Seminar Sophus Lie
1
(1991) 109{122
Lie
Sup
eralgebras
and
Lie
Sup
ergroups,
I
Helmut Boseck
1. Graded Algebras
Let
K
denote the base eld.
K
is assumed to be of characteristic zero.
In the examples
K
is the eld of real or complex numbers.
Let
Z
denote the additive group of the integers. A
Z
-graded linear space
is a
K
-linear space
V
and a family of subspaces
V
k
, (
k
= 0 1 2
:::
) such
that
V
=
M
k
2Z
V
k
:
A
Z
-graded associative algebra
is an associative
K
-algebra
A
, which is
Z
-graded
as a
K
-linear space
A
=
M
k
2Z
A
k
such that
A
k
A
l
A
k
+
`
(
k l
2
Z
)
:
The elements of
A
k
are called
homogeneous of degree
k
.
Example 1
.
Let
V
=
L
k
2Z
V
k
be a
Z
-graded linear space. A linear operator
A
:
V
k
;
!
V
k
+
l
(
k
2
Z
) is called
homogeneous of degree
`
. If
L
`
(
V
) denotes the
space of linear operators of degree
`
, then
L
0
(
V
) =
M
k
2Z
L
k
(
V
)
is a
Z
-graded associative algebra with unit. If
V
is nite dimensional we have
L
(
V
) =
M
k
2Z
L
k
(
V
)
:
Assume
V
to be nite dimensional and
V
k
=
f
0
g
, (
k
=
;
1 2 3
:::
),
V
=
V
0
V
1
:
Then the following relations hold
L
0
(
V
) =
L
(
V
0
)
L
(
V
1
)
L
;1
(
V
) =
L
(
V
1
V
0
)
L
1
(
V
) =
L
(
V
0
V
1
) and
L
k
(
V
) =
f
0
g
(
k
= 2 3
:::
)
:
110
Boseck
Chosing a basis of homogeneous elements in
V
we may represent the elements
of
L
0
(
V
) by diagonal block matrices
(1)
A
0
0
0
A
1
:
The elements of
L
;1
(
V
) and
L
1
(
V
) become represented by the block matrices
(2)
0
B
1
0 0
and 0 0
B
0
0
respectively.
Example 2
.
Let (
n
) = (
y
1
::: y
n
) denote the exterior or
Grassmann
algebra with
n
generators, which are assumed to be homogeneous of degree 1 .
Put
0
=
K
1
= span
f
y
1
::: y
n
g
k
= span
f
y
i
1
y
i
k
: 1
i
1
<
< i
k
n
g
(
k
= 2
::: n
;
1)
n
= span
f
y
1
y
n
g
`
=
f
0
g
if
`
=
;
1
;
2
:::
or
`
=
n
+ 1
n
+ 2
:::
then we have
(
n
) =
0
1
n
and (
n
) becomes a
Z
-graded associative algebra with unit. The
Z
-graded
algebra (
n
) is (
graded
)
commutative
, i.e., for homogeneous elements
a
k
2
k
,
a
`
2
`
the following equation holds:
a
`
a
k
= (
;
1)
k`
a
k
a
`
:
Example 3
.
Let
A
=
L
k
2Z
A
k
be a
Z
-graded associative algebra with unit.
We dene a new product by
a
k
a
`
] =
a
k
a
`
;
(
;
1)
k`
a
`
a
k
a
k
2
A
k
:
With respect to that multiplication
A
becomes a
Z
-graded
Lie
algebra
A
L
=
M
k
2Z
A
k
:
We have
(3)
a
`
a
k
] =
;
(
;
1)
k`
a
k
a
`
]
and the modied
Jacobi
identity
(4)
(
;
1)
km
a
k
a
`
a
m
]] + (
;
1)
`k
a
`
a
m
a
k
]] + (
;
1)
m`
a
m
a
k
a
`
]] = 0
:
Boseck
111
A
Z
-graded
Lie
algebra is a
Z
-graded linear space
L
=
L
k
2Z
L
k
with a bracket multiplication compatible with the grading
L
k
L
`
]
L
k
+
`
and
satisfying (3) and (4) for homogeneous elements.
Let
Z
2
=
Z
=
(2) =
f
0 1
g
denote the additive group of two elements. A
Z
2
-graded linear space is a
K
-linear space
V
with two distinguished subspaces
V
0
and
V
1
such that
V
=
V
0
V
1
holds. The elements of
V
0
are called
even
,
those of
V
1
are called
odd
. A homogeneous element is either even or odd.
Every
Z
-graded linear space admits a canonical
Z
2
-gradation by
V
0
=
L
k
2Z
V
2
k
,
V
1
=
L
k
2Z
V
2
k
+1
.
