Chevalley-Eilenberg algebra (def. ) which is closed.
the cocycle is non-trivial if it is not -exact
hene the super-Lie algebra cohomology of (with coefficients in ) 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 to its Chevalley-Eilenberg algebra (def. ) 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 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.
(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
rep on | ||
---|---|---|
vector vector | ||
vector vector | ||
vector vector vector | ||
spinor vector | ||
spinor vector | ||
adjoint |
For type 1 the -grading lifts to an -grading with .
rep on | ||
---|---|---|
vector vector | ||
vector vector | ||
vector |
reviewed e.g. in (Farmer 84, p. 25,26, Minwalla 98, section 4.1).
A dg-Lie algebra may be understood equivalently as a super Lie algebra
equipped with
a lift of the -grading of the underlying vector space to a -graded vector space through the projection , hence
such that
an element
such that
(See also at “NQ-supermanifold”.)
Given this, define the differential to be the adjoint action by :
That this differential squares to 0 follows by the super-Jacobi identity (1) and by the nilpotency (2):
and the derivation-property of the differential over the bracket follows again with the super Jacobi identity (1):
Some obvious but important classes of examples are the following:
every -graded vector space becomes a super Lie algebra (def. , prop. ) 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. , prop. ) 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 Poincaré 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.
For every connected topological space, the Whitehead product makes its homotopy groups into a super Lie algebra over the ring of integers – the Whitehead super Lie algebra.
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
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).
Let be a finite-dimensional vector space over some ground field .
Define a -graded vector space
concentrated in degrees , recursively as follows:
For we set
For , the component space in degree is taken to be the vector space of linear maps from to the component space in degree :
Hence:
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 we set
For and we set
Finally, for and we set
By (4) the definition (5) is equivalent to
Hence (5) is already implied by (4) if the bracket is to satisfy the super Jacobi identity (1).
It remains to show that:
Def. indeed gives a super Lie algebra in that the bracket (5) satisfies the super Jacobi identity (1).
We proceed by induction:
By Remark we have that the super Jacobi identity holds for for all triples with .
Now assume that the super Jacobi identity has been shown for triples and , for any . The following computation shows that then it holds for :
(Fine, but is this sufficient to induct over the full range of all three degrees?)
For (3) we have that the bracket on in Def. restricts to
This is the Lie bracket of the general linear Lie algebra , as indicated on the right in (3).
(embedding tensors are square-0 elements in )
Let be a ground field of characteristic zero.
An element in degree -1 of the super Lie algebra from Def. ,
which by Example is identified with a linear map
from to the general linear Lie algebra on , is square-0 (2) precisely if it is an embedding tensor, in that:
Here on the right, denotes the Lie bracket in , while denotes the canonical Lie algebra action of on .
(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 induces a dg-Lie algebra .
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.
According to V. Kac 1977b the definition of super Lie algebra is originally due to:
(in Russian)
See also:
Isaiah Kantor, Graded Lie algebras, Trudy Sem. Vektor. Tenzor. Anal 15 (1970): 227-266.
Felix A. Berezin (edited by Alexandre A. Kirillov): Lie Superalgebras, chapters I.5 and II.1 in: Introduction to Superanalysis, Mathematical Physics and Applied Mathematics 9, Springer (1987) [doi:10.1007/978-94-017-1963-6_6, doi:10.1007/978-94-017-1963-6_7]
The original references on the classification of super Lie algebras:
Victor Kac, Lie superalgebras, Advances in Math. 26 1 (1977) 8-96 [doi:10.1016/0001-8708(77)90017-2]
Victor Kac: A sketch of Lie superalgebra theory, Comm. Math. Phys. 53 1 (1977) 31-64 [euclid:1103900590, doi:10.1007/BF01609166]
See also:
Werner Nahm, Vladimir Rittenberg, Manfred Scheunert, The classification of graded Lie algebras, Physics Letters B 61 4 (1976) 383-384 [doi:10.1016/0370-2693(76)90594-3]
M. Parker, Classification Of Real Simple Lie Superalgebras Of Classical Type, J.Math.Phys. 21 (1980) 689-697 (spire)
Further discussion specifically of classification of supersymmetry:
Werner Nahm, Supersymmetries and their Representations, Nucl. Phys. B 135 (1978) 149 [spire, pdf]
Steven Shnider, The superconformal algebra in higher dimensions, Letters in Mathematical Physics 16 4 (1988) 377-383 [doi:10.1007/BF00402046]
Victor Kac, Classification of supersymmetries, Proceedings of the ICM, Beijing 2002, vol. 1, 319–344 (arXiv:math-ph/0302016)
Introductions and surveys:
Vladimir Rittenberg: A guide to Lie superalgebras, in P. Kramers, A. Rieckers (eds.): Group Theoretical Methods in Physics, Lecture Notes in Physics 79 Springer (1978) 3-21 [doi:10.1007/3-540-08848-2_1, pdf]
Manfred Scheunert, The theory of Lie superalgebras. An introduction, Lect. Notes Math. 716 (1979) [doi:10.1007/BFb0070929]
Dimitry A. Leites, Lie Superalgebras, J. Soviet Math. 30 (1985) 2481-2512 [doi:10.1007/BF02249121]
Leonardo Castellani, Riccardo D'Auria, Pietro Fré, section II.2 of: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, chII.2: pdf]
(with an eye towards supergravity)
Richard J. Farmer, Orthosymplectic superalgebras in mathematics and science, PhD Thesis (1984) [eprints:19542, pdf]
Luc Frappat, Antonio Sciarrino, Paul Sorba: Structure of basic Lie superalgebras and of their affine extensions, Comm. Math. Phys. 121 3 (1989) 457-500 [euclid:cmp/1104178142]
Luc Frappat, Antonio Sciarrino, Paul Sorba: Dictionary on Lie Superalgebras, Academic Press (2000) [arXiv:hep-th/9607161, ISBN:978-0122653407]
Groeger, Super Lie groups and super Lie algebras, lecture notes 2011 (pdf)
D. Westra, Superrings and supergroups (pdf)
Shiraz Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 781 [arXiv:hep-th/9712074]
Discussion in the presheaf topos over superpoints:
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 September 2, 2024 at 15:57:47. See the history of this page for a list of all contributions to it.