Download 12 Super Lie Groups and Super Lie Algebras

Differential Geometry of Supermanifolds
Josua Groeger, HU Berlin, WS 2010/2011
http://www.mathematik.hu-berlin.de/∼groegerj
Lecture 12, Version January 4, 2011
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Super Lie Groups and Super Lie Algebras
We introduce super Lie groups and super Lie algebras as natural generalisations of
classical Lie groups and Lie algebras. Our main example is the super translation group
in dimension 1|1.
12.1
Super Lie Groups
Since points on a supermanifold have only partial geometric significance, we need to
formulate the group axioms in terms of morphisms first.
12.1.1
Groups
Let G be some (ordinary) group. Starting with associativity, we first observe that the
map (a, b, c) 7→ a(bc) for elements a, b, c ∈ G is the concatenation of the two maps
id × m : G × G × G , (a, b, c) 7→ (a, bc) ;
m : G × G → G , (x, y) 7→ xy
where m (multiplication) is the group operation, and analogous for the map (a, b, c) 7→
(ab)c. Thus the associativity axiom can be stated in the form
(1)
m ◦ (id × m) = m ◦ (m × id)
Existence of a unit element 1 ∈ G such that a1 = 1a = a for a ∈ G can be formulated
as follows. There exists a (constant) map e : {0} → G , 0 7→ 1 (identified with 1) such
that
(2)
m ◦ (id × e) = pL ,
m ◦ (e × id) = pR
wherere pL : G × {0} → G denotes the (canonical) left projection and, similarly, pR :
{0} × G → G denotes the right projection.
By the inverse axiom, there is a map i : G → G , a 7→ a−1 (called the inverse map)
such that
(3)
m ◦ (id × i) ◦ d = m ◦ (i × id) ◦ d = e ◦ p
where d : G → G × G , a 7→ (a, a) is the diagonal map and p : G → {0} , a 7→ 0 is the
(unique) map with target {0}.
12.1.2
... and Group Objects
To summarise the discussion so far, a group can be defined as a tuple (G, m, e, i), consisting of a set G and maps m, e and i such that equations (1), (2) and (3) hold. In this
definition, two ingredients are important: First, we have (finite) products G × G defined
for sets and, second, we have the special set {0} (consisting of one element) which can
be defined by the property that, for every set X, there is a single map p : X → {0}. For
this property, {0} is also called terminal.
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There is a natural generalisation to categories. We have not discussed too much
category theory yet, but for our purposes it suffices to think of a category as consisting
of objects and morphisms, examples include the category of sets (being the objects) and
functions (being the morphisms), the category of smooth manifolds (with the smooth
maps as morphisms), and the category of supermanifolds.
Finite products exist in all three examples, and terminal objects are, respectively,
the set {0}, the manifold R0 ∼
= {0}, and the supermanifold R0|0 . In general categories,
terminal objects need not exist but if they exist, they are ”essentially” unique. Moreover,
left and right projections pL and pR as well as the diagonal map d come along the
definition of products. Consult books on category theory such as [Sch70] for details.
In the category of supermanifolds, d : G → G×G is the morphism defined by sending
a local supercoordinate ξ k in either copy of G on the right hand side to the same local
supercoordinate on the left hand copy of G. You may check that this prescription is
well-defined and the resulting morphisms is indeed the diagonal map induced by the
definition of products of supermanifolds (the latter is an exercise in category theory).
Definition 12.1. A group object in a category with finite products and a terminal
object T is a tuple (G, m, e, i), consisting of an object G and morphisms m, e and i such
that equations (1), (2) and (3) hold (with {0} replaced by T etc.). With respect to our
three example categories, we define:
(i) A group is a group object in the category of sets.
(ii) A Lie group is a group object in the category of smooth manifolds.
(iii) A super Lie group is a group object in the category of supermanifolds.
You should check that this definition of a Lie group coincides with the usual one and
write down the appropriate versions of equations (1), (2) and (3) for super Lie groups.
12.1.3
Supertranslations as an Example for Super Lie Groups
Vector addition on Rn admits a canonical generalisation to Rn|m : Consider three copies
of Rn|m and denote, for distinction, their respective canonical global supercoordinates
by (x1 , . . . , θ1 , . . .), (x′1 , . . . , θ′1 , . . .) and (x′′1 , . . . , θ′′1 , . . .). Using the symbolic way of
calculation, we define the map
(4)
+ : Rn|m × Rn|m → Rn|m ,
x′′j := xj + x′j ,
θ′′j := θj + θ′j
To make sense of this, remember how the symbolic way of calculation works. In the
case at hand, x′′j and θ′′j are local sections of On|m which are mapped, under the sheaf
morphism part φ+ of +, to sections φ+ (x′′j ) := xj + x′j and φ+ (θ′′j ) := θj + θ′j of
On+n|m+m (the sheaf of the product space Rn|m × Rn|m = Rn+n|m+m ). As proved in
lecture five, this prescription uniquely determines a morphism of supermanifolds, which
we denote by +. We further define unit and inverse morphisms as follows.
(5) e : R0|0 → Rn|m , xj := 0 , θj := 0 ;
i : Rn|m → Rn|m , x′j := −xj , θ′j := −θj
Lemma 12.2. (Rn|m , +, e, i) is a super Lie group.
Proof. This is an (instructive) exercise!
