ESI

The Erwin Schrödinger International Boltzmanngasse 9

Institute for Mathematical Physics A-1090 Wien, Austria

Graded Differential Geometry of
Graded Matrix Algebras

H. Grosse
G. Reiter

Vienna, Preprint ESI 708 (1999) June 1, 1999

Supported by Federal Ministry of Science and Transport, Austria
Available via anonymous ftp or gopher from FTP.ESI.AC.AT
or via WWW, URL: http://www.esi.ac.at

UWThPh-6-1999

Graded Differential Geometry of
Graded Matrix Algebras

H.Grossea;1 and G.Reitera;b;2;3

a Universität Wien, Institut für Theoretische Physik, Boltzmanngasse
5, A-1090 Wien, Austria
b Technische Universität Graz, Institut für Theoretische Physik, Pe-
tersgasse 16, A-8010 Graz, Austria

Abstract

We study the graded derivation-based noncommutative differential geometry of the Z2-graded algebra M (njm) of complex (n + m) ? (n + m)-matrices with the \usual block matrix grading" (for n 6= m). Beside the (infinite-dimensional) algebra of graded forms the graded Cartan calculus, graded symplectic structure, graded vector bundles, graded connections and curvature are introduced and investigated. In particular we prove the universality of the graded derivation-based first-order differential calculus and show, that M (njm) is a \noncommutative graded manifold" in a stricter sense: There is a natural body map and the cohomologies of M (njm) and its body coincide (as in the case of ordinary graded manifolds).

1991 MSC: 17B56, 17B70, 46L87, 58A50, 58B30, 58C50, 81T60
PACS: 02.40.-k, 11.30.Pb
Keywords: Supermanifolds, Lie superalgebras, noncommutative differential
geometry, matrix geometry

1 Introduction

The basic idea of noncommutative geometry [2], that is the formulation of differential geometric concepts on more general algebras than the algebras of C1-functions on differentiable manifolds, is at least conceptionally rooted in the fact, that all the information about the 1Tel.: +43 1 31367 3413, Fax: +43 1 317 2220, E-mail: grosse@doppler.thp.univie.ac.at
2Research partly supported by the \Fonds zur Förderung der wissenschaftlichen Forschung (FWF)" through the research project P11783-PHY \Quantum Field Theory on Noncommutative Manifolds".
3Tel.: +43 316 873 8670, Fax: +43 316 873 8678, E-mail: reiter@itp.tu-graz.ac.at

differentiable manifold and its sheaf of differentiable functions is encoded in its algebra of global C1-functions such that differential geometry can be formulated in terms of the latter algebras. Although the Z2-graded algebra of global sections of the structure sheaf of a graded manifold is a \baby-noncommutative geometry" the differential geometry of graded manifolds is treated and interpreted in the spirit of classical differential and algebraic geometry. So graded manifolds should not be seen as specific noncommutative geometries to which the general methods of noncommutative geometry applies but rather as a conceptional starting point of a \super-generalization" of noncommutative geometry. Because graded manifolds are completely determined by the Z2-graded algebra of global sections of their structure sheafs [1] the natural class of objects to which such a generalization applies are Z2-graded real respectively complex algebras.
There exist already several articles and books in the literature dealing with various aspects of Z2-graded C -algebras, supersymmetry and noncommutative geometry. Without being complete let us just mention [20, 27], where notions as cyclic cohomology and Fredholm modules are treated in the Z2-graded setting, [10], where supersymmetry is employed to establish metric, Kähler and symplectic structures in noncommutative geometry, [19], where the concept of a spectral triple is extended to algebras which contain bosonic and fermionic degrees of freedom and [8, 21], where a possibility of generalizing matrix geometry to the Z2-graded framework is presented. Here we want to adopt a somewhat different point of view.
If O(X) is the Z2-graded algebra of global sections of the (complexified) structure sheaf of some graded manifold, (complex) global graded vector fields on the graded manifold are by definition graded derivations of O(X). All global graded vector fields Vg(X) form a complex Lie subsuperalgebra and a graded module over the graded center Zg(O(X)). (Complex) global graded p-forms for p 2 N are defined as p-fold Zg(O(X))-graded-multilinear, gradedalternating maps from Vg(X) to O(X) and one can form the N0 ? Z2-bigraded C -vector space ?g(X) of global graded forms as direct sum of all Z2-graded C -vector spaces ?g;p(X) of global graded p-forms. The graded wedge product as well as the whole graded Cartan calculus on ?g(X) can be introduced (see [1, 22] for example) by employing only the facts, that Vg(X) is a C -Lie superalgebra and Z2-graded Zg(O(X))-module and that O(X) is a Z2-graded C -algebra. The important feature of the recipe for the construction of the graded deRham complex and the graded Cartan calculus formulated above is, that it uses only the graded algebra structure of O(X). In particular it does not use the graded commutativity of O(X) and we can define on arbitrary Z2-graded C -algebras noncommutative graded differential calculi.
What we have just described is mutatis mutandis the basic idea of the so-called derivationbased differential calculi [5, 6, 9] transposed to the Z2-graded setting. Such graded derivationbased differential calculi were investigated for arbitrary, but graded-commutative Z2-graded algebras in the framework of Z2-graded Lie-Cartan pairs [13, 20, 27].

