Seminar Sophus Lie 1 (1991) 109–122

Lie Superalgebras and Lie Supergroups, I

Helmut Boseck

1. Graded Algebras

Let K denote the base field. K is assumed to be of characteristic zero. In the examples K is the field 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 = ⨁_{k∈ℤ} V_{k}.

A ℤ-graded associative algebra is an associative K -algebra A , which is ℤ-graded as a K -linear space

A = ⨁_{k∈ℤ} A_{k},

such that

A_{k}A_{l} ⊆ A_{k+ℓ}, (k, l ∈ ℤ).

The elements of A_{k} are called homogeneous of degree k.

Example 1^{⋄}. Let V = ⨁_{k∈ℤ} V_{k} be a ℤ-graded linear space. A linear operator A: V_{k} ⟶ V_{k+l} (k ∈ ℤ) is called homogeneous of degree ℓ. If L_{ℓ}(V) denotes the space of linear operators of degree ℓ , then

L′(V) = ⨁_{k∈ℤ} L_{k}(V)

is a ℤ -graded associative algebra with unit. If V is finite dimensional we have

L(V) = ⨁_{k∈ℤ} L_{k}(V).

Assume V to be finite dimensional and V_{k} = {0}, (k = −1, ±2, ±3, …), V = V_{0} ⊕ V_{1}.

Then the following relations hold

\begin{aligned}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) & = \{0\}, (k = ±2, ±3, …). \end{aligned}

Chosing a basis of homogeneous elements in V we may represent the elements of L_{0}(V) by diagonal block matrices

(1)  ( \begin{matrix}A_{0} & 0 \\ 0 & A_{1} \end{matrix} ) .

The elements of L_{−1}(V) and L_{1}(V) become represented by the block matrices

(2)  ( \begin{matrix}0 & B_{1} \\ 0 & 0 \end{matrix} ) and ( \begin{matrix}0 & 0 \\ B_{0} & 0 \end{matrix} ) , respectively.

Example 2^{⋄}. Let Λ(n) = Λ(y_{1}, …, y_{n}) denote the exterior or Grassmann algebra with n generators, which are assumed to be homogeneous of degree 1 . Put

Λ_{0} = K,
Λ_{1} = span{y_{1}, …, y_{n}},
Λ_{k} = span{y_{i_{1}} ⋯ y_{i_{k}} : 1 ≤ i_{1} < ⋯ < i_{k} ≤ n}, (k = 2, …, n − 1),
Λ_{n} = span{y_{1} ⋯ y_{n}},
Λ_{ℓ} = {0}, if ℓ = −1, −2, … or ℓ = n + 1, n + 2, …,

then we have

Λ(n) = Λ_{0} ⊕ Λ_{1} ⊕ ⋯ ⊕ Λ_{n},

and Λ(n) becomes a ℤ-graded associative algebra with unit. The ℤ-graded algebra Λ(n) is (graded) commutative, i.e., for homogeneous elements a_{k} ∈ Λ_{k}, a_{ℓ} ∈ Λ_{ℓ} the following equation holds:

a_{ℓ}a_{k} = (−1)^{kℓ}a_{k}a_{ℓ}.

Example 3^{⋄}. Let A = ⨁_{k∈ℤ} A_{k} be a ℤ-graded associative algebra with unit. We define a new product by

[a_{k}, a_{ℓ}] = a_{k}a_{ℓ} − (−1)^{kℓ}a_{ℓ}a_{k}, a_{k} ∈ A_{k}.

With respect to that multiplication A becomes a ℤ-graded Lie algebra

A_{L} = ⨁_{k∈ℤ} A_{k}.

We have

(3)  [a_{ℓ}, a_{k}] = −(−1)^{kℓ}[a_{k}, a_{ℓ}],

and the modified Jacobi identity

(4) (−1)^{km}[a_{k}, [a_{ℓ}, a_{m}]] + (−1)^{ℓk}[a_{ℓ}[a_{m}, a_{k}]] + (−1)^{mℓ}[a_{m}[a_{k}, a_{ℓ}]] = 0.

A ℤ-graded Lie algebra is a ℤ -graded linear space L = ⨁_{k∈ℤ} L_{k} with a bracket multiplication compatible with the grading [L_{k}, L_{ℓ}] ⊆ L_{k+ℓ} and satisfying (3) and (4) for homogeneous elements.

