Cohomological Approach to the Graded Berezinian

TIFFANYCOVOLO

Abstrat. Wedevelopthetheoryoflinearalgebraovera

( Z ₂ ) ⁿ

-ommutativealgebra(

n ∈ N

),whihinludes thewell-knownsuperlinearalgebraasaspeialase(

n = 1

^{). Examples}

ofsuhgraded-ommutativealgebrasaretheCliordalgebras,inpartiularthe quaternion

algebra

H

. Following a ohomologial approah,we introdueanalogues of the notionsof traeand determinant. Our onstrutionreduesinthe lassial ommutativease to the

oordinate-free desriptionofthe determinantbymeansoftheationofinvertiblematries

onthetopexteriorpower,andinthesuperommutativeaseitoinideswiththewell-known

ohomologialinterpretationoftheBerezinian.

MathematisSubjetClassiation(2010).16W50, 17A70,11R52,15A15,15A66,16E40.

Keywords.Gradedlinearalgebra,gradedtraeandBerezinian,Quaternions,Cliordalgebra.

Introdution

Remarkable series of algebras, suh as the algebra of quaternions and, more generally,

Cliord algebras turn out to be graded-ommutative. Originated in [1℄ and [2℄, this idea

wasdevelopedin [7 ℄and[8 ℄. Thegradinggroup inthisaseis

( Z ₂ ) ⁿ⁺¹

^,where

n

^{isthe number}

of generators,andthe graded-ommutativity reads as

a · b = (−1) ^{h e a,e bi} b · a

⁽¹⁾

where

e a, e b ∈ ( Z ₂ ) ⁿ⁺¹

^{denote the degrees of the respetive} homogeneous elements

a

^and

b

^,

and

h., .i : ( Z ₂ ) ⁿ⁺¹ × ( Z ₂ ) ⁿ⁺¹ → Z ₂

is the standard salar produt of binary

(n + 1)

^-vetors

(see Setion 1). This hoie of the graded-ommutativity has various motivations. First, it

is the intuitive extension of the well-known superalgebra, whih orresponds to this

( Z ₂ ) ⁿ

^-

ommutativityfor

n = 1

^,

h. , .i

^{beinginthisasejustlassial}multipliation. Seondly, itwas provedin[8 ℄thatsuh

( Z ₂ ) ⁿ

-ommutativityisuniversalamonggraded-ommutative algebras.

Thatis, if

Γ

^{isa nitely generated Abelian group, then for an arbitrary}

Γ

-graded-ommutative algebra

A

^with graded-ommutativity of the form

a · b = (−1) ^{β( a,e e b)} b · a

^{, with}

β : Γ × Γ → Z ₂

a bilinear symmetri map, it exists

n ∈ N

suh that

A

^is

( Z ₂ ) ⁿ

-ommutative (in the sense of (1)).

Firststepstowardsthe

( Z ₂ ) ⁿ

^{-gradedversionoflinearalgebraweredonein[3 ℄. Thenotionof}

gradedtraeforallendomorphismsandthatofgradedBerezinianfor

0

^-degree automorphisms were introdued in the most general framework of an arbitrary free module (of nite rank)

over a

( Z ₂ ) ⁿ

-ommutative algebra.

Inthispaper,we developaohomologial approah tothe notionofgradedBerezinianand

graded trae. In the super ase, this approah is originally due to O. V. Ogievetskii and I.

B. Penkov ([9 ℄), but we will mostly refer to the desription given in [6 ℄. Similarly to this

latter, we dene agraded analogue ofthe Koszul omplex and the gradedBerezinianmodule

assoiated to a given freemoduleof nite rank. We believe this to be the rst step towards

the oneptionof ageneralization ofthe Berezinian integral over a possible multi-graded (i.e.

( Z ₂ ) ⁿ

^{-graded) manifold.}

The paper is organized as follows. We reall the basi notions of graded linear algebra in

Setion1andderivethegradedmatrixalulusinSetion 2. InSetion3,wepresentourrst

main result, a ohomologial interpretation ofthe gradedBerezinian. In Setion 4,we give a

similardesriptionofthegradedtrae. Itisworthnotiingthattheohomologialdesription

of the gradedtrae of arbitrary even matries leads to interesting restritionsfor the grading

group

( Z ₂ ) ⁿ

^{, namely}

n

^{has to be odd. F}urthermore, the parity hanging operator has to be hosenin a anonialwayand orrespondsto the element

(1, 1, . . . , 1)

^of

( Z ₂ ) ⁿ

^.