Motivated by the rich differential geometric structure of ordinary matrix algebras [5, 6, 24] and by the our previous investigation of the fuzzy supersphere [15, 16], where each truncated supersphere was a graded matrix algebra in particular, we will investigate especially the differential calculus based on all graded derivations on the Z2-graded C -algebra M (njm) of complex

(n+m)?(n+m)-matrices (n; m 2 N0; n+m 2 N), Z2-graded by declaring the vector subspace

M (njm)0 :=

(

M =

M1 0
0 M4

! fififi M1 2 M (n); M4 2 M (m)
)

(1)

of M (njm) as even and the vector subspace

M (njm)1 :=

(

M =

0 M2
M3 0

! fififi M2 2 M (n; m); M3 2 M (m; n)
)
(2)

of M (njm) as odd. Here M (n; m) and M (n) denote the vector space of n ? m- respectively the algebra of n ? n-matrices and we will always assume n 6= m.
Chapter 2 is devoted to the precise definition of the graded derivation-based differential calculus on M (njm) as described above and its immediate consequences. The resulting complexes are nothing else than the complexes of Lie superalgebra cohomology with values in M (njm) and typically infinite. The latter fact shows in particular, that the complex is completely different to that proposed in [8, 21].
In chapter 3 we continue the investigation of the differential calculus using the facts, that there exist graded-commutative homogeneous bases in the Z2-graded M (njm)-modules of all graded p-forms and that all graded derivations of M (njm) are inner. Especially we construct an invariant graded 1-form, which determines the differential of graded matrices in terms of graded commutators (within the graded algebra of graded forms) and show, that the firstorder differential calculus is universal.
Associated with every graded manifold there exists an even, surjective algebra homomorphism fiX from the Z2-graded algebra of global sections of the (complexified) structure sheaf of the graded manifold O(X) to the algebra C1(X) of (complex) C1-functions on its body manifold X. We call this map, which is the key to all further developments in graded manifold theory, the body map. In chapter 4 we show, that there exists a natural noncommutative analogue to the body map. It induces an isomorphism between the graded derivation-based cohomology of M (njm) and the derivation-based cohomology of its body, such that the situation described by a theorem of Kostant [22] is generalized to the noncommutative case.
In chapter 5 we study the graded symplectic geometry of M (njm). As for ordinary matrix geometry [5, 6], which is included as special case, there exists a graded symplectic structure such that the induced graded Poisson bracket on M (njm) is (i times) the graded commutator on M (njm).
In the last chapter we investigate the noncommutative generalization of graded vector bundles over graded manifolds. Graded vector bundles over a graded manifold (X; O) are usually introduced as locally graded-free O-modules [1, 17, 22]. In the spirit of noncommutative geometry [2, 14] we concentrate on the module of global sections and introduce graded vector bundles over M (njm) as Z2-graded, finitely generated (graded-projective) M (njm)-modules. Concepts like connections and curvature can be generalized to the Z2-graded noncommutative setting. In addition we have included an appendix in which we analyze the associative product of supertrace-free, graded matrices. The results of this analysis are used for a minimality proof in chapter 3.

There will appear lots of Z2-graded objects. If the object is denoted by A its even part is denoted by A0, its odd part by A1. If a is some homogeneous element of such an object its degree will be denoted by a. Speaking of grading in the context of an ungraded object we mean, that the object is endowed with its trivial graduation. If for some construction the Z2-grading is indicated by an index \g" we omit this index in the case of trivial graduation.

2 Graded derivation-based differential calculus on graded matrix algebras

We will interpret the C -Lie superalgebra and Z2-graded Zg(M (njm))-module Derg(M (njm)) of all graded derivations of M (njm) as \`noncommutative graded vector fields" on M (njm). Because M (njm) is graded-central,

Zg(M (njm)) = Zg(M (njm))0 = C 1n+m ?= C ; (3)

the concept of graded Zg(M (njm))-multilinearity reduces to ordinary C -multilinearity and we can employ ideas and results of Lie superalgebra cohomology (see [11, 28]) for the construction of the graded derivation-based differential calculus on M (njm).

For every natural number p 2 N let us denote by Homp(Derg(M (njm)); M (njm)) the Z2-graded C -vector space of all p-linear maps Derg(M (njm))? p: : : ?Derg(M (njm)) ?! M (njm) and by Sp the symmetric group of p letters. Introducing a commutation factor flp : Sp ? Z2? p: : : ?Z2 ?! f?1g via
flp(ö; i1; ? ? ? ; ip) :=

Y

r;s=1;???;p;r<s
ö?1(r)>ö?1(s)

(?1)iris ; (4)

we can define a representation ß of Sp on Homp(Derg(M (njm)); M (njm)) by

(ßö!) (D1; ? ? ? ; Dp) := flp(ö; D1; ? ? ? ; Dp)!(Dö(1); ? ? ? ; Dö(p)) (5)

for all ! 2 Homp(Derg(M (njm)); M (njm)), all homogeneous D1; ? ? ? ; Dp 2 Derg(M (njm)) and all ö 2 Sp. Now by definition a p-linear map ! 2 Homp(Derg(M (njm)); M (njm)) is called graded-alternating if
ßö! = sgnö ! (6)

is fulfilled for all ö 2 Sp and we interpret such maps as graded p-forms on M (njm). All graded p-forms on M (njm) form a graded vector subspace of Homp(Derg(M (njm)); M (njm)), which we will denote by ?g;p(M (njm)).
A general graded form on M (njm) is an element of the direct sum

?g(M (njm)) :=
M

p2N0
?g;p(M (njm)) ; (7)

where we set ?g;0(M (njm)) := M (njm). Employing the multiplicative structure of M (njm) we can proceed exactly as in the case of graded manifolds [1, 22] (respectively graded Lie-Cartan pairs [13, 27]) to introduce a graded wedge product on ?g(M (njm)). So we first define for all

p; p0 2 N0; i; i0 2 Z2 a bilinear map ^ : ?g;p(M (njm))i ? ?g;p(M (njm))i0 ?!
?g;p+p0(M (njm))i+i0 by

?! ^ !0? (D1; ? ? ? ; Dp+p0) := 1
p!p0!
X

ö2Sp+p0
sgnö flp+p0(ö; D1; ? ? ? ; Dp+p0) ? (8)

?(?1)i0 Ppl=1 Dö(l)!(Dö(1); ? ? ? ; Dö(p))!0(Dö(p+1); ? ? ? ; Dö(p+p0))

for all homogeneous D1; ? ? ? ; Dp+p0 2 Derg(M (njm)) and extend these by bilinearity to ?g(M (njm)). With respect to it ?g(M (njm)) becomes a N0 ? Z2-bigraded C -algebra.