Let ℤ_{2} = ℤ/(2) = \{\overline{0}, \overline{1}\} denote the additive group of two elements. A ℤ_{2} -graded linear space is a K -linear space V with two distinguished subspaces V\overline{0} and V\overline{1} such that V = V\overline{0} ⊕ V\overline{1} holds. The elements of V\overline{0} are called even , those of V\overline{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\overline{0} = ⨁_{k∈ℤ} V_{2k}, V\overline{1} = ⨁_{k∈ℤ} V_{2k+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\overline{0} ⊕ A\overline{1},

such that the multiplication satisfies

\begin{aligned}A\overline{0}A\overline{0} & ⊆ A\overline{0}, A\overline{1}A\overline{1} ⊆ A\overline{0}, \\ A\overline{0}A\overline{1} & ⊆ A\overline{1}, A\overline{1}A\overline{0} ⊆ A\overline{1}. \end{aligned}

Every ℤ-graded algebra A admits a canonical Z_{2} -gradation by

A\overline{0} = ⨁_{k∈ℤ} A_{2k}, A\overline{1} = ⨁_{k∈ℤ} A_{2k+1}.

Example 4^{⋄}. Let V = V\overline{0} ⊕ V\overline{1} denote a ℤ_{2} -graded linear space. A linear operator A on V is called paritiy preserving or even, if it satisfies A: V\overline{0} → V\overline{0} and A: V\overline{1} → V\overline{1} . The operator A is called parity reversing or odd if A: V\overline{0} → V\overline{1} and A: V\overline{1} → V\overline{0} . By L\overline{0}(V) and L\overline{1}(V) we denote the linear space of even and odd linear operators on V , respectively.

L′(V) = L\overline{0}(V) ⊕ L\overline{1}(V)

is a ℤ_{2} -graded associative algebra with unit. If V is finite dimensional then

L(V) = L\overline{0}(V) ⊕ L\overline{1}(V)

holds and

\begin{matrix}L\overline{0}(V) ≅ L(V\overline{0}) ⊕ L(V\overline{1}), \\ L\overline{1} & ≅ L(V\overline{0}, V\overline{1}) ⊕ L(V\overline{1}, V\overline{0}). \end{matrix}

Choosing a basis of homogeneous elements in V , the elements of L\overline{0}(V) are represented by diagonal block matrices as in (1), while the elements of L\overline{1}(V) are represented by block matrices of the following type

( \begin{matrix}0 & B_{1} \\ B_{0} & 0 \end{matrix} ) .

Example 5^{⋄}. The Grassmann algebra Λ(n) admits a canonical ℤ_{2} -grading with respect to a chosen system of generators Λ(n) = Λ\overline{0} ⊕ Λ\overline{1} with Λ\overline{0} = Λ_{0} ⊕ Λ_{2} ⊕ ⋯, Λ\overline{1} = Λ_{1} ⊕ Λ_{3} ⊕ ⋯ Now Λ(n) is a ℤ_{2} -graded associative and (graded) commutative algebra with unit or, equivalently, an associative, commutative superalgebra with unit.

Example 6^{⋄}. Let A = A\overline{0} ⊕ A\overline{1} denote a ℤ_{2} -graded associative algebra with unit. We define brackets by [a, b] = ab − (−1)^{|a||b|}ba for homogeneous elements a, b ∈ A. Define the parity |a| as follows:

|a| = \{ \begin{matrix}0 & if a ∈ A\overline{0}, \\ 1 & if a ∈ A\overline{1}. \end{matrix}

With respect to the brackets, A becomes a ℤ_{2} -graded Lie algebra or,equivalently, a Lie superalgebra A_{L} = A\overline{0} ⊕ A\overline{1}. For homogeneous elements a, b, c ∈ A the following equations hold:

(4)  [b, a] = −(−1)^{|a||b|}[a, b],

(5)  (−1)^{|a||c|}[a, [b, c]] + (−1)^{|b||a|}[b, [c, a]] + (−1)^{|c||b|}[c, [a, b]] = 0

(the modified Jacobi equation). A ℤ_{2} -graded Lie algebra or Lie superalgebra is a ℤ_{2} -graded linear space L = L\overline{0} ⊕ L\overline{1} with a bracket multiplication [·, ·] compatible with the gradation

\begin{aligned}[L\overline{0}, L\overline{0}] & ⊆ L\overline{0}, [L\overline{1}, L\overline{1}] ⊆ L\overline{0}, \\ [L\overline{0}, L\overline{1}] & ⊆ L\overline{1}, [L\overline{1}, L\overline{0}] ⊆ L\overline{1} \end{aligned}

and satisfying (4) and (5).

2. LIE Superalgebras: The series A, B, C, D, Q

Let L = L\overline{0} ⊕ L\overline{1} denote a ℤ_{2} -graded Lie algebra. Its even part L\overline{0} is a Lie algebra. In view of [L\overline{0}, L\overline{1}] ⊆ L\overline{1} , multiplication of odd elements by even ones defines a representation ad\overline{0} of the Lie algebra L\overline{0} on the linear space L\overline{1}

(ad\overline{0} x\overline{0})x\overline{1} = [x\overline{0}, x\overline{1}] x\overline{0} ∈ L\overline{0}, x\overline{1} ∈ L\overline{1}.

ad\overline{0} is called the adjoint representation of the even part L\overline{0} on the odd part L_{1}.