A
Z
2
-graded associative algebra or an associative
superalgebra
is a
K
-
algebra, which is
Z
2
-graded as a linear space
A
=
A
0
A
1
such that the multiplication satises
A
0
A
0
A
0
A
1
A
1
A
0
A
0
A
1
A
1
A
1
A
0
A
1
:
Every
Z
-graded algebra
A
admits a canonical
Z
2
-gradation by
A
0
=
M
k
2Z
A
2
k
A
1
=
M
k
2Z
A
2
k
+1
:
Example 4
.
Let
V
=
V
0
V
1
denote a
Z
2
-graded linear space. A linear
operator
A
on
V
is called
paritiy preserving
or
even
, if it satises
A
:
V
0
!
V
0
and
A
:
V
1
!
V
1
. The operator
A
is called
parity reversing
or
odd
if
A
:
V
0
!
V
1
and
A
:
V
1
!
V
0
. By
L
0
(
V
) and
L
1
(
V
) we denote the linear space of even and
odd linear operators on
V
, respectively.
L
0
(
V
) =
L
0
(
V
)
L
1
(
V
)
is a
Z
2
-graded associative algebra with unit. If
V
is nite dimensional then
L
(
V
) =
L
0
(
V
)
L
1
(
V
)
holds and
L
0
(
V
)
=
L
(
V
0
)
L
(
V
1
)
L
1
=
L
(
V
0
V
1
)
L
(
V
1
V
0
)
:
Choosing a basis of homogeneous elements in
V
, the elements of
L
0
(
V
)
are represented by diagonal block matrices as in (1), while the elements of
L
1
(
V
)
are represented by block matrices of the following type
0
B
1
B
0
0
:
Example 5
.
The
Grassmann
algebra (
n
) admits a canonical
Z
2
-grading
with respect to a chosen system of generators (
n
) =
0
1
with
0
=
0
2
,
1
=
1
3
Now (
n
) is a
Z
2
-graded associative
and (graded) commutative algebra with unit or, equivalently, an associative,
commutative superalgebra with unit.
112
Boseck
Example 6
.
Let
A
=
A
0
A
1
denote a
Z
2
-graded associative algebra with
unit. We dene brackets by
a b
] =
ab
;
(
;
1)
j
a
jj
b
j
ba
for homogeneous elements
a b
2
A
. Dene the parity
j
a
j
as follows:
j
a
j
=
0 if
a
2
A
0
,
1 if
a
2
A
1
.
With respect to the brackets,
A
becomes a
Z
2
-graded
Lie
algebra or,equivalently,
a
Lie
superalgebra
A
L
=
A
0
A
1
:
For homogeneous elements
a b c
2
A
the
following equations hold:
(4)
b a
] =
;
(
;
1)
j
a
jj
b
j
a b
]
(5)
(
;
1)
j
a
jj
c
j
a
b c
]] + (
;
1)
j
b
jj
a
j
b
c a
]] + (
;
1)
j
c
jj
b
j
c
a b
]] = 0
(the modied
Jacobi
equation). A
Z
2
-graded
Lie
algebra or
Lie
superalgebra
is a
Z
2
-graded linear space
L
=
L
0
L
1
with a bracket multiplication
]
compatible with the gradation
L
0
L
0
]
L
0
L
1
L
1
]
L
0
L
0
L
1
]
L
1
L
1
L
0
]
L
1
and satisfying (4) and (5).
2. L
IE
Superalgebras:
The series A, B, C, D, Q
Let
L
=
L
0
L
1
denote a
Z
2
-graded
Lie
algebra. Its even part
L
0
is
a
Lie
algebra. In view of
L
0
L
1
]
L
1
, multiplication of odd elements by even
ones denes a representation ad
0
of the
Lie
algebra
L
0
on the linear space
L
1
(ad
0
x
0
)
x
1
=
x
0
x
1
]
x
0
2
L
0
x
1
2
L
1
:
ad
0
is called the
adjoint representation of the even part
L
0
on the odd part
L
1
:
A
Z
2
-graded
Lie
algebra
L
=
L
0
L
1
is called
simple
, if there are no
nontrivial
Z
2
-graded ideals: If
I
=
I
0
I
1
denotes a
Z
2
-graded ideal of
L
then
we have
I
=
f
0
g
or
I
=
L
. A simple
Z
2
-graded
Lie
algebra is called
classical
, if
the representation ad
0
is completely reducible. A simple
Z
2
-graded
Lie
algebra
is
classical
i its even part
L
0
is a reductive
Lie
algebra.