Via

(LD0!) (D1; ? ? ? ; Dp) := D0 (!(D1; ? ? ? ; Dp)) ? (9)

?

p
X

l=1
(?1)D0(!+Pl?1
l0=1 Dl0)!(D1; ? ? ? ; [D0; Dl]g ; ? ? ? ; Dp) ;

({D1!) (D2; ? ? ? ; Dp) := !(D1; D2; ? ? ? ; Dp) (10)

and

d!(D0; ? ? ? ; Dp) =

p

X

l=0

(?1)l+Dl(!+Pl?1

l0=0 Dl0)LDl

?

!(D0; ? ? ? ;

_

Dl; ? ? ? ; Dp)

?

+ (11)

+

X

0<=l<l0<=p

(?1)l0+Dl0 Pl0?1
l00=l+1 Dl00 !(D0; ? ? ? ; Dl?1; [Dl; Dl0]g ; ? ? ? ;
_
Dl0 ; ? ? ? ; Dp)

for all homogeneous D0; D1; ? ? ? ; Dp 2 Derg(M (njm)) and all homogeneous ! 2 ?g;p(M (njm)) (_ denotes omission), one defines homogeneous endomorphisms ?g(M (njm)) ?! ?g(M (njm)) of bidegree (0; D0), (?1; D0) and (1; 0) respectively. The assignments D 7! {D and D 7! LD extend to C -linear maps Derg(M (njm)) ?! End(?g(M (njm))) and L is a graded representation of Derg(M (njm)) in particular. Furthermore the relations

d ffi d = 0
d ffi LD = LD ffi d (12)

as well as

{D ffi {D0 + (?1)DD0{D0 ffi {D = 0
(LD ffi {D0 ? {D0 ffi LD) ! = (?1)D!{[D;D0]g! (13)

({D ffi d + d ffi {D) ! = (?1)D!LD!

for all homogeneous D; D0 2 Derg(M (njm)) and all bihomogeneous ! 2 ?g(M (njm)) are known from Lie superalgebra cohomology [28].
By analogy with the case of graded manifolds we call d; LD and {D exterior derivative, Lie derivative and interior product (with respect to a graded vector field D 2 Derg(M (njm))). (12) and (13) tell us, that they fulfill exactly the same relations as in the \graded-commutative case", but this observation remains also true for the graded wedge product (8) of graded forms.

Proposition 1 The relations

LD
?! ^ !0? = (LD!) ^ !0 + (?1)D!! ^ LD!0

{D
?! ^ !0? = (?1)D!0 ({D!) ^ !0 + (?1)p! ^ {D!0 (14)

d ?! ^ !0? = (d!) ^ !0 + (?1)p! ^ d!0

are fulfilled for all homogeneous D 2 Derg(M (njm)), ! 2 ?g;p(M (njm)), !0 2 ?g;p0(M (njm)).

Proof : This can be shown exactly as in the case of graded manifolds. That is, one starts with a direct proof of the second relation and proofs the other equations inductively using the last two relations (13). 2

Because we interpret d as exterior derivative, the Lie superalgebra cohomology of
Derg(M (njm)) with values in M (njm),

H(M (njm)) ?

M

p2N0
Hp(M (njm)) := ker d
im d ; (15)

has to be seen as analogue to the graded deRham-cohomology on graded manifolds. Via

[!] ^ [!0] := [! ^ !0] (16)

the above graded derivation-based cohomology of M (njm) becomes a N0 ? Z2-bigraded C - algebra and we will continue to study it in chapter 4.

3 Homogeneous bases and the canonical graded 1-form

Whereas the definitions and results of the preceding considerations apply to each Z2-graded, graded-central C -algebra we shall now employ more specific properties of M (njm). There will result similar formulas as in \ordinary" matrix geometry [5, 6, 24], which is included as special case.

The sets ?g;p
Zg(M (njm)) of graded p-forms with values in the graded center of M (njm) form

graded vector subspaces of ?g;p(M (njm)) for all p 2 N and one can introduce

?gZg(M (njm)) :=

M

p2N0

?g;p
Zg(M (njm)) (17)

with ?g;0
Zg(M (njm)) = Zg(M (njm)). ?gZg(M (njm)) is a bigraded subalgebra of ?g(M (njm)), whose product fulfills
! ^ !0 = (?1)pp0+!!0!0 ^ ! (18)

for all homogeneous ! 2 ?g;p
Zg(M (njm)); !0 2 ?g;p0
Zg (M (njm)) and which is stable with respect to the whole Cartan calculus.
Now let us introduce a homogeneous basis f@AgA=1;???;n0+m0 of Derg(M (njm)) with @1; ? ? ? ; @n0 2 Derg(M (njm))0, @n0+1; ? ? ? ; @n0+m0 2 Derg(M (njm))1, where we set n0 := dimC Derg(M (njm))0

and m0 := dimC Derg(M (njm))1. If f?AgA=1;???;n0+m0 denotes the dual basis to f@AgA=1;???;n0+m0 we can introduce a homogeneous basis fÀgA=1;???;n0+m0 of ?g;1
Zg(M (njm)) by

À(D) := ?A(D)1n+m (19)

for all D 2 Derg(M (njm)). Employing the standard isomorphisms between graded-alternating maps and the graded exterior algebra [1, 28] one deduces, that
n

À1 ^ ? ? ? ^ Àp fififi (A1; ? ? ? ; Ap) 2 In0jm0
p

o

(20)

with

In0jm0
p :=

n
(k1; ? ? ? ; kp0; ffp0+1; ? ? ? ; ffp)

fififi 0 <= p0 <= p; k1; ? ? ? ; kp0 = 1; ? ? ? ; n0; (21)

ffp0+1; ? ? ? ; ffp = n0 + 1; ? ? ? ; n0 + m0; k1 < k2 < ? ? ? < kp0 < ffp0+1 <= ? ? ? <= ffp?1 <= ffp
o

is a homogeneous basis of ?g;p
Zg(M (njm)); p 2 N.
Because of
M ^ ! = (?1)M!! ^ M (22)

for all homogeneous M 2 M (njm) and all bihomogeneous ! 2 ?gZg(M (njm)), the N0 ? Z2- bigraded C -algebras ?g(M (njm)) and M (njm)^?C ?gZg(M (njm)), where ^? denotes the tensor product of Z2-graded algebras, are canonically isomorphic. In particular we can conclude:

Proposition 2 The Z2-graded M (njm)-bimodules ?g;p(M (njm)) are graded-free for both multiplications and for all p 2 N0. The set (20) determines a homogeneous basis of the left (right), Z2-graded M (njm)-module ?g;p(M (njm)). 2

Consequently every ! 2 ?g;p(M (njm)) can be written as

! =

X

(A1;???;Ap)2In0 jm0
p

!A1???Ap ^ À1 ^ ? ? ? ^ Àp (23)

with unique coefficients !A1???Ap 2 M (njm). Explicitely these coefficients are given by

!A1???Ap = (?1) 12 p00(p00?1) 1
Qn0+m0
l=1 Nl!

!(@A1 ; ? ? ? ; @Ap) ; (24)

where p00 is the number of entries in (A1; ? ? ? ; Ap) greater than n0 and Nl is the number of entries in (A1; ? ? ? ; Ap) being equal l.

In order to investigate graded derivations of M (njm) (we include the case n = m for the moment) let us denote by gl(njm) the (complex) general linear Lie superalgebra and by sl(njm) the (complex) special linear Lie superalgebra. The adjoint representation of gl(njm) is at the same time a Lie superalgebra homomorphism ad : gl(njm) ?! Derg(M (njm)) and, as we will see, the structure of Derg(M (njm)) and its Lie subsuperalgebras is determined by this homomorphism.

Proposition 3 If L is a Lie subsuperalgebra of gl(njm) then

Lad := im adjL (25)

is a Lie subsuperalgebra of Derg(M (njm)). Contrary every Lie subsuperalgebra of
Derg(M (njm)) is of this form. There are two different cases:

(i:) For n 6= m the restriction of ad to sl(njm) is an Lie superalgebra isomorphism onto Derg(M (njm)) and the various restrictions of ad induce a bijective correspondence between Lie subsuperalgebras of sl(njm) and Lie subsuperalgebras of Derg(M (njm)).

(ii:) For n = m there is no Lie subsuperalgebra L of gl(njn) such that the restriction of ad to L becomes an Lie superalgebra isomorphism onto Derg(M (njn)).

Proof : An even graded derivation of M (njm) is just an ordinary derivation of the C -algebra M (n + m) and these are inner, because the first Hochschild cohomology group of M (n + m) with values in M (n + m) is trivial [26]. Introducing

? :=

1n 0
0 ?1m

!

we find for some D 2 Derg(M (njm))1 and all homogeneous M 2 M (njm)

DM = ad

?1

2(D?)?

?
(M) ;

from which we can conclude, that D is inner. Consequently, if D is a Lie subsuperalgebra of Derg(M (njm)), then L := ad?1(D) is a Lie subsuperalgebra of gl(njm) with Lad = D. (i.) and (ii.) are consequences of 1n+m 62 sl(njm) for n 6= m respectively 12n 2 ?gl(njn)1; gl(njn)1
?
g. 2

The ultimate reason for restricting our geometric investigation to the case n 6= m lies in the existence of the Lie superalgebra isomorphism ad : sl(njm) ?! Derg(M (njm)). The elements of every homogeneous basis f@AgA=1;???;n0+m0 of Derg(M (njm)) are given by

@A = ad EA ; (26)

where fEAgA=1;???;n0+m0 is a homogeneous basis of sl(njm) and we have n0 = n2 +m2 ?1; m0 = 2nm in particular. Moreover, the structure constants cCAB appearing in

[@A; @B]g =

(n+m)2?1
X

C=1

cCAB@C (27)

are the structure constants of the homogeneous sl(njm)-basis fEAgA=1;???;(n+m)2?1 and one deduces the nice formulas

dEA = ?

(n+m)2?1
X

B;C=1

cCABEC ^ ?B (28)

and

dÀ = 1

2

(n+m)2?1
X

B;C=1

cABC?C ^ ?B : (29)

The even graded 1-form

? :=

(n+m)2?1
X

A=1
EA ^ À (30)

will be called canonical graded 1-form, because it plays a distinguished role.

Proposition 4 The definition of ? is independent of the choice of the homogeneous basis of Derg(M (njm)). ? is (Derg(M (njm))-)invariant and this property determines ? up to constant multiples. Furthermore its exterior differential fulfills

d? = ? ^ ? (31)

and the exterior differential of each M 2 M (njm) can be expressed according to

dM = [?; M ]g ? ? ^ M ? M ^ ? : (32)

Proof : Beside the uniqueness statement only simple calculations are involved (for which one can use (28) and (29) advantageously). The irreducibility of the adjoint representation of sl(njm) [4] guarantees, that
LD! = 0 ; ! 2 ?g;1(M (njm)) ;

for all D 2 Derg(M (njm)) implies ! = c?; c 2 C . 2

Finally we note, that ?g(M (njm)) is in a certain sense minimal.

Proposition 5 (28) can be inverted according to

À = 4(n ? m)2

(n+m)2?1
X

B;C;D=1
(?1)EBEDKABKCDECEB ^ dED ; (33)

where K is the Killing form of sl(njm) and KAB denote the components of the inverse matrix of (K(EA; EB)). Consequently, if ? is differential subalgebra of ?g(M (njm)) containing M (njm), then ? = ?g(M (njm)).

