Lie Superalgebras and Lie Supergroups, II
Helmut Boseck
6. The Hopf Dual.
Let H = H0 H1 denote an a ne Hopf superalgebra, i.e. a Z -graded
2
commutative, nitely generated Hopf algebra. The set G (H) = homalg(H K)
of algebra homomorphisms mapping the superalgebra H onto the base eld is a
group with respect to convolution, called the structure group of H:
The convolution is de ned by =( ) . It holds 2 G (H)
1 2 1 2 1 2
if 2 G (H) , and " = = " , and ( ) = " =( ) :
1 2
By an easy calculation we have f.i. " =( ") = (idH ") = idH =
, and ( ) =(( ) ) = ( )( idH) = ( idH) = " = ":
Let H = H=(H1) denote the quotient of H by the homogeneous ideal
generated by H1 , which is a coideal too: (H) H1 H0 + H0 H1 "(H1) = 0 ,
and (H1) H1:
H is an a ne Hopf algebra and there is a canonical isomorphism between
the structure groups of H and H
G (H) G (H ):
=
Let denote a Grassmann algebra and let G (H ) = homalg(H )
denote the set of algebra homomorphisms mapping H into , then G (H ) is
a group with respect to convolution. The equation ( ) = p0 with 2
G (H ) de nes an epimorphism mapping G (H ) onto G (H) . Moreover, by
the inclusion K we have a canonical embedding of G (H) into G (H ) .
The diagram
G (H) ;! G (H ) ;! G (H)
implies a semidirect product structure of G (H )
G (H ) G (H) K(H ):
=
The normal subgroup K(H ) = ker is a unipotent group. In the representa-
tion by block matrices from Mat(m n ) the group K(H ) consists of matrices
of the form
!
Em B1( )
:
B0( ) En
A di erentiation : H ;! K of H is a linear mapping satisfying the
equation
(XY) = (X)"(Y) + "(X) (Y) X Y 2 H:
Let and denote homogeneous di erentiations of H . Then
1 2
1 2
[ ] = ; (;1)j j j j
1 2 1 2 2 1
is a homogeneous di erentiation of H:
Let L0(H) and L1(H) denote the linear space of even and odd di erentia-
tions of H respectively.
The linear space L(H) = L0(H) L1(H) is a Lie superalgebra with respect
to the brackets de ned above.
L( H ) is called the Liesuperalgebra of the Hopf superalgebra H:
Let : H;! denote an even linear mapping satisfying the equation
(XY) = (X)"(Y)+"(X) (Y) X Y 2 H: We shall call it a -di erentiation
of H . Then it holds
The linear space L(H ) of - di erentiations of H is a Lie algebra with
respect to the brackets [ ] = ; : The Lie algebra L(H )
1 2 1 2 2 1
is isomorphic to the Grassmann-hull of the Lie superalgebra L(H):
The algebraic dual H0 = H0 H0 of the Hopf superalgebra H is an
0 1
associative superalgebra with unit. The product of H0 is the convolution, i.e. the
0
restriction of the dual mapping : (H H)0 ;!H0 of the coproduct to the
subspace H0 H0 of (H H)0 .
Let H denote the subalgebra of H0 consisting of those linear functionals
' 2 H0 which vanish on a homogeneous ideal of nite codimension in H . Then it
0
is known, that the restriction of the dual mapping of the product map
of H to H maps H into H H and hence de nes a coproduct in
0
H . Moreover, the restriction of the dual map of the antipode of H to H
is an antipode of H . At last, the counit of H is the evaluation of the linear
functionals at the unit of H:
H = H 0 H 1 is a cocommutative Hopf superalgebra. It is called the
Hopfdual of H:
The structure group G (H) consists of the group-like elements of H : 2
G (H) i ( ) = :
The structure group of H is the group of units in H :
The Lie superalgebra L(H) consists of the primitive or Liealgebra-like
elements of H : 2 L(H) i ( ) = " + " .
It is easy to verify the following equations :
< ( ) X Y > = < XY >
= < X >< Y > = < X Y >
and < ( ) X Y > = < XY >
= < X >< " Y > + < " X >< Y >
= < " + " X Y > X Y 2 H:
Structure Theorem of cocommutative Hopf superalgebras .
(Sweedler, Kostant) Let H denote a cocommutative Hopf superalgebra, G its
group of group-like elements and L its Lie superalgebra of primitive elements. Let
K(G) denote the group algebra of G and U(L) the e nveloping superalgebra of L.