Denote by Mat(
m n
) the
Z
2
-graded associative algebra with unit con-
sisting of (
m
+
n
)
(
m
+
n
) block matrices with entries from
K
A
0
B
1
B
0
A
1
:
Boseck
113
We observe Mat(
m n
) = Mat
;1
(
m n
)
Mat
0
(
m n
)
Mat
1
(
m n
) (compare (1)
and (2)).
The
Z
2
-graded
Lie
algebra dened by Mat(
m n
) is denoted by
gl
(
m n
)
or
pl
(
m n
). It is called the
general linear
Lie
superalgebra
.
gl
(
m n
) admits a
Z
-gradation
gl
(
m n
) =
gl
;1
(
m n
)
gl
0
(
m n
)
gl
1
(
m n
)
implying the
Z
2
-gradation in the natural way. We have
gl
0
(
m n
) =
gl
0
(
m n
)
=
gl
(
m
)
gl
(
n
)
gl
1
(
m n
) =
gl
;1
(
m n
)
gl
1
(
m n
)
:
We shall discuss several subalgebras of
gl
(
m n
).
Example 1
.
sl
(
m n
) or
spl
(
m n
) denotes the subalgebra consisting of those
block matrices for which the diagonal blocks have equal trace, i.e., satisfying
the equation Tr(
A
0
) = Tr(
A
1
). It is called the
special linear
Lie
superalgebra.
sl
(
m n
) admits the
Z
-gradation induced by
gl
(
m n
) and the corresponding
Z
2
-
grading. We have
sl
0
(
m n
)
=
sl
(
m
)
sl
(
n
)
K
.
Example 2
.
osp
(
m n
) consists of those block matrices satisfying the follow-
ing relations
A
0
>
+
A
0
= 0
B
1
>
;
I
n
B
0
= 0
A
1
>
I
n
+
I
n
A
1
= 0
:
Here
A
>
denotes the transpose of
A
, and
n
is assumed to be even
n
= 2
`
, and
I
n
=
0
E
`
;
E
`
0
where
E
`
denotes the
`
`
unit matrix. The matrices of
osp
(
m
2
`
) may be
written in the following form
0
@
A
0
B
1
B
2
;
B
2
>
A
1
A
12
B
1
>
A
21
;
A
1
>
1
A
with
A
0
>
=
;
A
0
A
>
12
=
A
12
and
A
>
21
=
A
21
:
Note that
osp
(
m n
) is a
Z
2
-graded
Lie
algebra. It is called the
orthogonal-
symplectic
Lie
superalgebra. For the even part one has
osp
0
(
m n
)
=
o
(
m
)
sp
(
n
)
:
Example 3
.
e
gl
(
m
) consists of those block matrices satisfying
m
=
n
and
A
1
=
A
0
and
B
1
=
B
0
:
It is called the
general linear
Lie
superalgebra of the
second kind
.
e
gl
(
m
) is a
Z
2
-graded
Lie
algebra and we have
e
gl
0
(
m
)
=
gl
(
m
).
114
Boseck
Example 4
.
e
sl
(
m
) consists of those block matrices satisfying
m
=
n
and
A
1
=
A
0
,
B
1
=
B
0
, and Tr(
B
1
) = 0. It is called the
special linear
Lie
superalgebra of the second kind
.
e
sl
(
m
) is a
Z
2
-graded
Lie
algebra, a subalgebra
of
e
gl
(
m
), and the following relation holds:
e
sl
0
(
m
) =
e
gl
0
(
m
)
=
gl
(
m
).
The
Lie
superalgebra
sl
(
m n
)
is simple if
m
6
=
n
,
m
1
,
n
1
.
The multiples of the unit matrix
f
E
2
m
2
K
g
make up a
Z
2
-graded
ideal of
sl
(
m m
), namely, the center.
The quotient algebra
sl
(
m m
)
=
f
E
2
m
:
2
K
g
is simple if
m >
1.
The
A
-series of simple
Lie
superalgebras is dened in analogy with the
usual classication of simple
Lie
algebras by
A
(
m n
) =
sl
(
m
+ 1
n
+ 1)
m
6
=
n m
0
n
0
:
A
(
m m
) =
sl
(
m
+ 1
m
+ 1)
=
f
E
2
m
+2
:
2
K
g
m >
0
:
The orthogonal-symplectic
Lie
superalgebra
osp
(
m n
)
is simple if
m
1
,
n >
1
.
The series
B C D
are dened as follows
B
(
m n
) =
osp
(2
m
+ 1 2
n
)
m
0
n >
0
:
C
(
n
) =
osp
(2 2
n
;
2)
n
2
:
D
(
m n
) =
osp
(2
m
2
n
)
m
2
n >
0
:
As in
sl
(
m m
), the multiples of unity
f
E
2
m
2
K
g
make up a
homogeneous ideal in
e
sl
(
m
), namely, its center.
