Lie

Sup

eralgebras

and

Lie

Sup

ergroups,

I

I

Helmut Boseck

6. The

Hopf

Dual.

Let

H

=

H

0

H

1

denote an ane

Hopf

superalgebra, i.e. a

Z

2

-graded

commutative, nitely generated

Hopf

algebra. The set

G

(

H

) = hom

alg

(

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 dened by

1

2

= (

1

2

). It holds

1

2

2

G

(

H

)

if

1

2

2

G

(

H

), and

" = = "

, and

(

) = " = ()

:

By an easy calculation we have f.i.

" = (

") = (id

H

") = id

H

=

, and ()

= (()

) = (

)(

id

H

) =

(

id

H

) =

" = ":

Let

H

=

H

=(

H

1

) denote the quotient of

H

by the homogeneous ideal

generated by

H

1

, which is a coideal too: (

H

)

H

1

H

0

+

H

0

H

1

"(

H

1

) = 0,

and

(

H

1

)

H

1

:

H

is an ane

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

) = hom

alg

(

H

)

denote the set of algebra homomorphisms mapping

H

into , then

G

(

H

) is

a group with respect to convolution. The equation

(

) =

p

0

with

2

G

(

H

) denes 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(

mn) the group

K

(

H

) consists of matrices

of the form

E

m

B

1

()

B

0

()

E

n

!

:

A

di erentiation

:

H

;

!

K of

H

is a linear mapping satisfying the

equation

(XY ) = (X)"(Y ) + "(X)(Y ) XY

2

H

:

Let

1

and

2

denote homogeneous dierentiations of

H

. Then

1

2

] =

1

2

;

(

;

1)

j

1

jj

2

j

2

1

is a homogeneous dierentiation of

H

:

Let

L

0

(

H

) and

L

1

(

H

) denote the linear space of even and odd dierentia-

tions of

H

respectively.

The linear space

L

(

H

) =

L

0

(

H

)

L

1

(

H

)

is a

Lie

superalgebra with respect

to the brackets dened above.

L

(

H

) is called the

Lie

superalgebra of the

Hopf

superalgebra

H

:

Let

:

H

;

!

denote an even linear mapping satisfying the equation

(

XY ) =

(

X)"(Y )+"(X)

(

Y ) XY

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

1

2

] =

1

2

;

2

1

:

The

Lie

algebra

L

(

H

)

is isomorphic to the

Grassmann

-hull of the

Lie

superalgebra

L

(

H

)

:

The algebraic dual

H

0

=

H

0

0

H

0

1

of the

Hopf

superalgebra

H

is an

associative superalgebra with unit. The product of

H

0

is the convolution, i.e. the

restriction of the dual mapping

0

: (

H

H

)

0

;

!

H

0

of the coproduct to the

subspace

H

0

H

0

of (

H

H

)

0

.

Let

H

denote the subalgebra of

H

0

consisting of those linear functionals

'

2

H

0

which vanish on a homogeneous ideal of nite codimension in

H

. Then it

is known, that the restriction of the dual mapping

0

of the product map

of

H

to

H

maps

H

into

H

H

and hence denes a coproduct in

H

. Moreover, the restriction of the dual map

0

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

Hopf

dual

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

Lie

algebra-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 >XY

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 enveloping 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. Ane Algebraic Supergroups.

Let

G denote a group. A representative function on G is a K-valued

function

f on G with the property that span

f

f

g

g

2

G

g

( or span

f

g

fg

2

G

g

)

is nite dimensional. It is

f

g

(

g

0

) =

f(g

0

g) and

g

f(g

0

) =

f(g

;1

g

0

).

Proposition.

(Hochschild)

The representative functions of

G

make up

an ane

Hopf

algebra

R

(

G)

. The coproduct

is dened by the equation

e

f(g

1

g

2

) =

f(g

1

g

2

)

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 dened by

(f)(g) = f(g

;1

).

Denition.

(Hochschild)

The

structure of an ane 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

L

0

=

L

(

P

) is the

Lie

algebra of the ane algebraic group (

G

P

).

Proposition.

(Hochschild)

Let

H

denote an ane

Hopf

algebra, then

(

G

(

H

)

H

)

is an ane algebraic group structure.

Denition.

The

structure of an ane algebraic supergroup

is a pair (

G

P

)

consisting of a group

G and an ane

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 ane 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

=

P

0

P

1

is called the

superalgebra of polynomial functions in commuting

and anticommuting variables on

G .