A ℤ_{2} -graded Lie algebra L = L\overline{0} ⊕ L\overline{1} is called simple , if there are no nontrivial ℤ_{2} -graded ideals: If I = I\overline{0} ⊕ I\overline{1} denotes a ℤ_{2} -graded ideal of L then we have I = {0} or I = L. A simple ℤ_{2} -graded Lie algebra is called classical, if the representation ad\overline{0} is completely reducible. A simple ℤ_{2} -graded Lie algebra is classical iff its even part L\overline{0} is a reductive Lie algebra.

Denote by Mat(m, n) the ℤ_{2} -graded associative algebra with unit consisting of (m + n) × (m + n) block matrices with entries from K

( \begin{matrix}A_{0} & B_{1} \\ B_{0} & A_{1} \end{matrix} ) .

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 defined 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

\begin{aligned}𝔤𝔩\overline{0}(m, n) & = 𝔤𝔩_{0}(m, n) ≅ 𝔤𝔩(m) × 𝔤𝔩(n), \\ 𝔤𝔩\overline{1}(m, n) & = 𝔤𝔩_{−1}(m, n) ⊕ 𝔤𝔩_{1}(m, n). \end{aligned}

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\overline{0}(m, n) ≅ 𝔰𝔩(m) × 𝔰𝔩(n) × K.

Example 2^{•}. 𝔬𝔰𝔭(m, n) consists of those block matrices satisfying the following relations

\begin{aligned}A_{0}^{⊤} + A_{0} & = 0, \\ B_{1}^{⊤} − I_{n}B_{0} & = 0, \\ A_{1}^{⊤}I_{n} + I_{n}A_{1} & = 0. \end{aligned}

Here A^{⊤} denotes the transpose of A, and n is assumed to be even n = 2ℓ, and

I_{n} = ( \begin{matrix}0 & E_{ℓ} \\ −E_{ℓ} & 0 \end{matrix} ) ,

where E_{ℓ} denotes the ℓ × ℓ unit matrix. The matrices of 𝔬𝔰𝔭(m, 2ℓ) may be written in the following form

( \begin{matrix}A_{0} & B_{1} & B_{2} \\ −B_{2}^{⊤} & A_{1} & A_{12} \\ B_{1}^{⊤} & A_{21} & −A_{1}^{⊤} \end{matrix} ) 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 orthogonalsymplectic Lie superalgebra. For the even part one has

𝔬𝔰𝔭\overline{0}(m, n) ≅ 𝔬(m) × 𝔰𝔭(n).

Example 3^{•}. 𝔤̃𝔩(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. 𝔤̃𝔩(m) is a ℤ_{2} -graded Lie algebra and we have 𝔤̃𝔩\overline{0}(m) ≅ 𝔤𝔩(m).

Example 4^{•}. 𝔰̃𝔩(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. 𝔰̃𝔩(m) is a ℤ_{2} -graded Lie algebra, a subalgebra of 𝔤̃𝔩(m), and the following relation holds: 𝔰̃𝔩\overline{0}(m) = 𝔤̃𝔩\overline{0}(m) ≅ 𝔤𝔩(m).

The Lie superalgebra 𝔰𝔩(m, n) is simple if m ≠ n, m ≥ 1, n ≥ 1.

The multiples of the unit matrix {αE_{2m}; α ∈ K} make up a ℤ_{2} -graded ideal of 𝔰𝔩(m, m), namely, the center.

The quotient algebra 𝔰𝔩(m, m)/{αE_{2m} : α ∈ K} is simple if m > 1.

The A-series of simple Lie superalgebras is defined in analogy with the usual classification of simple Lie algebras by

\begin{aligned}A(m, n) & = 𝔰𝔩(m + 1, n + 1), m ≠ n, m ≥ 0, n ≥ 0. \\ A(m, m) & = 𝔰𝔩(m + 1, m + 1)/\{αE_{2m+2} : α ∈ K\}, m > 0. \end{aligned}

The orthogonal-symplectic Lie superalgebra 𝔬𝔰𝔭(m, n) is simple if m ≥ 1, n > 1.

The series B, C, D are defined as follows

\begin{aligned}B(m, n) & = 𝔬𝔰𝔭(2m + 1, 2n), m ≥ 0, n > 0. \\ C(n) & = 𝔬𝔰𝔭(2, 2n − 2), n ≥ 2. \\ D(m, n) & = 𝔬𝔰𝔭(2m, 2n), m ≥ 2, n > 0. \end{aligned}

As in 𝔰𝔩(m, m), the multiples of unity {αE_{2m}; α ∈ K} make up a homogeneous ideal in 𝔰̃𝔩(m), namely, its center.