The quotient algebra
e
sl
(
m
)
=
f
E
2
m
:
2
K
g
is simple if
m
3
.
The
Q
-series is dened by
Q
(
m
) =
e
sl
(
m
+ 1)
=
f
E
2
m
+2
:
2
K
g
m
2
:
The
Lie
superalgebras
Q
(
m
) are often called
the
f
;
d
-algebras of
Michal
and
Radicati
.
The
Lie
superalgebras of the series
A B C D
and
Q
are classical
Lie
superalgebras.
The decomposition of the even parts in a direct product of simple
Lie
algebras is given by the following isomorphisms
A
0
(
m n
)
=
A
(
m
)
A
(
n
)
K m
6
=
n
A
0
(
m m
)
=
A
(
m
)
A
(
m
)
B
0
(
m n
)
=
B
(
m
)
C
(
n
)
C
0
(
m
)
=
C
(
m
;
1)
K
D
0
(
m n
)
=
D
(
m
)
C
(
n
)
Q
0
(
m
)
=
A
(
m
)
:
Boseck
115
3. The G
RASSMANN
-hull
The
Grassmann
-hull is a construction, which enables us to make a
Z
2
-
graded
Lie
algebra into a
Lie
algebra. Let denote a nitely generated
Grass-
mann
algebra and let
L
be a
Z
2
-graded
Lie
algebra. Taking the tensorproduct
L
of
Z
2
-graded algebras we have
(
L
)
0
=
0
L
0
+
1
L
1
(
L
)
1
=
0
L
1
+
1
L
0
:
Writing the elements of
L
for simplicity as
x
, the brackets are de-
ned for homogeneous elements as follows:
x
x
] = (
;
1)
x
x
],
2
Z
2
. If
=
and
=
, then we have
x
x
] = (
;
1)
x
x
]
= (
;
1)
+
+
(
;
1)
x
x
]
=
;
(
;
1)
+
+
+
x
x
]
Since
+
=
+
= 0 we have
x
x
] =
;
x
x
]. The even
part (
L
)
0
of the tensor product
L
is a
Lie
algebra. It is called the
Grassmann
-
hull
of the
Lie
superalgebra
L
.
Let
L
denote one of the
Lie
superalgebras
gl
(
m n
),
sl
(
m n
),
osp
(
m n
),
e
gl
(
m
), or
e
sl
(
m
). The
Grassmann
-hull consists of block matrices
(5)
A
0
(
0
)
B
1
(
1
)
B
0
(
1
)
A
1
(
0
)
:
The entries of the diagonal blocks
A
0
and
A
1
belong to
0
, while the entries
of the matrices
B
0
and
B
1
are from
1
. We denote the
Grassmann
-hulls
by
gl
(
m n
),
sl
(
m n
),
osp
(
m n
),
e
gl
(
m
),
e
sl
(
m
), respectively. Note
that
sl
(
m n
) is the
Lie
algebra of block matrices of type (5) such that
Tr(
A
0
(
0
)) = Tr(
A
1
(
0
)). Further,
osp
(
m
2
l
) is the
Lie
algebra of block
matrices
0
@
A
0
(
0
)
B
1
(
1
)
B
2
(
1
)
;
B
>
2
(
1
)
A
1
(
0
)
A
12
(
0
)
B
>
1
(
0
)
A
21
(
0
)
;
A
>
1
(
0
)
1
A
satisfying the relations
A
>
0
(
0
) =
A
0
(
0
),
A
>
12
(
0
) =
A
12
(
0
),
A
>
21
(
0
) =
A
21
(
0
). Finally,
e
sl
(
m
) consists of block matrices
A
(
0
)
B
(
1
)
B
(
1
)
A
(
0
)
satisfying the relation Tr
B
(
1
) = 0.
116
Boseck
In all of our considerations denotes an arbitrary nitely generated
Grassmann
algebra. Later it will become clear, that it is necessary to assume,
that the number of generators "is not to small" with respect to
m
and
n
.
Let
p
0
:
!
0
=
K
denote the canonical projection of the
Z
-graded
algebra onto its zero component. Then
p
0
denes a canonical projection of
the
Grassmann
-hull (
L
)
0
of a
Lie
superalgebra
L
onto its even part
L
0
.
We denote it once more by
p
0
, so that
p
0
(
x
) =
p
0
(
)
x
. With respect to the
matrix
Lie
algebra
gl
(
m n
) and its subalgebras we have
p
0
A
0
(
0
)
B
1
(
1
)
B
0
(
1
)
A
1
(
0
)
=
A
0
0
0
A
1
with entries from
K
in the diagonal blocks
A
0
and
A
1
.