L

(

P

) =

L

=

L

0

L

1

is called the

Lie

superalgebra of the ane algebraic

supergroup

(

G

P

) .

Evidently holds: dim

L

0

= 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

L

1

=

dim

W .

Proposition.

Let

H

=

H

0

H

1

denote an ane

Hopf

superalgebra, then

(

G

(

H

)

H

)

is the structure of an ane algebraic supergroup.

If

H

=

H

(

W), then it holds

G

(

H

)

=

G

(

H

) canonically and

(

G

(

H

)

H

) is the structure of an ane algebraic group, the

underlying algebraic

group

of the algebraic supergroup (

G

(

H

)

H

).

Assume

L

=

L

0

L

1

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 (

L

0

)

(

L

0

1

)

:

Proposition.

The following statements are equivalent

(i)

U (

L

)

is an ane

Hopf

superalgebra

(ii)

L

0

L

0

] =

L

0

(iii)

L

(

U (

L

))

=

L

:

If one of the statements (i) - (iii) holds, then

e

G =

G

(

U (

L

))

=

G

(

U (

L

0

))

is the connectend and simply connected algebraic

Lie

group associated to

L

0

and

(

e

GU (

L

)) is the structure of an ane algebraic supergroup.

Corollary.

Let

L

denote a nite dimensional

Lie

superalgebra with semi-

simple even part

L

0

,

and let

e

G

denote the connected, simply connected, semisimple

Lie

group corresponding to

L

0

, then

(

e

GU (

L

0

))

is the structure of an ane

algebraic supergroup.

8. Representations.

Let us start with a motivation.

Let

L

=

L

0

L

1

denote a

Lie

superalgebra, and let

V = V

0

V

1

denote

a nite dimensional linear superspace. Assume that

:

L

;

!

L

L

(

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

u

jj

v

j

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

denes an

U (

L

)

-left su-

percomodule structure on the representation superspace

V

.

Denition.

Let

H

=

H

0

H

1

denote a

Hopf

superalgebra. A

H

-

left

supercomodule

is a linear superspace

V = V

0

V

1

endowed with an

even linear mapping

: V

;

!

H

V

satisfying the following equations:

(

id

V

)

= (id

H

) ("

id

V

)

= id

V

:

Example 1

. Let

H

=

H

0

H

1

denote a

Hopf

superalgebra. Put

V =

H

and

= .

H

is a

H

-left (-right) supercomodul.

Example 2

. Assume

H

=

S

P

(

mn)

= KXX

0

]

=(detX

;

detX

0

)

(

YY

0

)

X

= X Y

0

Y X

0

!

L

=

sl

(

mn):

Take

V = K

mn

=

K

m

0

K

0

n

, and let

e

1

:::e

m

and

f

1

:::f

n

denote the canonical

basis of

K

m

0

and

K

0

n

respectively. Write

E

=

0

B

B

B

B

B

B

B

B

B

B

@

e

1

...

e

m

f

1

...

f

n

1

C

C

C

C

C

C

C

C

C

C

A

and

(

E

) =

X

E

:

In more detail

(e

) =

m

X

j

=1

X

j

e

j

+

n

X

k

=1

Y

0

k

f

k

(f

) =

m

X

j

=1

Y

j

e

j

+

n

X

k

=1

X

0

k

f

k

:

It holds (

id

V

)

(

E

) = (

id

V

)(

X

E

) =

X

E

= (

X

X

)

E

=

X

(

X

E

) = (

id

H

)(

E

)(

"

id

V

)

(

E

) = (

"

id

V

)(

X

E

) =

"(

X

)

E

=

E

E

: K

mn

is a

S

P

(

mn)-

left supercomodule

.

Let

H

=

H

0

H

1

denote a

Hopf

superalgebra. A block matrix

X

2

Mat(

mn

H

) is called

multiplicative

, if it holds

(

X

) =

X

X

:

Proposition.

(Manin)

The

H

-left supercomodules

(

K

mn

)

are in one-

to-one correspondence to the multiplicative block matrices

X

2

Mat(

mn

H

)

:

Given

: K

mn

;

!

H

K

mn

we may compute

X

from

(

E

) =

X

E

and given a multiplicative

X

2

Mat(

mn

H

)

is dened by the same equation.

A linear mapping

f : K

mn

;

!