The quotient algebra 𝔰̃𝔩(m)/{αE_{2m} : α ∈ K} is simple if m ≥ 3.

The Q-series is defined by

Q(m) = 𝔰̃𝔩(m + 1)/{αE_{2m+2} : α ∈ K}, 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\overline{0}(m, n) ≅ A(m) × A(n) × K, m ≠ n
A\overline{0}(m, m) ≅ A(m) × A(m)
B\overline{0}(m, n) ≅ B(m) × C(n)
C\overline{0}(m) ≅ C(m − 1) × K
D\overline{0}(m, n) ≅ D(m) × C(n)
Q\overline{0}(m) ≅ A(m).

3. The GRASSMANN-hull

The Grassmann-hull is a construction, which enables us to make a ℤ_{2} - graded Lie algebra into a Lie algebra. Let Λ denote a finitely generated Grassmann algebra and let L be a ℤ_{2} -graded Lie algebra. Taking the tensorproduct Λ ⊗ L of ℤ_{2} -graded algebras we have

\begin{aligned}(Λ ⊗ L)\overline{0} & = Λ\overline{0} ⊗ L\overline{0} + Λ\overline{1} ⊗ L\overline{1}, \\ (Λ ⊗ L)\overline{1} & = Λ\overline{0} ⊗ L\overline{1} + Λ\overline{1} ⊗ L\overline{0}. \end{aligned}

Writing the elements of Λ ⊗ L for simplicity as λx, the brackets are defined for homogeneous elements as follows: [λ_{α}x_{β}, λ_{γ}x_{δ}] = (−1)^{βγ}λ_{α}λ_{γ}[x_{β}, x_{δ}], α, β, γ, δ ∈ ℤ_{2} . If α = β and γ = δ, then we have

[λ_{γ}x_{δ}, λ_{α}x_{β}] = (−1)^{αδ}λ_{γ}λ_{α}[x_{δ}, x_{β}]
= (−1)^{αδ+αγ+βδ}(−1)λ_{α}λ_{γ}[x_{β}, x_{δ}]
= −(−1)^{αδ+αγ+βδ+βγ}[λ_{α}x_{β}, λ_{γ}x_{δ}]

Since αδ + βδ = αγ + βγ = \overline{0} we have [λ_{γ}x_{δ}, λ_{α}x_{β}] = −[λ_{α}x_{β}, λ_{γ}x_{δ}]. The even part (Λ ⊗ L)\overline{0} of the tensor product Λ ⊗ 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), 𝔤̃𝔩(m), or 𝔰̃𝔩(m). The Grassmann-hull consists of block matrices

(5)  ( \begin{matrix}A_{0}(Λ\overline{0}) & B_{1}(Λ\overline{1}) \\ B_{0}(Λ\overline{1}) & A_{1}(Λ\overline{0}) \end{matrix} ) .

The entries of the diagonal blocks A_{0} and A_{1} belong to Λ\overline{0} , while the entries of the matrices B_{0} and B_{1} are from Λ\overline{1} . We denote the Grassmann-hulls by 𝔤𝔩(m, n; Λ), 𝔰𝔩(m, n; Λ), 𝔬𝔰𝔭(m, n; Λ), 𝔤̃𝔩(m; Λ), 𝔰̃𝔩(m; Λ), respectively. Note that 𝔰𝔩(m, n; Λ) is the Lie algebra of block matrices of type (5) such that Tr(A_{0}(Λ\overline{0})) = Tr(A_{1}(Λ\overline{0})). Further, 𝔬𝔰𝔭(m, 2l; Λ) is the Lie algebra of block matrices

A_{0}(Λ\overline{0}) B_{1}(Λ\overline{1}) B_{2}(Λ\overline{1})
−B_{2}^{⊤}(Λ\overline{1}) A_{1}(Λ\overline{0}) A_{12}(Λ\overline{0})
B_{1}^{⊤}(Λ\overline{0}) A_{21}(Λ\overline{0}) −A_{1}^{⊤}(Λ\overline{0})

satisfying the relations A_{0}^{⊤}(Λ\overline{0}) = A_{0}(Λ\overline{0}), A_{12}^{⊤}(Λ\overline{0}) = A_{12}(Λ\overline{0}), A_{21}^{⊤}(Λ\overline{0}) = A_{21}(Λ\overline{0}). Finally, 𝔰̃𝔩(m; Λ) consists of block matrices

( \begin{matrix}A(Λ\overline{0}) & B(Λ\overline{1}) \\ B(Λ\overline{1}) & A(Λ\overline{0}) \end{matrix} )

satisfying the relation Tr B(Λ\overline{1}) = 0.