In the case of
osp
(
m n
) the matrices
A
0
make up the
Lie
algebra
o
(
m
) while the matrices
A
1
are the matrices of
sp
(
n
).
In the same way we may dene the
Grassmann
-hull of a
Z
2
-graded
associative algebra. The
Grassmann
-hull of the matrix algebra Mat(
m n
) is
denoted by Mat(
m n
) = (
Mat(
m n
))
0
, the elements are the block matrices
(5).
4. G
RASSMANN
L
IE
groups
The series GL, SL, OSp,
f
GL,
f
SL. We dene matrix
Lie
groups
corresponding to the
Grassmann
-hulls of matrix
Lie
superalgebras. First we
answer the question of invertibility of a matrix (5) from Mat(
m n
).
A block matrix of type
(5)
is invertible i the matrices
A
0
=
p
0
(
A
0
(
0
))
and
A
1
=
p
0
(
A
1
(
0
))
are invertible.
It follows that the diagonal blocks
A
0
(
0
) and
A
1
(
0
) are invertible for
invertible block matrices. The inverse of a block matrix is written
A
(;1)
0
(
0
)
B
(;1)
1
(
1
)
B
(;1)
0
(
1
)
A
(;1)
1
(
0
)
satisfying the equations
A
(;1)
0
(
0
) = (
A
0
(
0
)
;
B
1
(
1
)
A
1
(
0
)
;1
B
0
(
1
))
;1
A
(;1)
1
(
0
) = (
A
1
(
0
)
;
B
0
(
1
)
A
0
(
0
)
;1
B
1
(
1
))
;1
B
(;1)
1
(
1
) =
;
A
0
(
0
)
;1
B
1
(
1
)(
A
1
(
0
)
;
B
0
(
1
)
A
0
(
0
)
;1
B
1
(
1
))
;1
B
(;1)
0
(
1
) =
;
A
1
(
0
)
;1
B
0
(
1
)(
A
0
(
0
)
;
B
1
(
1
)
A
1
(
0
)
;1
B
0
(
1
))
;1
:
Let GL(
m n
) denote the group of units in Mat(
m n
). It consists
of the invertible block matrices of type (5) which for simplicity are written
A
() =
A
0
()
B
1
()
B
0
()
A
1
()
:
Boseck
117
The
superdeterminant
or
Berezin
ian
of
A
is dened by sdet
A
() =
det
;
A
0
()
;
B
1
()
A
1
()
;1
B
0
()
det
A
;1
1
(). The superdeterminant is dened
on the
Grassmann Lie
group GL(
m n
). It is multiplicative, i.e., sdet is a
homomorphism of GL(
m n
) into the group of units
K
of .
Also, SL(
m n
) is the subgroup of GL(
m n
) dened by
sdet
A
() = 1
or, equivalently, by
det(
A
0
()
;
B
1
()
A
1
()
;1
B
0
()) = det
A
1
()
:
Using the projection
p
0
:
!
K
we get
det
A
0
=
p
0
(det
A
0
()) =
p
0
(det
A
1
()) = det
A
1
:
We observe that GL(
m n
) and SL(
m n
) are the -matrix groups
corresponding to the -matrix
Lie
algebras
gl
(
m n
) and
sl
(
m n
), respec-
tively.
OSp(
m n
) is the subgroup of GL(
m n
) dened by the following
relations
A
0
()
>
A
0
()
;
B
0
()
>
I
n
B
0
() =
E
m
A
0
()
>
B
1
()
;
B
0
()
>
I
n
A
1
() = 0
B
1
()
>
B
1
() +
A
1
()
>
I
n
A
1
() =
I
n
:
Applying the projection
p
0
to these equations, we get
A
>
0
A
0
=
E
m
and
A
>
1
I
n
A
1
=
I
n
hence
p
0
:OSp(
m n
)
!
O(
m
)
Sp(
n
).
OSp(
m n
) is the -matrix group corresponding to the -matrix
Lie
algebra
osp
(
m n
). The projection
p
0
is a homomorphism mapping the
Grassmann Lie
group OSp(
m n
) onto the
Lie
group O(
m
)
Sp(
n
) corre-
sponding to the even part of
osp
(
m n
).
We observe that
f
GL(
m
) denotes the group of block matrices
e
A
() =
A
()
B
()
B
()
A
()
with
A
() =
A
(
0
) and
B
() =
B
(
1
).
f
SL(
m
) is the subgroup of
f
GL(
m
) consisting of those block matrices
e
A
() satisfying
g
sdet
e
A
= 1. Here the
superdeterminant of the second kind
g
sdet
is dened on the
Grassmann Lie
group
f
GL(
m
) by
(6)
g
sdet
e
A
() = 1 + Trlog(
E
m
+
A
()
;1
B
())
:
118
Boseck
The relation (6) may be rewritten by
Trlog(
E
m
+
A
()
;1
B
()) =
X
1
2
+ 1 Tr(
A
()
;1
B
())
2
+1
= 0
:
Notice that the entries of the product matrix
A
()
;1
B
() belong to
1
which
implies that the series of the logarithm is nite.