K

m

0

n

0

is a morphism of the supercomodules

(

K

mn

)

and

(

K

m

0

n

0

0

)

, if there is a block matrix

F

interchanging the corre-

sponding multiplicative block matrices

X

and

X

0

:

F

X

0

=

X

F

and

f(

E

) =

F

E

0

.

Let

H

=

H

0

H

1

denote a

Hopf

superalgebra and let (

V) denote a

H

-left supercomodule ( not necessarily nite dimensional). If

'

2

H

0

=

H

0

0

H

0

1

then the equation

0

(

') = ('

id

V

)

denes a representation

0

of the associative superalgebra

H

0

over

V .

By suitable restrictions of

0

we get representations of the superalgebra

H

,

of the structure group

G

(

H

), and of the

Lie

superalgebra

L

(

H

):

(') = ('

id

V

)

'

2

H

G

(

) = (

id

V

)

2

G

(

H

)

L

(

) = (

id

V

)

2

L

(

H

)

:

Let (

G

P

) denote an ane algebraic supergroup and assume

V =

P

as

P

-left supercomodule (

P

). The corresponding representation

L

of the

Lie

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 ane 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 ane algebraic supergroup (

G

P

). If (

K

mn

) 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:

G

(

) = (

X

) =

(X) 0

0

(X

0

)

!

L

(

) = (

X

) =

(X) (Y

0

)

(Y ) (X

0

)

!

:

We may also get representations of the group

G

=

G

(

P

) and of the

Lie

algebra

L

=

L

(

P

) by matrices :

G

(

) =

(

X

) =

(

X)

(

Y

0

)

(

Y )

(

X

0

)

!

L

(

) =

(

X

) =

(

X)

(

Y

0

)

(

Y )

(

X

0

)

!

:

References

1]

Abe, E., \

Hopf

algebras," Cambridge University Press 1977.

2]

Berezin, F. A., \Introduction to algebra and analysis with anticommuting

variables," (Russian) Moscow University 1983.

3]

Boseck H.,

On representative functions of

Lie

superalgebras

, Math.Nachr.

123

(1985), 61{72

Correction to my paper: `On representative functions...'

,

Math. Nachr.

130

(1987), 137{138.

4]

|,

Ane

Lie

supergroups

, Math. Nachr.

143

(1987), 303{327.

5]

|,

Classical

Lie

supergroups

, Math. Nachr.

148

(1990), 81{115.

6]

|,

Lie

superalgebras and

Lie

supergroups, I

Seminar Sophus Lie (Hel-

dermann Verlag Berlin),

2

(1992), 109{122.

7]

Corwin, L., Y. Ne'eman, and S. Sternberg,

Graded

Lie

algebras in Math-

ematics and Physics (

Bose

-

Fermi

-symmetry)

, Rev. of Mod. Physics

47

(1975), 57{63.

8]

Heynemann, R. G., and M. E. Sweedler

Ane

Hopf

algebras I

, J. of

Algebra

13

(1969), 192{241.

9]

Hochschild, G.,

Algebraic groups and

Hopf

algebras

, Illinois J. Math.

14

(1970), 52{65.

10]

|, \Introduction to Ane Algebraic Groups," Holden Day Inc., San

Francisco etc., 1971.

11]

|, \Basic Theory of Algebraic Groups and

Lie

Algebras," Graduate Texts

in Math.

75

, Springer Verlag Berlin etc., 1981.

12]

Kac, V. G.,

Lie

superalgebras

, Adv. in Math.

26

(1977), 8{96.

13]

Kostant, B.,

Graded manifolds, graded

Lie

theory, and prequantization

,

Lect. Notes in Math.

570

(1977), 177{306.

14]

Manin, Y., \Quantum Groups and Noncommutative Geometry," CRM

Universite de Montreal 1988.

15]

Milnor J., and J. Moore,

On the structure of

Hopf

algebras

, Ann. of Math.

81

(1965), 211{264.

16]

Scheunert M., \The Theory of

Lie

Superalgebras," Lect. Notes in Math.

716

, Springer Verlag, Berlin etc., 1979.

17]

Sweedler M. E. "

Hopf

Algebras", Benjamin Inc., New York, 1969.

Fachrichtungen

Mathematik/Informatik

Ernst Moritz Arndt Universitat

Greifswald

Friedrich Ludwig Jahn-Strae 15a

O-2200 Greifswald, Germany

Received March 31, 1992