Proof : The minimality statement follows from (33) because of proposition 2. In order to show (33) one uses (28) and expands the threefold product of the basis elements EA according to (A4). Using the results of proposition A (33) follows. 2

The second part of proposition 5 can be stated differently: The canonical even algebra homomorphisms from the (intrinsic) Z2-graded universal differential envelope of M (njm) to ?g(M (njm)) (see [3, 20] for a precise definition) is onto. The restriction of this homomorphism to the corresponding first-order differential calculi is an isomorphism.

4 Cohomology and the noncommutative body map

We will call the even, surjective C -linear map

fi : M (njm) ?! M (n) with n :=

(
n if n > m
m if n < m (34)

defined by

fi(M) ? fi

M1 M2
M3 M4

!!

:=

(
M1 if n = n
M4 if n = m (35)

canonical body map of M (njm). A justification for choosing this terminology will result from the investigation of its properties: They are completely analogous to the corresponding map of graded manifolds if one takes the noncommutativity of M (njm) and its \body" M (n) appropriately into account. In order to distinguish between \objects" on M (njm) and corresponding \objects" on the body we underline the latter.

The restriction of fi to Zg(M (njm)) is an even algebra homomorphism onto Z(M (n)) and by

?(M) :=

8

>

>
>

>
>
>
>
<

>

>
>

>

>

>

>

:

M 0
0 1nTr(M)1m

!

if n = n

1mTr(M)1n 0

0 M

!

if n = m

; (36)

we can introduce an even, injective C -linear map ? : M (n) ?! M (njm), which is right-inverse to fi on the one hand and whose restriction to Z(M (n)) is an even algebra homomorphism into Zg(M (njm)) on the other hand.
Analogous to the body map of graded manifolds fi induces a Lie algebra homomorphism ^fi : Derg(M (njm))0 ?! Der(M (n)) via
^fi(D)fi(M) := fi(DM) (37)

for all M 2 M (njm). ^fi is surjective because of
^fi(ad E) = ad fi(E) (38)

for all E 2 sl(njm)0 and in addition ^? : Der(M (n)) ?! Derg(M (njm))0,

^?(ad E) := ad ?(E) (39)

is a Lie algebra homomorphism right-inverse to ^fi.
Now we can introduce even C -linear maps fi(p) : ?g;p(M (njm)) ?! ?p(M (n)); p 2 N, by
?

fi(p)(!)

??

D1; ? ? ? ; Dp

?

:= fi

?

!

?

^?(D1); ? ? ? ; ^?(Dp)

??
(40)

for all D1; ? ? ? ; Dp 2 Der(M (n)). sl(njm)0 is canonically isomorphic to sl(n)?gl(1)?sl(m) and we can choose a homogeneous basis fEAgA=1;???;(n+m)2?1 of sl(njm) such that E1; ? ? ? ; En2?1 lie

in the isomorphic copy of sl(n) and En2 ; ? ? ? ; En2+m2?1 in the isomorphic copy of gl(1) ? sl(m) with m := minfn; mg. Then the elements fi(Ek) := Ek ; k = 1; ? ? ? ; n2 ?1, form a basis of sl(n). Denoting the elements of the basis of ?1(M (n)) corresponding with fEkgk=1;???;n2?1 according to (19) and (26) with ?k the action of the maps fi(p) can be described alternatively by

fi(p)(!) ? fi(p)

0

B@

X

(A1;???;Ap)2In2+m2?1j2nm
p

!A1???Ap ^ À1 ^ ? ? ? ^ Àp

1

CA =

=

X

(k1;???;kp)2In2?1j0

p

fi

?
!k1???kp
?
^ ?k1 ^ ? ? ? ^ ?kp : (41)

If we set fi(0) ? fi the maps fi(p); p 2 N0, extend uniquely to a bihomogeneous, C -linear map ?g(M (njm)) ?! ?(M (n)) of bidegree (0; 0) , which we again denote by fi. Because of (41) fi is onto and its restriction to ?gZg(M (njm)) is an surjective homomorphism of bigraded C -algebras onto ?Z(M (n)). Furthermore fi fulfills

fi ffi LD = L^fi(D) ffi fi (42)

for all D 2 Derg(M (njm))0 as well as

fi ffi d = d ffi fi : (43)

Consequently fi induces a homomorphism H(fi) : H(M (njm)) ?! H(M (n)) of cohomologies in the usual way. Analogous to graded manifold theory [22] this map is an isomorphism.

Proposition 6 H(fi) is an isomorphism of bigraded C -algebras, such that both cohomologies H(M (njm)) and H(M (n)) are isomorphic to the Lie algebra cohomology H(sl(n); C ) of sl(n) with trivial coefficients.

Proof : Using the results of [28] as well as Derg(M (njm)) = sl(njm)ad we find the sequence

H(M (njm)) ?= H(sl(njm); M (njm)) ?= H(sl(njm); C 1n+m) ? H(sl(njm); sl(njm)) ?=
?= H(sl(njm); C 1n+m) ?= H(sl(njm); C )

of natural isomorphisms between Lie superalgebra cohomologies. In particular we have H(M (n)) ?= H(sl(n); C ) (as N0-graded C -algebra), which is well-known from matrix geometry [5, 6, 9]. Combining the above result with the calculations of the cohomology of sl(njm) with trivial coefficients [11, 12] one can conclude, that H(fi) is an isomorphism of bigraded C -algebras. 2

5 Noncommutative graded symplectic geometry

Generalizing the situation on graded manifolds [22] we call an even, closed graded 2-form ! 2 ?g;2(M (njm)) graded symplectic structure on M (njm), if the equation

!(D; DM) = DM (44)

for all D 2 Derg(M (njm)) possesses a unique solution DM 2 Derg(M (njm)) for each M 2 M (njm). The graded vector fields DM 2 Derg(M (njm)) are called Hamiltonian and the set of all graded Hamiltonian vector fields is denoted by Hamg(!).
If ! 2 ?g;2(M (njm)) is graded symplectic structure on M (njm) the assignment M 7! DM defines an even C -linear map D! : M (njm) ?! Hamg(!) ? Derg(M (njm)) and one can conclude, that (44) is equivalent to

{DM! + dM = 0 : (45)

Using (13) we find
LDM! = 0 (46)

for all DM 2 Hamg(!), that is a graded symplectic structure on M (njm) is - as usual - invariant with respect to all graded Hamiltonian vector fields.
Via ?M;M 0?!

g := !(DM ; DM 0) (47)

for all M; M 0 2 M (njm) we can introduce a graded Poisson bracket, which has the analogous properties as its graded-commutative pendant.