Wehavetonotethatthereisanalternativeapproahto thegeneralizationofsuperalgebras

and related notions, whih makesuse of ategory theory. This approah follows from results

by Sheunert in [10 ℄(inthe Lie algebrassetting) andNekludova (inthe ommutative algebra

setting). An expliit desription of the results of the latter an be found in [5 ℄. This other

methodto treatthe problemand itsonsequenesin the

( Z ₂ ) ⁿ

-ommutative ase, will be the objetof aseparate work.

Aknowledgments. The author is pleased to thank Norbert Ponin whose suggestions

initiate the paper, Jean-Philippe Mihel for enlightening disussions and Dimitry Leiteswho

made her aware of Nekludova's work. The author is grateful to Valentin Ovsienko for the

ontributions he gavethrough the development and the nalization of the paper.

TheauthorthankstheLuxembourgianNRFforsupportviaAFRPhDgrant2010-1786207.

1. Graded linear algebra

In this setion, we give a brief survey of the main notions of linear algebra over graded-

ommutative algebras.

Consider anAbelian group

(Γ, +)

^{endowed with asymmetri}bi-additive map

h , i : Γ × Γ → Z ₂ .

We all

Γ

^{the grading group. This group admits anatural splitting}

Γ = Γ ¯ 0 ⊕ Γ ¯ 1

^{, where}

Γ ¯ 0

^is

the subgroup haraterized by

hγ, γi = 0

^{for all}

γ ∈ Γ ¯ 0

^{, andwhere}

Γ ¯ 1

^{isthe set} haraterized by

hγ, γi = 1

^{for all}

γ ∈ Γ ¯ 1

^{. We all}

Γ ¯ 0

^and

Γ ¯ 1

^{the even subgroup and odd part,} respetively.

In this paper, we restrit the onsiderations to the ase

Γ = ( Z ₂ ) ⁿ

^{, for some xed}

n ∈ N

, equipped with the standard salarprodut

hx, yi = X

1≤i≤n

x i y i

of

n

^{-vetors, dened over}

Z ₂

. Our main example arethe Cliord algebras equipped with the grading desribed in Example1 (seenext setion).

1.1. Graded-Commutative Algebras. A graded vetor spae isa diret sum

V = M

γ∈Γ

V ^γ

of vetor spaes

V ^γ

^{over a ommutative eld}

K

(that we always assume of harateristi 0).

An endomorphism of

V

^{is a}

K

-linear mapfrom

V

^to

V

^{that preservesthe degree; we denote}

by

End K (V )

^{the spae of}endomorphisms.

A

Γ

^{-graded algebra is an algebra}

A

^{whih has a struture of a}

Γ

^{-graded vetor spae}

A = L

γ∈Γ A ^γ

^{suh thatthe operation of}multipliation respets the grading:

A ^α A ^β ⊂ A ^α+β , ∀α, β ∈ Γ .

We always assume

A

^{assoiative andunital.}

An element

a ∈ A ^γ

^isalled homogeneous of degree

γ

^{. For every} homogeneous element

a

^,

we denote by

e a

^{its degree. Beause of the even-odd splitting of the grading group, one also}

has

A = A ¯ 0 ⊕ A ¯ 1 ,

where

a ∈ A ¯ 0

^or

A ¯ 1

^if

e a

^{is even or odd,} respetively. For simpliity, in most formulas below the involved elements are assumedto be homogeneous. Theseexpressions arethen extended

to arbitraryelements bylinearity.

A

Γ

^{-graded algebra isalled}

Γ

-ommutative if

a b = (−1) ^{h e a,e bi} b a.

⁽²⁾

Inpartiular, every odd element squares to zero.

Example1. Aswehavementionedinthe Introdution,aCliordalgebra

Cl _n

^of

n

^generators

(over

R

or

C

) isa

( Z ₂ ) ⁿ⁺¹

-ommutative algebra ([8℄). The gradingis givenon the generators

e _i

^of

Cl _n

^asfollows

e

e _i = (0, . . . , 0, 1, 0, . . . 0, 1) ∈ ( Z ₂ ) ⁿ⁺¹

where

1

^{isat the}

i

^{-thandat the last position.}

1.2. Graded Modules. A graded left module

M

^{over a}

Γ

-ommutative algebra

A

^{is a left}

A

^{-module with a}

Γ

^{-graded vetor spae struture}

M = L

γ∈Γ M ^γ

^{suh that the}

A

^-module

struture respets the grading,i.e.