Then H is a smashed product of K(G) and U(L) .
As a corollary from this theorem we mention
The Hopf dual H of a Hopf superalgebra H is generated by its structure
group G (H) and its Lie superalgebra L(H):
7. A ne Algebraic Supergroups.
Let G denote a group. A representative function on G is a K-valued
function f on G with the property that spanffg g 2 Gg ( or spanfgf g 2 Gg )
is nite dimensional. It is fg(g0) = f(g0g) and f(g0) = f(g;1 g0).
g
Proposition.(Hochschild) The representative functions of G make up
an a ne Hopf algebra R (G) . The coproduct is de ned by the equation
e
f(g1 g2) = f(g1 g2) using the isomorphism of R (G G) with R (G) R (G) .
The counit " is the evaluation at the identity : " (f) = f(e) and the antipode
is de ned by (f)(g) = f(g;1 ).
De nition.(Hochschild) The structure of an a ne algebraic group is
a pair (G P ) consisting of a group G and a sub Hopf algebra P of R (G)
satisfying the following properties
i.) P separates the points of G
ii.) every algebra homomorphism : P ;! K is the evaluation at a group
element.
The algebra P is called the algebra of polynomial functions on G .
The properties i.) and ii.) imply a canonical isomorphism G G (P ) and
=
L0 = L(P ) is the Lie algebra of the a ne algebraic group (G P ).
Proposition.(Hochschild) Let H denote an a ne Hopf algebra, then
(G (H ) H ) is an a ne algebraic group structure.
De nition. The structure of an a ne algebraic supergroup is a pair (G P)
consisting of a group G and an a ne Hopf superalgebra P satisfying the following
property:.
There is a sub Hopf algebra P of R (G) and a nite dimensional vector
space W such that it holds
i.) P P (W) is an isomorphism of associative superalgebras
=
ii.) the canonical projection p : P ;!P is a morphism of supercoalgebras
compatible with the antipodes
iii.) (G P ) is the structure of an a ne algebraic group.
Property ii.) is equivalent to the equations : (p p) = p " =
" p and p = p: Property iii.) implies canonical isomorphisms G (P) =
G (P ) G . Every algebra homomorphism : P ;! K is of the following type:
=
projection by p and evaluation at a group element.
P = P0 P1 is called the superalgebra ofpolynomial functions in commuting
and anticommuting variables on G .
L(P) = L = L0 L1 is called the Lie superalgebra of the a ne algebraic
supergroup (G P) .
Evidently holds: dim L0 = dim(G P ) = degree of transcendency of
Q(P )=K Q(P ) denotes the quotient eld of P provided the algebraic group
structure (G P ) is irreducible, i.e. P is an integral domain and dim L1 =
dim W .
Proposition. Let H = H0 H1 denote an a ne Hopf superalgebra, then
(G (H) H) is the structure of an a ne algebraic supergroup.
If H H (W) , then it holds G (H) G (H ) canonically and
= =
(G (H) H ) is the structure of an a ne algebraic group, the underlying algebraic
group of the algebraic supergroup (G (H) H).
Assume L = L0 L1 to be a nite dimensional Lie superalgebra. Denote
by U(L) its enveloping algebra, U(L) is a cocommutative Hopf superalgebra:
( ) = " + " 2 L " denotes the unit element of U(L). Its Hopf
dual U (L) is a commutative Hopf superalgebra.
It holds the following isomorphism of associative, commutative superalge-
bras:
U (L) U (L0) (L0 ):
=
1
Proposition. The following statements are equivalent
(i) U (L) is an a ne Hopf superalgebra
(ii) [L0 L0] = L0
(iii) L(U (L)) L:
=
e
If one of the statements (i) - (iii) holds, then G = G (U (L)) G (U (L0))
=
is the connectend and simply connected algebraic Lie group associated to L0 and
e
(G U (L)) is the structure of an a ne algebraic supergroup.
Corollary. Let L denote a nite dimensional Lie superalgebra with semi-
e
simple even part L0 , and let G denote the connected, simply connected, semisimple
e
Lie group corresponding to L0 , then (G U (L0)) is the structure of an a ne
algebraic supergroup.
8. Representations.
Let us start with a motivation.