In all of our considerations Λ denotes an arbitrary finitely 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}: Λ → Λ_{0} = K denote the canonical projection of the ℤ-graded algebra Λ onto its zero component. Then p_{0} defines a canonical projection of the Grassmann-hull (Λ ⊗ L)\overline{0} of a Lie superalgebra L onto its even part L\overline{0} . We denote it once more by p_{0} , so that p_{0}(λx) = p_{0}(λ)x. With respect to the matrix Lie algebra 𝔤𝔩(m, n; Λ) and its subalgebras we have

p_{0} ( \begin{matrix}A_{0}(Λ\overline{0}) & B_{1}(Λ\overline{1}) \\ B_{0}(Λ\overline{1}) & A_{1}(Λ\overline{0}) \end{matrix} ) = ( \begin{matrix}A_{0} & 0 \\ 0 & A_{1} \end{matrix} )

with entries from K in the diagonal blocks A_{0} and A_{1} .

In the case of 𝔬𝔰𝔭(m, n; Λ) 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 define 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; Λ) = (Λ⊗Mat(m, n))\overline{0} , the elements are the block matrices (5).

4. GRASSMANN LIE groups

The series GL, SL, OSp, G̃L, S̃L. We define 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; Λ).

A block matrix of type (5) is invertible iff the matrices A_{0} = p_{0}(A_{0}(Λ\overline{0})) and A_{1} = p_{0}(A_{1}(Λ\overline{0})) are invertible.

It follows that the diagonal blocks A_{0}(Λ\overline{0}) and A_{1}(Λ\overline{0}) are invertible for invertible block matrices. The inverse of a block matrix is written

( \begin{matrix}A_{0}^{(−1)}(Λ\overline{0}) & B_{1}^{(−1)}(Λ\overline{1}) \\ B_{0}^{(−1)}(Λ\overline{1}) & A_{1}^{(−1)}(Λ\overline{0}) \end{matrix} )

satisfying the equations

A_{0}^{(−1)}(Λ\overline{0}) = (A_{0}(Λ\overline{0}) − B_{1}(Λ\overline{1})A_{1}(Λ\overline{0})^{−1}B_{0}(Λ\overline{1}))^{−1},
A_{1}^{(−1)}(Λ\overline{0}) = (A_{1}(Λ\overline{0}) − B_{0}(Λ\overline{1})A_{0}(Λ\overline{0})^{−1}B_{1}(Λ\overline{1}))^{−1},
B_{1}^{(−1)}(Λ\overline{1}) = −A_{0}(Λ\overline{0})^{−1}B_{1}(Λ\overline{1})(A_{1}(Λ\overline{0}) − B_{0}(Λ\overline{1})A_{0}(Λ\overline{0})^{−1}B_{1}(Λ\overline{1}))^{−1},
B_{0}^{(−1)}(Λ\overline{1}) = −A_{1}(Λ\overline{0})^{−1}B_{0}(Λ\overline{1})(A_{0}(Λ\overline{0}) − B_{1}(Λ\overline{1})A_{1}(Λ\overline{0})^{−1}B_{0}(Λ\overline{1}))^{−1}.

Let GL(m, n; Λ) denote the group of units in Mat(m, n; Λ). It consists of the invertible block matrices of type (5) which for simplicity are written

A(Λ) = ( \begin{matrix}A_{0}(Λ) & B_{1}(Λ) \\ B_{0}(Λ) & A_{1}(Λ) \end{matrix} ) .

The superdeterminant or Berezinian of A is defined by sdet A(Λ) = det (A_{0}(Λ)−B_{1}(Λ)A_{1}(Λ)^{−1}B_{0}(Λ)) det A_{1}^{−1}(Λ). The superdeterminant is defined on the Grassmann Lie group GL(m, n; Λ). It is multiplicative, i.e., sdet is a homomorphism of GL(m, n; Λ) into the group of units K^{×} of Λ.

Also, SL(m, n; Λ) is the subgroup of GL(m, n; Λ) defined by

sdet A(Λ) = 1,

or, equivalently, by

det(A_{0}(Λ) − B_{1}(Λ)A_{1}(Λ)^{−1}B_{0}(Λ)) = det A_{1}(Λ).

Using the projection p_{0}: Λ → K we get

det A_{0} = p_{0}(det A_{0}(Λ)) = p_{0}(det A_{1}(Λ)) = detA_{1}.

We observe that GL(m, n; Λ) and SL(m, n; Λ) are the Λ-matrix groups corresponding to the Λ-matrix Lie algebras 𝔤𝔩(m, n; Λ) and 𝔰𝔩(m, n; Λ), respectively.