A ^α M ^β ⊂ M ^α+β , ∀α, β ∈ Γ .

The notion of graded right module over

A

^{is dened} analogously. Thanks to the graded- ommutativityof

A

^,aleft

A

^{-modulestrutureinduesaompatible right}

A

^{-modulestruture}

given by

m a := (−1) ^{h e a, mi e} a m ,

⁽³⁾

and vie-versa. Hene,weidentify the twoonepts.

An

A

^{-moduleisalledfree oftotalrank}

r ∈ N

ifitadmitsabasisofrhomogeneouselements

{e s } s=1,...,r

^{. Inthis ase, everyelement}

m ∈ M

^{an bedeomposedin a basiseither with left}

or right oeients, whih arelearly relatedthrough(3).

A morphism of

A

^{-modules, is a map}

ℓ : M → N

^{whih is}

A

^{-linear of degree zero (i.e.}

ℓ(M ^α ) ⊂ N ^α

^,

∀α ∈ Γ

^{). We will denote the set of suh maps by}

Hom A (M, N )

^{. We usually}

refertothis setasthe ategorial

Hom

^sine

A

^{-modules withthese}degree-preserving

A

^-linear

maps forma ategoryGr

Γ

^Mod

A

^.

We remark that the

Hom

^{set of graded}

A

^{-modules is not a graded}

A

^{-module itself. The}

internal

Hom

of the ategory Gr

Γ

^Mod

A

^is

Hom _{A (M, N ) :=} ^M

γ∈Γ

Hom ^γ

A (M, N) ,

where eah

Hom ^γ

A (M, N )

^onsistsof

A

^{-linear maps}

ℓ : M → N

^{of degree}

γ

^{, thatis additive}

maps satisfying

ℓ(am) = (−1) ^{hγ, e ai} a ℓ(m)

^or equivalently,

ℓ(ma) = ℓ(m)a

(4)

and

ℓ(M ^α ) ⊂ N ^α+γ .

The

A

^{-module strutureof}

Hom _{A (M, N )}

isgiven by

(a · ℓ)(−) := (−1) ^{he ℓ, e ai} ℓ(a · −)

^or equivalently,

(ℓ · a)(−) = ℓ(a · −)

.

⁽⁵⁾

Remark 1. The ategorial

Hom

^{oinides with the}

0

^{-degree part of the internal}

Hom

, i.e.

Hom _A (M, N) = Hom ⁰ _{A (M, N )}

. By abuse of notation, we also refer to the elements of the internal

Hom

as morphisms. To make lear the distintion between ategorial and internal hom, we oftenaddthe adjetive graded in the latterase.

We dene gradedendomorphisms and gradedautomorphisms of an

A

^-module

M

^by

End _{A (M ) :=} Hom _{A (M, M)}

and

Aut _{A (M ) := {ℓ ∈} End _{A (M) : ℓ}

invertible

} ,

and theirdegree-preserving analogues by

End A (M) := End ⁰ _{A (M )}

and

Aut A (M ) := Aut ⁰ _{A (M ) .}

Insituationswhereitisnotmisleading, wewilldropthesubsriptandjustwrite

Hom _{(M, N )}

,

End _(M)

,

Aut _{(M )}

, et.

The dual of a graded

A

^-module

M

^{is the graded}

A

^-module

M ^∗ := Hom _{(M, A)}

. As for lassial modules, if

M

^{is free with basis}

{e i } _i=1,...,r

^{then its dualmodule}

M ^∗

^{is also free of}

same rank. Itsbasis

{ε ⁱ } _i=1,...,r

^{isdened asusual by}

ε ⁱ (e j ) = δ _j ⁱ , ∀ i, j

where

δ _j ⁱ

^{is the Kroneker delta. Notethatthis implies}

ε e ⁱ = e e i

^{for all}

i

^.

1.3. Lie algebras, Derivations. A

Γ

^{-olored Lie algebra}

A

^{is a}

Γ

^{-graded algebra in whih}

the multipliationoperation (denoted

[ · , · ]

^{) verifythe followingtwo onditions,for allhomo-}

geneous elements

a, b, c ∈ A

^.