The superdeterminant of the second kind is multiplicative. Thus
g
sdet is
a homomorphism of
f
GL(
m
) into the group
K
of units in . The projection
p
0
maps the
Grassmann Lie
group
f
SL(
m
) onto the
Lie
group GL(
m
) which
corresponds to the even part of
e
sl
(
m
).
f
GL(
m
) and
f
SL(
m
) are the -matrix groups corresponding to the
-matrix
Lie
algebras
e
gl
(
m
) and
e
sl
(
m
), respectively.
The center of the -matrix group
f
SL(
m
) consists of the even multiples
of the unit matrix, and the quotient
f
SL(
m
+ 1)
=
f
0
E
2
m
+2
:
0
2
0
g
is a
Grassmann Lie
group, which corresponds to the
Grassmann
-hull of the
Lie
superalgebra
Q
(
m
).
5. H
OPF
Superalgebras
Let
H
=
H
0
H
1
denote a
Z
2
-graded
Hopf
algebra. Here
H
is a
Z
2
-
graded associative algebra with unit|the product and the unit are considered
as linear mappings
:
H
H
!
H
and
:
K
!
H
, respectively|endowed
with a coproduct :
H
!
H
H
, a counit
"
:
H
!
K
, and an antipode
:
H
!
H
. Here and
"
are homomorphisms of the corresponding
Z
2
-graded
algebras,
is an antiautomorphism of
H
. The following relations are satised
(
id
H
)
= (id
H
)
, called the
coassociativity
of the coproduct,
(
"
id
H
)
= id
H
= (id
H
"
)
, and
(
id
H
)
=
"
=
(id
H
)
.
Let
:
H
H
!
H
H
denote the
twist
homomorphism given by
(
h
1
h
2
) =
(
;
1)
j
h
1
jj
h
2
j
h
2
h
1
for homogeneous elements
h
1
and
h
2
in
H
. Then
H
is called
commutative
or
cocommutative
if the relations
=
or
=
hold, respectively.
Example 1 .
Put
H
=
K
X
1
::: X
m
]
(
Y
1
::: Y
n
). Then we have
H
0
=
K
X
1
::: X
m
]
0
(
Y
1
::: Y
n
)
H
1
=
K
X
1
::: X
m
]
1
(
Y
1
::: Y
n
)
:
Now
H
is a
Hopf
algebra with respect to the usual product, the usual
unit , the coproduct (
X
) = 1
X
+
X
1, (
Y
) = 1
Y
+
Y
1,
(1) = 1
1, the counit
"
(
X
) =
"
(
Y
) = 0,
"
(1) = 1, and the antipode
(
X
) =
;
X
,
(
Y
) =
;
Y
,
(1) = 1, for
= 1
::: m
,
= 1
::: n
. The
Hopf
superalgebra
H
is commutative and cocommutative.
Boseck
119
Example 2 .
Let
G
= GL(
m n
) denote the -matrix group dened in
Section 4. We dene a block matrix of
commuting
and
anticommuting variables
by
X
=
X Y
0
Y X
0
:
We assume
X
= (
X
ij
)
i
=1
:::m
j
=1
:::m
Y
0
= (
Y
0
i`
)
i
=1
::: m
`
=1
::: n
Y
= (
Y
kj
)
k
=1
::: n
j
=1
:::m
X
0
= (
X
0
k`
)
k
=1
::: n
`
=1
::: n
:
Put
H
(
G
) =
K
X
ij
x X
0
k`
x
0
]
(
Y
0
i`
Y
kj
)
=
(
x
det
j
X
ij
j
;
1
x
0
det
j
X
0
k`
j
;
1)
then
H
(
G
) is an associative and commutative
Z
2
-graded algebra with unit.