OSp(m, n; Λ) is the subgroup of GL(m, n; Λ) defined by the following relations

A_{0}(Λ)^{⊤}A_{0}(Λ) − B_{0}(Λ)^{⊤}I_{n}B_{0}(Λ) = E_{m},
A_{0}(Λ)^{⊤}B_{1}(Λ) − B_{0}(Λ)^{⊤}I_{n}A_{1}(Λ) = 0,
B_{1}(Λ)^{⊤}B_{1}(Λ) + A_{1}(Λ)^{⊤}I_{n}A_{1}(Λ) = 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; Λ) → O(m) × Sp(n).

OSp(m, n; Λ) is the Λ-matrix group corresponding to the Λ-matrix Lie algebra 𝔬𝔰𝔭(m, n; Λ). The projection p_{0} is a homomorphism mapping the Grassmann Lie group OSp(m, n; Λ) onto the Lie group O(m) × Sp(n) corresponding to the even part of 𝔬𝔰𝔭(m, n).

We observe that G̃L(m; Λ) denotes the group of block matrices

Ã(Λ) = ( \begin{matrix}A(Λ) & B(Λ) \\ B(Λ) & A(Λ) \end{matrix} ) ,

with A(Λ) = A(Λ\overline{0}) and B(Λ) = B(Λ\overline{1}).

S̃L(m; Λ) is the subgroup of G̃L(m; Λ) consisting of those block matrices Ã(Λ) satisfying s̃detÃ = 1. Here the superdeterminant of the second kind s̃det is defined on the Grassmann Lie group G̃L(m; Λ) by

(6)  s̃detÃ(Λ) = 1 + Tr log(E_{m} + A(Λ)^{−1}B(Λ)).

The relation (6) may be rewritten by

Tr log(E_{m} + A(Λ)^{−1}B(Λ)) = ∑_{ν} \frac{1}{2ν + 1} Tr(A(Λ)^{−1}B(Λ))^{2ν+1} = 0.

Notice that the entries of the product matrix A(Λ)^{−1}B(Λ) belong to Λ\overline{1} which implies that the series of the logarithm is finite.

The superdeterminant of the second kind is multiplicative. Thus s̃det is a homomorphism of G̃L(m; Λ) into the group K^{×} of units in Λ. The projection p_{0} maps the Grassmann Lie group S̃L(m; Λ) onto the Lie group GL(m) which corresponds to the even part of 𝔰̃𝔩(m).

G̃L(m; Λ) and S̃L(m; Λ) are the Λ-matrix groups corresponding to the Λ-matrix Lie algebras 𝔤̃𝔩(m; Λ) and 𝔰̃𝔩(m; Λ), respectively.

The center of the Λ-matrix group S̃L(m; Λ) consists of the even multiples of the unit matrix, and the quotient

S̃L(m + 1; Λ)/\{λ\overline{0}E_{2m+2} : λ\overline{0} ∈ Λ\overline{0}\}

is a Grassmann Lie group, which corresponds to the Grassmann -hull of the Lie superalgebra Q(m).

5. HOPF Superalgebras

Let H = H\overline{0} ⊕ H\overline{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 μ: H ⊗ H → H and ι: K → H, respectively—endowed with a coproduct Δ : H → H ⊗ H, a counit ε: H → K, and an antipode σ: H → H. Here Δ and ε are homomorphisms of the corresponding ℤ_{2} -graded algebras, σ is an antiautomorphism of H. The following relations are satisfied (Δ ⊗ id_{H})∘Δ = (id_{H} ⊗ Δ) ∘ Δ, called the coassociativity of the coproduct, (ε⊗id_{H})∘Δ = id_{H} = (id_{H} ⊗ε)∘Δ, and μ∘(σ ⊗id_{H})∘Δ = ιε = μ∘(id_{H} ⊗σ)∘Δ. Let ν: H ⊗ H → H ⊗ H denote the twist homomorphism given by ν(h_{1} ⊗ h_{2}) = (−1)^{|h_{1}||h_{2}|}h_{2}h_{1} for homogeneous elements h_{1} and h_{2} in H. Then H is called commutative or cocommutative if the relations

μ ∘ ν = μ or ν ∘ Δ = Δ

hold, respectively.

Example 1 . Put H = K[X_{1}, ..., X_{m}] ⊗ Λ(Y_{1}, ..., Y_{n}). Then we have

\begin{aligned}H\overline{0} & = K[X_{1}, ..., X_{m}] ⊗ Λ\overline{0}(Y_{1}, ..., Y_{n}), \\ H\overline{1} & = K[X_{1}, ..., X_{m}] ⊗ Λ\overline{1}(Y_{1}, ..., Y_{n}). \end{aligned}

Now H is a Hopf algebra with respect to the usual product, the usual unit , the coproduct Δ(X_{μ}) = 1 ⊗ X_{μ} + X_{μ} ⊗ 1, Δ(Y_{ν}) = 1 ⊗ Y_{ν} + Y_{ν} ⊗ 1, Δ(1) = 1 ⊗ 1, the counit ε(X_{μ}) = ε(Y_{ν}) = 0, ε(1) = 1, and the antipode σ(X_{μ}) = −X_{μ}, σ(Y_{ν}) = −Y_{ν}, σ(1) = 1, for μ = 1, …, m, ν = 1, …, n. The Hopf superalgebra H is commutative and cocommutative.