1) Graded skew-symmetry:

[a, b] = −(−1) ^{h e a,e bi} [b, a] ;

2) Graded Jaobyidentity:

[a, [b, c]] = [[a, b], c] + (−1) ^{h e a,e bi} [b, [a, c]] .

Naturalexamplesof oloredLie algebrasare

Γ

^{-gradedassoiative algebraswith thegraded}

ommutator

[a, b] = ab − (−1) ^{h e a,e bi} ba .

⁽⁶⁾

If

A

^isa

( Z ₂ ) ⁿ

^{-olored Liealgebra,its}

0

^{-degree part}

A ⁰

^{is alassial Liealgebra.}

Anhomogeneousderivation ofdegree

γ

^ofa

Γ

^{-gradedalgebra}

A

^overaeld

K

,isa

K

-linear mapof degree

γ D ∈ End ^γ _{K (A)}

whih veriesthe graded Leibniz rule

D(ab) = D(a) b + (−1) ^{h e a,γi} a D(b)

for allhomogeneous elements

a, b ∈ A

^.

We denote the set of derivations of degree

γ

^of

A

^by

Der ^γ (A)

^{. Then, the set ofall graded}

derivations of

A

Der(A) := M

γ∈Γ

Der ^γ (A)

is a

Γ

^{-graded vetor spae. It is also an}

A

^{-module, with the}

A

^{-module struture given by}

(aD)(x) = a D(x)

^{. Moreover, onsidering ompositionof derivations,}

Der(A)

^{isalsoa olored}

Liealgebra for the ommutator (6).

1.4. Graded Tensor and Symmetri Algebras. Let

A

^{be a}

Γ

-ommutative algebra.

The tensor produt of two graded

A

^-modules

M

^and

N

^{an be dened asfollows. Let us}

forget,for themoment, thegradedstrutureof

A

^{, seeingitsimplyasa}non-ommutative ring, and onsider

M

^and

N

respetively as right and left modules. In this situation, the notion of tensor produt is well-known (see [4℄), and the obtained objet

M ⊗ A N

^{is a}

Z

-module.

Then, reonsidering the graded struture of the initial objets, we see that

M ⊗ _A N

^admits

an indued

Γ

^{-graded struture}

M ⊗ A N = M

γ∈Γ

(M ⊗ A N ) ^γ = M

γ∈Γ

M

α+β=γ

nX m ⊗ A n m ∈ M ^α , n ∈ N ^β o

.

⁽⁷⁾

Moreover, beause of the atual two-sided modulestruture ofboth

M

^and

N

^{, the resulting}

objet

M ⊗ A N

^{have also right and left}

A

^{-module strutures, whih are by onstrution}

ompatible (inthe sense of (3)).

As usual, tensor produt of graded

A

^{-modules an be} haraterized asa universal objet.

Alllassialresultsandonstrutionsrelatedto thetensor produt anthen betransferred to

the gradedase withoutmajor diulties.

Set

M ^⊗k = M ⊗ A . . . ⊗ A M

⁽

n

^fators

M

^,

n ≥ 1

^{) and}

M ^⊗0 := A

^{. Thegraded}

A

^-module

T _A ^• M := M

k∈ N M ^⊗k

isan assoiative graded

A

^{-algebra, alled graded tensor algebra,with} multipliation

⊗ A : M ^⊗r × M ^⊗s → M ^⊗r ⊗ A M ^⊗s ≃ M ^⊗(r+s) .

The graded

A

^-algebra

T _A ^• M

^{is in fat bi-graded: it has the lassial}

N

-grading (given by the number offatorsin

M

^{)thatweall weight,andanindued}

Γ

^{-grading (see(7)) alleddegree.}

Taking the quotient of

T _A ^• M

^{bythe ideal}

J S

^{generated bythe elementsof the form}

m ⊗ m ^′ − (−1) ^{h m, e m f ′ i} m ^′ ⊗ m, ∀m, m ^′ ∈ M

we obtain a

Γ

-ommutative

A

^-algebra

S _A ^• (M )

^{alled the graded symmetri algebra.}

Astheir lassialanalogues, boththese notions satisfyuniversalproperties.

1.5. Change of Parity Funtors. Unlike the lassial super ase (i.e.

Γ = Z ₂

), in general we havemany dierent parity reversion funtors.