Using the matrix product we dene the coproduct as follows:
(
X
ij
) =
m
X
j
0
=1
X
ij
0
X
j
0
j
+
n
X
`
0
=1
Y
0
i`
0
Y
`
0
`
(
X
0
k`
) =
m
X
j
0
=1
Y
kj
0
Y
0
j
0
`
+
n
X
`
0
=1
X
0
k`
0
X
0
`
0
l
(
Y
0
i`
) =
m
X
j
0
=1
X
ij
0
Y
0
j
0
`
+
n
X
`
0
=1
Y
0
i`
0
X
0
`
0
`
(
Y
kj
) =
m
X
j
0
=1
Y
kj
0
X
j
0
j
+
n
X
`
0
=1
X
0
k`
0
Y
`
0
j
(
x
) =
x
x
(
x
0
) =
x
0
x
0
:
For short we may write (
X
) =
X
X
. The counit is given by
"
(
X
ij
) =
ij
"
(
X
0
k`
) =
kl
"
(
Y
0
i`
) = 0
:
"
(
Y
kj
) = 0
"
(
x
) = 1
"
(
x
0
) = 1
:
For short we may write
"
(
X
) =
E
m
+
n
. The coproduct in
H
(
G
) mirrors the
matrix product, i.e., the product in the group
G
, the counit represents the
evaluation at the unit matrix, i.e., the evaluation at the identity of the group
G
. Now we dene the antipode, which mirrors the inverse of matrices or group
elements
(
X
) = (
X
;
Y
0
X
0;1
Y
)
;1
(
X
0
) = (
X
0
;
Y X
;1
Y
0
)
;1
(
Y
0
) =
;
X
;1
Y
0
(
X
(
Y
) =
;
X
0;1
Y
(
X
0
)
(
x
) = det
X
(
x
0
) = det
X
0
:
120
Boseck
For short we may write
(
X
) =
X
;1
. Now
H
(
G
) is a commutative and not
cocommutative
Hopf
superalgebra, which we shall denote by
P
(
m n
)
:
Example 3 .
Assume
G
= SL(
m n
). Put
H
(
G
) =
P
(
m n
)
=
(sdet
X
;
1)
=
P
(
m n
)
=
(det(
X
;
Y
0
X
0;1
Y
)
;
det
X
0
)
:
It follows from sdet(
X
1
X
2
) = sdet
X
1
sdet
X
2
that the homogeneous ideal gener-
ated by sdet
X
;
1 is a coideal, too. We have
(sdet
X
;
1) = sdet
X
;
1
1
= sdet(
X
X
)
;
1
1
= sdet
X
sdet
X
;
1
1
= sdet
X
sdet
X
;
sdet
X
1 + sdet
X
1
;
1
1
= sdet
X
(sdet
X
;
1) + (sdet
X
;
1)
1
:
Moreover,
"
(sdet
X
) = 1 and
(sdet
X
) = sdet
X
;1
, i.e., the ideal generated
by sdet
X
;
1 is contained in the kernel of
"
and invariant under
. Hence it
is possible to factorize ,
"
,
, and
H
(
G
) becomes a commutative and not
cocommutative
Hopf
superalgebra, which is denoted by
S
P
(
m n
).
A -matrix group, i.e, a subgroup of GL(
m n
) is called
algebraic
, if
it is the annihilator set of a
Z
2
;
graded
ideal of
P
(
m n
).
Let
G
denote an algebraic
-matrix group, and let
I
(
G
)
denote its an-
nihilator ideal in
P
(
m n
)
, then
H
(
G
) =
P
(
m n
)
=I
(
G
)
is a
Hopf
superalgebra.
The coproduct, the counit, and the antipode of
H
(
G
)
are induced by factoriza-
tion of the coproduct, the counit, and the antipode of
P
(
m n
)
. The annihilator
ideal
I
(
G
)
is a coideal of
P
(
m n
)
, it is contained in the kernel of
"
, and it is
invariant under
.
Example 4 .
The -matrix group OSp(
m n
) is an algebraic -matrix
group. Its annihilator ideal is generated by the \polynomials"
X
>
X
;
Y
>
I
n
Y
;
E
m
,
X
>
Y
0
;
Y I
n
X
0
, and
Y
0>
Y
0
+
X
0>
I
n
X
0
;
I
n
. The corresponding
Hopf
superalgebra is denoted by
O S
p
P
(
m n
).
Example 5 .
The -matrix group
f
GL(
m
) is an algebraic -matrix group.
Its
Hopf
superalgebra is denoted by
e
P
(
m
):
e
P
(
m
)
=
K
X
ij
x
]
(
Y
ij
)
=
(
x
det
j
X
ij
j
;
1)
:
Example 6 .
The -matrix group
f
SL(
m
) is an algebraic -matrix group.
Its
Hopf
superalgebra is denoted by
g
S
P
(
m
)
:
A
Hopf
superalgebra
H
=
H
0
H
1
is called
ane
if it is commutative
and nitely generated.
Boseck
121
The
Hopf
superalgebra
H
(
G
) =
P
(
m n
)
=I
(
G
)
of an algebraic
-matrix
group
G
is ane.