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It would make perfect sense to call Rn|m with the morphisms just described the super
Lie group of super translations or super translation group. Yet, it is not. Instead, one
defines a super commutator [·, ·] on Rn ⊕ Rm by means of some bilinear form on a spinor
module, thus obtaining a super Lie algebra. Starting from there, one then constructs
a multiplication morphism on Rn|m by means of the usual addition on Rn ⊕ Rm and a
suitable version of the Baker-Campbell-Hausdorff formula for the exponential map. It
is clear that this construction does only work in dimensions where there is such a form
on the spinor module considered.
We will not go into detail here, cf. [Var04] and [DF99] for details. Instead, we
present the simplest example of a super translation group thus obtained. The underlying
supermanifold is G := R1|1 , and (super-)translation is defined as follows.
t : R1|1 × R1|1 → R1|1 ,
x′′ := x + x′ + θθ′ ,
θ′′ := θ + θ′
Lemma 12.3. (R1|1 , t, e, i) is a super Lie group with e and i as defined in (5). It is
called the super translation group of dimension 1|1.
12.2
Super Lie Algebras
Definition 12.4. A super Lie algebra is a super vector space g together with a morphism
[·, ·] : g ⊗R g → g of super vector spaces (that is an even (bi)linear map) which is (super)antisymmetric and satisfies the super Jacobi identity as follows.
[a, b] = −(−1)|a||b| [b, a]
0 = [a, [b, c]] + (−1)|a|(|b|+|c|) [b, [c, a]] + (−1)|c|(|a|+|b|) [c, [a, b]]
for all (homogeneous) a, b, c ∈ g.
The previous definition is such that super Lie algebras are related to super Lie groups
as in the classical theory. To formulate the statement, we need to make precise what
is a left invariant super vector field on a super Lie group. To motivate the following
definition, let G be an ordinary Lie group and recall that a vector field X ∈ Γ(T G)
is, by definition, left invariant if Xm(g,h) = dh lg (Xh ) holds for all g, h ∈ G, where
m : G × G → G is the group operation and lg := m(g, ·) : G → G denotes left translation
by g. Similarly, right translation by h is defined by rh := m(·, h) : G → G. The derivative
of m then decomposes as d(g,h) m = dh lg + dg rh such that
dh lg (Xh ) = d(g,h) m (id ⊗R X)(g,h)
where id ⊗ X is the vector field on G × G, defined in the obvious way by acting only on
the second component.
To reformulate left invariance, we let f ∈ C ∞ (G) and calculate
Xm(g,h) (f ) = (X(f ))(m(g, h)) = (m∗ ◦ X)(f )(g, h)
where m∗ denotes the pullback of m. On the other hand, we obtain
(dh lg (Xh ))(f ) = d(g,h) m (id ⊗ X)(g,h) (f )
= dm(g,h) f ◦ d(g,h) m (id ⊗ X)(g,h)
= d(g,h) (f ◦ m) (id ⊗ X)(g,h)
= ((id ⊗ X)(f ◦ m)) (g, h)
= ((id ⊗ X) ◦ m∗ ) (f )(g, h)
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and, therefore, X is left invariant if and only if
m∗ ◦ X = (id ⊗ X) ◦ m∗
Similarly, X is right invariant if and only if m∗ ◦ X = (X ⊗ id) ◦ m∗
Definition 12.5. Let G be a super Lie group with group operation (m, µ) : G×G → G.
A super vector field X ∈ SG is left invariant (respectively right invariant) if
µ ◦ X = (id ⊗ X) ◦ µ
respectively
µ ◦ X = (X ⊗ id) ◦ µ
Theorem 12.6. Let G be a super Lie group. Then the following holds.
(i) The super vector space of left invariant super vector fields on G, together with the
supercommutator of super vector fields, is a super Lie algebra, denoted g = Lie(G).
(ii) The map X 7→ X1 ∈ (SG)1 that sends X ∈ g to the tangent vector at the identity
point 1 = e(0) (i.e. the restriction of X to the stalk) is an isomorphism of super
vector spaces. In particular, the dimension n|m of g as a super vector space equals
the dimension of G as a supermanifold.
(iii) The even part of g is the Lie algebra of the classical Lie group G0 underlying
G = (G0 , OG , vG ), i.e. g0 = Lie(G0 ).
Proof. We leave the proof as an exercise. A sketch is provided in Sec. 7.1 of [Var04].
Moreover, it can be shown that a super Lie group is characterised in terms of a
Harish-Chandra pairs (G0 , g) consisting of a Lie group G0 and a super Lie algebra g
with Lie(G0 ) = g0 and a certain representation of G0 on g. Consult [Goe08] and the
references therein for details.
Exercise 12.7. Consider the super translation group R1|1 with the super translation
morphism t : R1|1 × R1|1 → R1|1 from Lem. 12.3. Show that the super vector fields
∂
∂
∂
,
Dθ :=
−θ
∂t
∂θ
∂t
are left invariant, and the super vector fields
∂
,
∂t
τQθ :=
∂
∂
+θ
∂θ
∂t
∂
and Dθ , thus characare right invariant. Calculate the supercommutator brackets of ∂t
terising the super Lie algebra of infinitesimal super translations.
References
[DF99] P. Deligne and D. Freed. Supersolutions. In P. Deligne et al., editor, Quantum
Fields and Strings: A Course for Mathematicians. American Mathematical
Society, 1999.
[Goe08] O. Goertsches. Riemannian supergeometry. Math. Z., 260(3):557–593, 2008.
[Sch70] H. Schubert. Kategorien I. Akademie-Verlag, 1970.
[Var04] V. Varadarajan. Supersymmetry for Mathematicians: An Introduction. Providence, R.I., 2004.
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