Forevery

π ∈ Γ ¯ 1

^{,wespeifyanendofuntoroftheategoryofmodulesovera}

Γ

-ommutative algebra

A

Π :

^Gr

_Γ

^Mod

A −→

^Gr

_Γ

^Mod

A

M 7→ ΠM

Hom _A (M, N ) ∋ f 7→ f ^Π ∈ Hom _A (ΠM, ΠN)

Theobjet

ΠM

^{isdened by}

(ΠM) ^α := M ^α+π , ∀α ∈ Γ ,

and graded

A

^{-modulestruture}

Π(m + m ^′ ) := Πm + Πm ^′

^and

Π(am) := (−1) ^{hπ, e ai} a Πm .

Themorphism

f ^Π ∈ Hom A (ΠM, ΠN )

^{is dened by}

f ^Π (Πm) := Π (f (m)) .

Clearly, the map

Π

^{whih sendsan}

A

^-module

M

^{to the}

A

^-module

ΠM

^{is an}

A

^{-linear map}

of degree

π

^{, i.e.}

Π ∈ Hom ^π _{(M, ΠM )}

.

2. Graded Matrix Calulus

A graded morphism

t : M → N

^{of free}

A

^{-modules of total rank}

r

^and

s

respetively, an berepresented bya matrix over

A

^{. Fixinga basis}

{e i } _i=1,...,r

^of

M

^and

{h j } _j=1,...,s

^of

N

^and

onsidering elements ofthe modules asolumn vetorsof right oordinates

M ∋ m = X

e _j a ^j ≃ m =



  a ¹ a ²

.

.

.



  ,

the gradedmorphism is dened bythe images ofthe basisvetors

t(m) = X

j

t(e _i ) a ⁱ = X

i,j

h _j t ^j _i a ⁱ

Hene, applying

t

^{orrespondsto left}multipliation bythe matrix

T = (t ^j _i ) ∈ M _{(s × r; A)}

,

t(m) ≃ T m .

We have asimilardesriptionwhenonsidering elementsofthe modules asrowvetorsofleft

oordinates. In this paper, we hoose the rst approah. This hoie is justiedby the fat

thatthegradedmorphismsareeasiertohandlewhenoneonsidertherightmodulestruture,

see(4).

Inwhatfollows,thegradedmodulesareimpliitlyassumedtobefree,exeptwhenexpliitly

stated.

2.1. Case

Γ = ( Z ₂ ) ⁿ

^.

( Z ₂ ) ⁿ

^{isaadditive groupofniteorder}

N := 2 ⁿ

^{andwean enumerate}

its elementsfollowing the standard order: the rst

q := 2 ⁿ⁻¹

^{elements being the even degrees}

ordered by lexiographial order, and the last ones being the remaining odd degrees, also

orderedlexiographially. E.g.

( Z ₂ ) ² = {(0, 0), (1, 1), (0, 1), (1, 0) }

^{. Inthe following,wedenote}

γ _i

^the

i

^{-thelement of}

( Z ₂ ) ⁿ

^{with respetto this standardorder.}

Thisallows to re-orderthe basisof the onsidered graded

A

^{-modules following the degrees}

of the elements. We all a basisordered in this waya standard basis. From now on, we only

onsiderthis type ofbasis.

The rank of a free gradedmodule

M

^{over a}

( Z ₂ ) ⁿ

-ommutative algebra isthen a

N

^-tuple

r = (r ₁ , . . . , r N ) ∈ N ^N

where eah

r i

^{is the number of basiselements of degree}

γ i

^{. Hene, a}

standard basis

{e _j } _j=1,...,r

^{of a graded}

A

^{-module of rank}

r

is suh that the rst

r ₁

^elements

areof degree

γ ₁

^{, the following}

r ₂

^{elements areof degree}

γ ₂

^{, et.}

Consequently, the matrix orresponding to an homogeneous gradedmorphism

t : M → N

of free

A

^{-modules of ranks}

r

and

s

respetively, writes asa blokmatrix

T =



 

 

 

T 11 . . . T 1N

. . . . . . . . . T _N1 . . . T N N



 

 

 

,

⁽⁸⁾

where eah blok

T uv

^{(of dimension}

s u × r v

^{) have} homogeneous entries of the same degree.