Example 2 . Let G = GL(m, n; Λ) denote the Λ-matrix group defined in Section 4. We define a block matrix of commuting and anticommuting variables by

X = ( \begin{matrix}X & Y′ \\ Y & X′ \end{matrix} ) .

We assume

\begin{aligned}X & = (X_{ij})_{{i=1,...m \atop j=1,...m}} , Y′ = (Y′_{iℓ})_{{i=1,...,m \atop ℓ=1,...,n}} , \\ Y & = (Y_{kj})_{{k=1,...,n \atop j=1,...m}} , X′ = (X′_{kℓ})_{{k=1,...,n \atop ℓ=1,...,n}} . \end{aligned}

Put

H(G) = K[X_{ij}, x, X′_{kℓ}, x′] ⊗ Λ(Y′_{iℓ}, Y_{kj})/(x det |X_{ij}| − 1, x′ det |X′_{kℓ}| − 1),

then H(G) is an associative and commutative ℤ_{2} -graded algebra with unit. Using the matrix product we define the coproduct as follows:

\begin{aligned}Δ(X_{ij}) & = ∑_{j′=1}^{m} X_{ij′} ⊗ X_{j′j} + ∑_{ℓ′=1}^{n} Y′_{iℓ′} ⊗ Y_{ℓ′ℓ}, \\ Δ(X′_{kℓ}) & = ∑_{j′=1}^{m} Y_{kj′} ⊗ Y′_{j′ℓ} + ∑_{ℓ′=1}^{n} X′_{kℓ′} ⊗ X′_{ℓ′l}, \\ Δ(Y′_{iℓ}) & = ∑_{j′=1}^{m} X_{ij′} ⊗ Y′_{j′ℓ} + ∑_{ℓ′=1}^{n} Y′_{iℓ′} ⊗ X′_{ℓ′ℓ}, \\ Δ(Y_{kj}) & = ∑_{j′=1}^{m} Y_{kj′} ⊗ X_{j′j} + ∑_{ℓ′=1}^{n} X′_{kℓ′} ⊗ Y_{ℓ′j}, \\ Δ(x) & = x ⊗ x, \\ Δ(x′) & = x′ ⊗ x′. \end{aligned}

For short we may write Δ(X) = X ⊗ X. The counit is given by

\begin{aligned}ε(X_{ij}) & = δ_{ij}, \\ ε(X′_{kℓ}) & = δ_{kl}, \\ ε(Y′_{iℓ}) & = 0. \\ ε(Y_{kj}) & = 0, \\ ε(x) & = 1, \\ ε(x′) & = 1. \end{aligned}

For short we may write ε(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 define the antipode, which mirrors the inverse of matrices or group elements

\begin{aligned}σ(X) & = (X − Y′X^{′−1}Y)^{−1}, \\ σ(X′) & = (X′ − Y X^{−1}Y′)^{−1}, \\ σ(Y′) & = −X^{−1}Y′σ(X, σ(Y) = −X^{′−1}Y σ(X′), \\ σ(x) & = det X, \\ σ(x′) & = det X′. \end{aligned}

For short we may write σ(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; Λ). Put

H(G) = P(m, n)/(sdet X − 1)
= P(m, n)/(det(X − Y′X^{′−1}Y) − det X′).

It follows from sdet(X_{1}X_{2}) = sdet X_{1} sdet X_{2} that the homogeneous ideal generated by sdet X − 1 is a coideal, too. We have

Δ(sdet X − 1) = Δ 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, ε(sdet X) = 1 and σ(sdet X) = sdet X^{−1} , i.e., the ideal generated by sdet X − 1 is contained in the kernel of ε and invariant under σ. Hence it is possible to factorize Δ, ε, σ, and H(G) becomes a commutative and not cocommutative Hopf superalgebra, which is denoted by SP(m, n).

A Λ-matrix group, i.e, a subgroup of GL(m, n; Λ) is called algebraic, if it is the annihilator set of a ℤ_{2} − graded ideal of P(m, n).

Let G denote an algebraic Λ-matrix group, and let I(G) denote its annihilator 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 factorization 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 ε, and it is invariant under σ.

Example 4 . The Λ-matrix group OSp(m, n; Λ) is an algebraic Λ-matrix group. Its annihilator ideal is generated by the “polynomials” X^{⊤}X − Y^{⊤}I_{n}Y − E_{m}, X^{⊤}Y′ − Y I_{n}X′, and Y^{′⊤}Y′ + X^{′⊤}I_{n}X′ − I_{n}. The corresponding Hopf superalgebra is denoted by OSpP(m, n).