Proposition 7 (M (njm); f?; ?g!g ) is a C -Lie superalgebra and the graded Poisson bracket fulfills in addition

?M;M 0M 00?!

g = ?M;M 0?!
g M 00 + (?1)MM 0
M 0 ?M; M 00?!
g
f1n+m; Mg!g = 0 (48)

for all homogeneous M; M 0; M 00 2 M (njm). Moreover, the map D! : M (njm) ?! Hamg(!) is a homomorphism of Lie superalgebras and

Hamg(!) = Derg(M (njm)) : (49)

Proof : The properties of f?; ?g!g and of D! result from the defining properties of the graded symplectic structure !. >From the irreducibility of the adjoint representation of sl(njm) one can deduce ker D! = C 1n+m on the one hand and the injectivity of D!jsl(njm) on the other hand. Then (49) follows because of Derg(M (njm)) = sl(njm)ad. 2

There exists an essentially unique graded symplectic structure on M (njm).

Proposition 8 d? is a graded symplectic structure on M (njm) and up to complex multiples it is the only one. The corresponding graded Poisson bracket is given by

?M;M 0?d?

g = ?M; M 0?
g (50)

for all M; M 0 2 M (njm).

Proof : The exact, even graded 2-form c d?; c 2 C n f0g induces via (44) a homomorphism
Dc d? : M (njm) ?! Derg(M (njm)),

Dc d?(M) = 1
cadM (51)

of Lie superalgebras and the corresponding graded Poisson bracket is given by fM; M 0gc d?
g = c?1 [M; M 0]g. The uniqueness property is a consequence of proposition 3, (49) and Schur's Lemma. 2

Consequently the extension of the body map fi maps a graded symplectic structure ! onto a symplectic structure fi(!). Moreover one has

^fi (D!(M)) = Dfi(!) (fi(M)) (52)

for all even graded (Hamiltonian) vector fields as well as

fi

??M;M 0?!

g

?
= ?fi(M); fi(M 0)?fi(!) (53)

for the graded Poisson bracket of M; M 0 2 M (njm)0. That is, the relation between M (njm) and its body is analogous to the one for graded symplectic manifolds and their respective underlying manifolds.

6 Graded vector bundles over graded matrix algebras

As a synthesis of the definition of graded vector bundles over graded manifolds [1, 17, 22] and the idea how to introduce vector bundles in noncommutative geometry [2, 14] we interpret left, Z2-graded, finitely generated (graded-projective) M (njm)-modules as graded vector bundles over M (njm) and even M (njm)-module homomorphisms between such modules as graded vector bundle homomorphisms. We note, that the specifying property of graded projectivity is redundant in the context of left, Z2-graded M (njm)-modules, because on the one hand gradedprojective means Z2-graded plus projective [25] and on the other hand every left M (n + m)- module is projective [26].
Let us denote by M (njm; rjs); r; s 2 N0; r + s 2 N; the C -vector space M (n + m; r + s) together with the Z2-grading defined by

M (njm; rjs)0 :=

(

v =

v1 0
0 v4

! fififi v1 2 M (n; r); v4 2 M (m; s)
)

M (njm; rjs)1 :=

(

v =

0 v2
v3 0

! fififi v2 2 M (n; s); v3 2 M (m; r)
)
: (54)

With respect to ordinary matrix multiplication M (njm; rjs) becomes a left, Z2-graded, finitely generated M (njm)-module and these examples constitute essentially all graded vector bundles over M (njm).

Proposition 9 If V is a graded vector bundle over M (njm) then there exist unique numbers r; s 2 N0; r + s 2 N and a graded vector bundle isomorphism Ö : V ?! M (njm; rjs). V is graded-free if and only if there are natural numbers p; q 2 N0; p + q 2 N; such that

pn + qm = r
pm + qn = s : (55)

Proof : The existence of the isomorphisms are implied by the graded simplicity of M (njm; 1j0) and M (njm; 0j1) and the fact, that every left, Z2-graded, finitely generated M (njm)-module is the homomorphic image of a left, Z2-graded, graded-free M (njm)-module with homogeneous basis of suitable cardinality pjq. Because all M (njm)-module isomorphisms are C -vector space isomorphisms in particular, the uniqueness statement and (55) follow. 2

After this \miniature-classification" we develop graded differential geometry on a fixed graded vector bundle V generalizing the treatment of noncommutative geometry [2, 5, 14, 24] on the one hand and the one of supergeometry [1] on the other hand.
So we first define the set ?g(V) of V-valued graded forms according to

?g(V) ?
M

p2N0

?g;p(V) := ?g(M (njm))^?M(njm)V : (56)

?g(V) is a left N0?Z2-bigraded ?g(M (njm))-module in a natural way and each ?g;p(V); p 2 N0, is a left, Z2-graded, finitely generated M (njm)-module in particular. The product will again be denoted by ^.
A connection on V is an even C -linear map r : V ?! ?g;1(V) such that

r (Mv) = dM ? v + M ^ rv (57)

is fulfilled for all M 2 M (njm); v 2 V . Connections always exist due to (graded) projectivity.

Proposition 10 Let V be a graded vector bundle over M (njm). Then there exists a graded-free vector bundle Vpjq over M (njm) with homogeneous basis f "A j "A 2 Vpjq
0 ; A = 1; ? ? ? ; p; "A 2 Vpjq
1 ; A = p+1; ? ? ? ; p+qg; p; q 2 N0; p+q 2 N, together with an even, idempotent endomorphism

P : Vpjq ?! Vpjq and an isomorphism ' : V ?! imP of Z2-graded M (njm)-modules. The map rd : Vpjq ?! ?g;1(Vpjq) defined by

rd(v) ? rd

p+q

X

A=1

vA"A

!