Let L = L0 L1 denote a Lie superalgebra, and let V = V0 V1 denote
a nite dimensional linear superspace. Assume that : L ;! LL(V) is a
representation of L and U( ) : U(L) ;! L(V) its lift to the enveloping algebra
U(L) . Put (v)(u) = (;1)j uj j vj U( )(u)v , with u 2 U(L) v 2 V and homogeneous.
(v) is a V -valued representative function of L : (v) 2 U (L) V . The map
: V ;! U (L) makes V an U (L) -left supercomodule.
Every representation of the Lie superalgebra L de nes an U (L) -left su-
percomodule structure on the representation superspace V .
De nition. Let H = H0 H1 denote a Hopf superalgebra. A H -left
supercomodule is a linear superspace V = V0 V1 endowed with an
even linear mapping : V ;!H V
satisfying the following equations:
( idV ) =(idH ) (" idV ) = idV :
Example 1 . Let H = H0 H1 denote a Hopf superalgebra. Put V = H
and = .
H is a H -left (-right) supercomodul.
Example 2 . Assume H = SP(m n) K[X X0]=(detX ; detX0)
=
(Y Y0)
!
X Y0
X = L = sl(m n):
Y X0
Take V = Km n = Km 0 K0 n , and let e1 ::: em and f1 ::: fn denote the canonical
basis of Km 0 and K0 n respectively. Write
0 1
e1
B C
.
B C
.
.
B C
B C
B C
em
B C
E = and (E) = X E:
B C
f1
B C
B C
.
B C
.
@ .
A
fn
In more detail
m n
X X
(e ) = X j ej + Y 0 fk
k
j=1 k=1
m n
X X
(f ) = Y j ej + X 0 fk:
k
j=1 k=1
It holds ( idV ) (E) = ( idV )(X E) = X E =(X X ) E = X
(X E) =(idH ) (E) (" idV) (E) = (" idV )(X E) = "(X ) E = E E : Km n
is a SP(m n)-left supercomodule .
Let H = H0 H1 denote a Hopf superalgebra. A block matrix X 2
Mat(m n H) is called multiplicative , if it holds
(X ) = X X :
Proposition.(Manin) The H -left supercomodules (Km n ) are in one-
to-one correspondence to the multiplicative block matrices X 2 Mat(m n H):
Given : Km n ;!H Km n we may compute X from
(E) = X E
and given a multiplicative X 2 Mat(m n H) is de ned by the same equation.
0
A linear mapping f : Km n ;! Km n0 is a morphism ofthe supercomodules
0
0
(Km n ) and (Km n0 ) , if there is a block matrix F interchanging the corre-
0 0
sponding multiplicative block matrices X and X : FX = XF and f(E) = FE0 .
Let H = H0 H1 denote a Hopf superalgebra and let (V ) denote a
H -left supercomodule ( not necessarily nite dimensional). If ' 2 H0 = H0 H0
0 1
then the equation
0
(') = (' idV )
0
de nes a representation of the associative superalgebra H0 over V .
0
By suitable restrictions of we get representations of the superalgebra H ,
of the structure group G (H) , and of the Lie superalgebra L(H):
(') = (' idV ) ' 2 H
( ) = ( idV ) 2 G (H)
G
( ) = ( idV ) 2 L(H):
L
Let (G P) denote an a ne algebraic supergroup and assume V = P as
P -left supercomodule (P ) . The corresponding representation of the Lie
L
superalgebra L = L(P) is called the left regular representation of L:
Proposition. The left regular representation of the Lie superalgebra L =
L(P) of the a ne algebraic supergroup (G P) is an isomorphism of L onto the
Lie superalgebra Der(r)P of right invariant derivations on P : L(P) Der(r)P .
=
Let us start with an a ne algebraic supergroup (G P). If (Km n ) is a P -
left supercomodule , and X denotes the corresponding multiplicative block matrix,
then we may get matrix representations of the group G and its Lie superalgebra
L = L(P) as well, in the following way:
!
(X) 0
( ) = (X ) =
G
0 (X0)
!
(X) (Y0)
( ) = (X ) = :
L
(Y) (X0)
We may also get representations of the group G = G (P ) and of the Lie
algebra L = L(P ) by matrices :
!
(X) (Y0)
( ) = (X ) =
G
(Y) (X0)
!
(X) (Y0)
( ) = (X ) = :
L
(Y) (X0)
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at
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Received March 31, 1992