Example 5 . The Λ-matrix group G̃L(m; Λ) is an algebraic Λ-matrix group. Its Hopf superalgebra is denoted by P̃(m):

P̃(m) ≅ K[X_{ij}, x] ⊗ Λ(Y_{ij})/(x det |X_{ij}| − 1).

Example 6 . The Λ-matrix group S̃L(m; Λ) is an algebraic Λ-matrix group. Its Hopf superalgebra is denoted by S̃P(m).

A Hopf superalgebra H = H\overline{0} ⊕ H\overline{1} is called affine if it is commutative and finitely generated.

The Hopf superalgebra H(G) = P(m, n)/I(G) of an algebraic Λ-matrix group G is affine.

The Hopf superalgebras P(m, n), SP(m, n), OSpP(m, n), P̃(m), and S̃P(m) are affine Hopf superalgebras.

We mention the following Structure Theorem. Let H denote an affine Hopf superalgebra. Then there exists an affine Hopf algebra H_{0} and odd elements W_{1}, ..., W_{s} such that H ≅ H_{0} ⊗ Λ(W_{1}, …, W_{s}). The isomorphism is an isomorphism of commutative superalgebras. The following relations hold:

\begin{aligned}(p ⊗ p) ∘ Δ & = Δ_{0} ∘ p, \\ ε & = ε_{0} ∘ p, \\ p ∘ σ & = σ_{0} ∘ p. \end{aligned}

Here p denotes the canonical projection annihilating all odd elements of H, i.e., p: H → H_{0} , and Δ_{0} ε_{0} , σ_{0} denote the coproduct, the counit, and the antipode of the Hopf algebra H_{0} , respectively.

The affine Hopf algebra H_{0} is the algebra of polynomial functions of an (affine) algebraic group G_{0} .

Example 2 (continued). Assume G = GL(m, n; Λ), H = H(G) = P(m, n). Then one has

H_{0} ≅ K[X_{ij}, x, X′_{kℓ}, x′]/(x det |X_{ij}| − 1, x′ det |X′_{kl}| − 1),

and

G_{0} ≅ GL(m) × GL(n) = p_{0}(GL(m, n; Λ)).

Example 3 (continued). Assume G = SL(m, n; Λ) H = H(G) = SP(m, n). Then one has

H_{0} ≅ K[X_{ij}, x, X′_{kℓ}, x′]/(x det |X_{ij}| − 1, x′ det |X′_{kℓ}| − 1, det |X_{ij}| − det |X′_{kℓ}|),

and

G_{0} ≅ SL(m) × SL(n) × K^{×} = p_{0}(SL(m, n; Λ).

Example 4 (continued). Assume G = OSp(m, n; Λ),

H = H(G) = OSpP(m, n).

Then one has

H_{0} ≅ K[X_{ij}, X′_{kℓ},]/I_{0},

and the ideal I_{0} is generated by

\begin{matrix}∑_{i′=1}^{m} X_{i′i}X_{i′j} − δ_{ij}, \\ ∑_{k′=0}^{n′−1}(X′_{n−k′,k}X′_{k′+1,ℓ} − X′_{n′−k′,k}X′_{n′+k′+1,ℓ} − δ_{kℓ}), (n = 2n′). \end{matrix}

This implies G_{0} ≅ O(m) × Sp(n) = p_{0}(OSp(m, n; Λ)).

Example 5 (continued). Assume G = G̃L(m; Λ), H = H(G) = P̃(m). Then one has

H_{0} ≅ K[X_{ij}, x]/(x det |X_{ij}| − 1)

and

G_{0} ≅ GL(m) = p_{0}(G̃L(m; Λ)).

Example 6 (continued). Assume G = S̃L(m; Λ), H = H(G) = S̃P(m). Then one has

H_{0} ≅ K[X_{ij}, x]/(x det |X_{ij}| − 1)

and

G_{0} ≅ GL(m) = p_{0}(S̃L(m; Λ)).

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

S̃P(m) ≅ K[X_{ij}, x] ⊗ Λ(Y_{ij})/I(S̃L(m; Λ)).

The annihilator ideal I(S̃L(m; Λ)) is generated by x det |X_{ij}|−1 and Tr log(E_{m}+ X^{−1}Y) = ∑_{ν} \frac{1}{2ν+1} Tr(X^{−1}Y)^{2ν+1} . Choosing instead of the Y_{ij} new odd variables W_{ij} defined 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, “Affine 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-Universität Greifswald Friedrich Ludwig Jahn-Straße 15a O-2200 Greifswald, Germany

Received July 31, 1991