:=

p+q
X

A=1
dvA ? "A (58)

is a connection on Vpjq and

rPd := Id?g;1(M(njm)) ? '?1 ffi Id?g;1(M(njm)) ? P ffi rd ffi ' (59)

is a connection on V. A map r : V ?! ?g;1(V) is a connection on V if and only if it is of the form
r = rPd + ff ; (60)

where ff : V ?! ?g;1(V) is an even homomorphism of Z2-graded M (njm)-modules.

Proof : Analogous to the ungraded case [2, 14]. 2

Quite generally we will denote the Z2-graded M (njm)-bimodule of graded (p + q) ? (p + q)- matrices over a Z2-graded bimodule B with M (pjq; B). It is a M (njm)-bimodule in a natural way and Z2-graded by declaring those matrices with even diagonal entries and odd off-diagonal entries as even and those with odd diagonal entries and even off-diagonal entries as odd. Adopting the notation of the above proposition we introduce homogeneous generators

?A := '?1 ffi P ("A) (61)

of V as well as an even matrix (PBA ) 2 M (pjq; M (njm))0 via

P ("A) =:

p+q
X

B=1

PBA "B : (62)

Then

(r ? rPd) (?A) = ff(?A) =:

p+q
X

B=1

ffBA ? ?B (63)

establishes a bijective correspondence between the set of all connections on V and the set PM (pjq; ?g;1(M (njm)))0P , which consists of those (ffBA) 2 M (pjq; ?g;1(M (njm)))0 fulfilling ffBA = Pp+q
C;D=1 PCA ^ ffDC ^ PBD (for V = Vpjq set P = ' = IdVpjq ). The graded 1-forms ffBA are called connection forms of the connection r = rPd + ff.
If r is a connection on a graded vector bundle V we can introduce a C -linear map ?g(V) ?! ?g(V), again denoted by r, via

r (! ? v) = d! ? v + (?1)p! ^ rv (64)

for all v 2 V ; ! 2 ?g;p(M (njm)); p 2 N0. This homogeneous map of bidegree (1; 0) extends the original connection if we identify V with ?g;0(V). Moreover it fulfills

r ?!0 ^ ! ? v? = d!0 ^ (! ? v) + (?1)p0!0 ^ r(! ? v) (65)

for all v 2 V ; ! 2 ?g;p(M (njm)); !0 2 ?g;p0(M (njm)); p; p0 2 N0, and this property determines the extension of the connection uniquely.
The curvature of a connection r on a graded vector bundle V is defined as

r2 ? r ffi r : V ?! ?g;2(V) : (66)

It is an even homomorphism of Z2-graded M (njm)-modules and one can describe its action on an element v = Pp+q
A=1 '(v)A?A 2 V according to

r2(v) =:

p+q
X

A;B=1

'(v)A ^ RBA ? ?B (67)

with a uniquely determined matrix (RBA) 2 PM (pjq; ?g;2(M (njm)))0P . The graded 2-forms RBA are called curvature forms and they can be expressed according to

RBA = ?

p+q

X

C=1

ffCA ^ ffBC +

p+q

X

C;D=1

?

PCA ^ dffDC ^ PBD ? PCA ^ dPDC ^ dPBD

?

(68)

in terms of the connection forms ffBA of the connection. Moreover they have to fulfill the Bianchi identity

p+q

X

C;D=1

PCA ^ dRDC ^ PBD ?

p+q

X

C=1

?
ffCA ^ RBC ? RCA ^ ffBC
?
= 0 : (69)

Let us finally analyze the space of flat connections, that is the set of all connections with vanishing curvature. We will not do this in complete generality but only for a graded-free vector bundle V1j0 with an even basis element ".

Proposition 11 A connection on V1j0 is flat if and only if its connection form
ff 2 ?g;1(M (njm))0 is either given by
ff = ? (70)

or by

ff = ? ?

(n+m)2?1
X

A=1

f(EA) ^ À ; (71)

where fEAg is the homogeneous basis of sl(njm) \corresponding" with fÀg and f is some automorphism of sl(njm).

Proof : Let us introduce an even graded 1-form æ = P(n+m)2?1
A=1 æA^À according to ff =: ??æ.

Using proposition 4 we find, that the curvature form is given by

R = 1

2

(n+m)2?1
X

A;B=1

?AB ^ À ^ ?B (72)

with

?AB = [æB; æA]g ?

(n+m)2?1
X

C=1

cCBAæC : (73)

Because the vanishing of the curvature is equivalent to ?AB = 0 the proposition follows from the simplicity of sl(njm). 2

That is, we have the same situation as in ordinary matrix geometry [5, 6]: There exist different \classes" of flat connections. Here \class" refers to the action of the group of automorphisms of the graded vector bundle on the space of connections, which can be introduced analogous to the ungraded case. The connection rd and the one associated with the connection form ? will lie in different classes, because the latter is invariant. However, if one does not restrict the space of connections by a suitable compatibility requirement with respect to a graded hermitian structure there will exist even more than two classes of flat connections.

7 Concluding remarks

We have developed the graded differential geometry of graded matrix algebras and shown that the results of matrix geometry [5, 6] carry over to the Z2-graded setting. In addition we found

a natural noncommutative analogue of the body map, which allows us to view graded matrix geometries as true noncommutative generalizations of graded manifolds.
Whereas in ordinary differential geometry one integrates forms this is not true in supergeometry. Except from the before mentioned body map, which plays a central role in the global theory of Berezin integration [18], we completely excluded the integral geometry of graded matrix algebras. We plan to treat this together with metric aspects in a separate work.
Beside its immediate application for the construction of (graded) differential calculi on fuzzy (super)manifolds [24, 16] the developments of this article offer another perspective. The extension of space-time by matrix geometries led to interesting new gauge models. In particular the existence of different gauge orbits of flat connections in matrix geometry is the origin of the appearance of the Higgs effect [5, 7]. The possibility of extending the structures of matrix geometry to Z2-graded matrix algebras suggests to think about similar \supersymmetric" noncommutative extensions of space-time.