The
Hopf
superalgebras
P
(
m n
)
,
S
P
(
m n
)
,
O S
p
P
(
m n
)
,
e
P
(
m
)
, and
g
S
P
(
m
)
are ane
Hopf
superalgebras.
We mention the following
Structure Theorem.
Let
H
denote an ane
Hopf
superalgebra. Then
there exists an ane
Hopf
algebra
H
0
and odd elements
W
1
::: W
s
such that
H
=
H
0
(
W
1
::: W
s
)
. The isomorphism is an isomorphism of commutative
superalgebras. The following relations hold:
(
p
p
)
=
0
p
"
=
"
0
p
p
=
0
p:
Here
p
denotes the canonical projection annihilating all odd elements of
H
, i.e.,
p
:
H
!
H
0
, and
0
"
0
,
0
denote the coproduct, the counit, and the antipode
of the
Hopf
algebra
H
0
, respectively.
The ane
Hopf
algebra
H
0
is the algebra of polynomial functions of an
(ane) algebraic group
G
0
.
Example 2 (continued).
Assume
G
= GL(
m n
),
H
=
H
(
G
) =
P
(
m n
).
Then one has
H
0
=
K
X
ij
x X
0
k`
x
0
]
=
(
x
det
j
X
ij
j
;
1
x
0
det
j
X
0
kl
j
;
1)
and
G
0
= GL(
m
)
GL(
n
) =
p
0
(GL(
m n
))
:
Example 3 (continued).
Assume
G
= SL(
m n
)
H
=
H
(
G
) =
S
P
(
m n
).
Then one has
H
0
=
K
X
ij
x X
0
k`
x
0
]
=
(
x
det
j
X
ij
j
;
1
x
0
det
j
X
0
k`
j
;
1 det
j
X
ij
j
;
det
j
X
0
k`
j
)
and
G
0
= SL(
m
)
SL(
n
)
K
=
p
0
(SL(
m n
)
:
Example 4 (continued).
Assume
G
= OSp(
m n
),
H
=
H
(
G
) =
O S
p
P
(
m n
)
:
Then one has
H
0
=
K
X
ij
X
0
k`
]
=I
0
and the ideal
I
0
is generated by
m
X
i
0
=1
X
i
0
i
X
i
0
j
;
ij
n
0
;1
X
k
0
=0
(
X
0
n
;
k
0
k
X
0
k
0
+1
`
;
X
0
n
0
;
k
0
k
X
0
n
0
+
k
0
+1
`
;
k`
) (
n
= 2
n
0
)
:
This implies
G
0
= O(
m
)
Sp(
n
) =
p
0
(OSp(
m n
)).
122
Boseck
Example 5 (continued).
Assume
G
=
f
GL(
m
),
H
=
H
(
G
) =
e
P
(
m
).
Then one has
H
0
=
K
X
ij
x
]
=
(
x
det
j
X
ij
j
;
1)
and
G
0
= GL(
m
) =
p
0
(
f
GL(
m
))
:
Example 6 (continued).
Assume
G
=
f
SL(
m
),
H
=
H
(
G
) =
g
S
P
(
m
).
Then one has
H
0
=
K
X
ij
x
]
=
(
x
det
j
X
ij
j
;
1)
and
G
0
= GL(
m
) =
p
0
(
f
SL(
m
))
:
Notice that the isomorphism of superalgebras stated in the structure
theorem is not a canonical one. In some sense it is the choice of a coordinate
system.
Example
6
(continued once more).
We have
g
S
P
(
m
)
=
K
X
ij
x
]
(
Y
ij
)
=I
(
f
SL(
m
))
:
The annihilator ideal
I
(
f
SL(
m
)) is generated by
x
det
j
X
ij
j ;
1 and Trlog(
E
m
+
X
;1
Y
) =
P
1
2
+1
Tr(
X
;1
Y
)
2
+1
. Choosing instead of the
Y
ij
new odd vari-
ables
W
ij
dened by the matrix equation
W
=
X
;1
Y
, then using the second
relation it is possible to eliminate one of the odd variables. In this case we have
s
=
m
2
;
1.
The last two sections, namely, Section 6, \Ane algebraic Supergroups"
and Section 7, \The
Hopf
dual. Representations", as well as the list of references
are postponed to the next seminar.
References
References to the literature will be given in the sequel to this article:
1]
Boseck, H.,
Lie superalgebras and Lie supergroups, II
, Seminar Sophus
Lie (Heldermann Verlag Berlin)
2
(1992), to appear.
Fachrichtungen Mathematik/Informatik
Ernst-Moritz-Arndt-UniversitatGreifswald
Friedrich Ludwig Jahn-Strae 15a
O-2200 Greifswald, Germany
Received July 31, 1991
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