A Associative product of supertrace-free, graded matrices

Let fEAjEA 2 sl(njm)0; A = 1; ? ? ? ; n2 +m2 ?1; EA 2 sl(njm)1; A = n2 +m2; ? ? ? ; (n+m)2?1g be a homogeneous basis of sl(njm); n 6= m. Our aim of this appendix is to investigate the associative product of the homogeneous matrices EA in a similar way as it was done in [23] for trace-free, hermitian matrices.

If we introduce a graded anticommutator

?M; M 0?+

g := MM 0 + (?1)MM 0
M 0M (A1)

of two homogeneous M; M 0 2 M (njm) we find the relations

h

M; ?M 0; M 00?

g

i

g ?

h

M; ?M 0; M 00?+

g

i+

g + (?1)M 0M 00 h?M; M 00?+
g ; M 0i+

g = 0

h?M; M 0?+

g ; M 00i

g ?

h

M; ?M 0; M 00?

g

i+

g ? (?1)M 0M 00 h?M; M 00?
g ; M 0i+
g = 0 (A2)

between the graded commutator and the graded anticommutator of homogeneous M; M 0; M 00 2 M (njm).
Because fEA; 1n+mgA=1;???;(n+m)2?1 forms a homogeneous basis of M (njm) the graded anticommutator of EA and EB can be written according to

[EA; EB]+g =

(n+m)2?1
X

C=1
dCABEC + gAB1n+m (A3)

with uniquely determined coefficients dCAB ; gAB 2 C . Then the associative product of EA and EB is given by

EAEB = 1

2

(n+m)2?1
X

C=1

(cCAB + dCAB)EC + 1
2gAB1n+m : (A4)

Independent of the specific choice of the homogeneous basis fEAgA=1;???;(n+m)2?1 there exist a lot of relations between the \structure constants" cCAB ; dCAB and gAB which we summarize in

Proposition A

(i:) cCAB and dCAB vanish if EA + EB + EC = 1 and gAB vanishes if EA + EB = 1.

(ii:) (n+m)2?1
X

B=1

(?1)EBcBAB = 0

(n+m)2?1
X

B=1

(?1)EBdBAB = 0 (A5)

(iii:) cABC and dABC, defined via

cABC :=

(n+m)2?1
X

D=1

cDABgDC

dABC :=

(n+m)2?1
X

D=1
dDABgDC ; (A6)

are totally antisymmetric respectively totally symmetric in the Z2-graded sense.

(iv:) (n+m)2?1

X

E=1

n

(?1)EAECcEBCcDAE + (?1)EBEAcECAcDBE + (?1)ECEBcEABcDCE

o

= 0

(n+m)2?1

X

E=1

n

cEBCcDAE ? dEABdDEC + (?1)EAEC+EBECdECAdDEB

o
+ (A7)

+2(?1)EAEC+EBECgCAffiDB ? 2gABffiDC = 0

(n+m)2?1

X

E=1

n
dEABcDEC ? cEBCdDAE ? (?1)EBECcEACdDEB
o
= 0

(v:) If KAB := K(EA; EB), where K is the Killing form of sl(njm), then

KAB = (n ? m)2gAB =

(n+m)2?1
X

C;D=1

(?1)ECcCADcDBC =

= (n ? m)2

(n ? m)2 ? 4

(n+m)2?1
X

C;D=1
(?1)ECdCADdDBC : (A8)

(vi:) Denoting by gAB the components of the matrix inverse to (gAB), then

(n+m)2?1
X

B;C=1

gBCcABC = 0

(n+m)2?1
X

B;C=1

gBCdABC = 0 : (A9)

(vii:) (n+m)2?1

X

C;D;E=1

gCDcACEcEDB = (n ? m)2ffiAB

(n+m)2?1
X

C;D;E=1

gCDcACEdEDB = 0 (A10)

(n+m)2?1

X

C;D;E=1

gCDdACEdEDB =

?
(n ? m)2 ? 4
?
ffiAB

(viii:) (n+m)2?1
X

D;E;F;G=1

(?1)EAEEgDEcFEBcGAF cCDG = 1
2(n ? m)2cCAB

(n+m)2?1
X

D;E;F;G=1

(?1)EAEEgDEcFEBcGAF dCDG = ?1
2(n ? m)2dCAB

(n+m)2?1

X

D;E;F;G=1

(?1)EAEEgDEcFEBdGAF dCDG = ?1

2

?

(n ? m)2 ? 4

?

cCAB (A11)

(n+m)2?1

X

D;E;F;G=1

(?1)EAEEgDEdFEBdGAF dCDG = 1

2

?
(n ? m)2 ? 12
?
dCAB

Proof : (i.) is a reformulation of the homogeneity of fEAg. The first line of (A8) as well as (iii.) result from KAB = 2(n ? m)Trs(EAEB). (ii.) is a consequence of Trs(adEA) = 0 and of P
B;C Trs(gBC [EB; EC]+g EA) = 0. (iv.) is a reformulation of the graded Jacobi identity and (A2). Using the second equation (A7) one deduces the second line of (A8). (vi.) follows from (ii.) and (iii.). The left hand side of the first equation (A10) is essentially the second-order Casimir operator of sl(njm) in the adjoint representation. The second part of (A10) follows from (iii.), whereas the third equation is a consequence of the first part together with (iv.) and (vi.). The relations (viii.) are results of calculations using (iii.),(iv.),(vi.) and (vii.). 2

Acknowledgment

The authors would like to thank W. Bulla, J. Madore, P. Pre<=snajder and L. Pittner for helpful discussions and the \Fonds zur Förderung der wissenschaftlichen Forschung (FWF)" for support funding.

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