This latter is given by

γ u + γ v + t e

^{, i.e. it depends on both the position}

(u, v)

^{of the blok}

and the degree of the matrix, whih is by denition the degree of the orresponding graded

morphism

t

^{. We will denoteby}

M ₍ s , r ; A) = L

γ∈( Z ₂ ) ⁿ M ^γ ₍ s , r ; A)

^{the spae ofsuh matries,}

also alled graded matries.

Example2. AsapartiularaseofCliordalgebras,the algebraofquaternions

H

isa

( Z ₂ ) ³

^-

ommutative algebra ([7℄). We assign to the generators

1

^,

i

^,

j

^and

k

^of

H

a degree following the standard orderof

( Z ₂ ) ³

^{, i.e.}

e 1 := (0, 0, 0) , e i := (0, 1, 1) , e j := (1, 0, 1)

^and

e k := (1, 1, 0) .

Note that, as every other Cliord algebra (with the gradation given as in Example 1), it is

purelyeven i.e. graded onlybythe even subgroup of

( Z ₂ ) ⁿ

^.

Everygraded endomorphismofa free

H

-module

M

^ofrank

r

then orrespondstoa matrix in

M ₍ r ; H )

^{, that isa}quaternioni matrix.

The

1

^-to-

1

orrespondene

M ^γ ₍ s , r ; A) ≃ Hom ^γ _{(M, N) , ∀γ ∈ (} Z ₂ ) ⁿ ,

permits to transfer the assoiative graded

A

^{-algebra struture of}

Hom _{(M, N )}

to

M ₍ s , r ; A)

^.

Thus,wegettheusualsumandmultipliation ofmatries,aswellasanunusualmultipliation

of matriesbysalars in

A

^{dened asfollows.}

aT =



 

 

 

(−1) ^{h e a,γ 1 i} aT ₁₁ . . . (−1) ^{h e a,γ 1 i} aT _1N . . . . . . . . . (−1) ^{h e a,γ N i} aT N 1 . . . (−1) ^{h e a,γ N i} aT N N



 

 

  ,

i.e. the

(i, j)

^{-th entry lyingin the}

(u, v)

^{-blok isgivenby}

(aT ) ⁱ _j = (−1) ^{h e a,γ u i} a t ⁱ _j .

Notethat the signappearing hereisa diret onsequene of(5).

Inpartiular, gradedendomorphisms

End _{(M )}

ofa freegraded

A

^-module

M

^{(of niterank}

r

) an be seen assquaregraded matries

M ₍ r ; A) := M ₍ r , r ; A)

^{. Withthe ommutator (6)it}

isanother example of

( Z ₂ ) ⁿ

^{-olored Liealgebra.}

2.2. Graded Transpose. Asintheprevioussetion, let

M

^and

N

^betwograded

A

^-modules

of ranks

r

and

s

respetively.

Thegradedtranspose

Γt T

^ofamatrix

T ∈ M ₍ s , r ; A)

^{, thatorrespondsto}

t ∈ Hom _{(M, N )}

, isdenedasthematrixorrespondingtothetranspose

t ^∗ ∈ Hom _(N ^∗ _{, M} ^∗ ₎

of

t

^{. Forsimpliity,}

we suppose

t

^tobehomogeneous of degree

e t

^.

We reall that the dual graded

A

^{-module of}

M

^is

M ^∗ := Hom _{(M, A)}

, so that the dual morphism

t ^∗

^{is naturally dened,for all}

n ^∗ ∈ N ^∗

^{and all}

m ∈ M

^,by

(t ^∗ (n ^∗ ), m) = (−1) ^{he t,f n ∗ i} (n ^∗ , t(m))

⁽⁹⁾

where

(−, −)

^{denotes the evaluationof the involved morphisms onthe} orrespondingsoure- moduleelement.

Let

{e k } _k=1,...,r

^(resp.

{h l } _l=1,...,s

^{) be the basis of}

M

^(resp.

N

^{) and let}

{ε ^k } _k=1,...,r

^(resp.

{η ^l } _l=1,...,s

^{) the} orrespondingdual basis.

Proposition 1. The graded transpose

Γt T

^{of a matrix}

T = (T uv ) ∈ M ^{e t} ₍ s , r ; A)

(onsidering here its blok form (8)) isgiven by

Γt T

vu = (−1) ^{hγ u +γ v ,e t+γ v i t} T uv .

where

t

isthe lassial transpose.

Proof. From the denition(9)of

t ^∗

^{, we havethat}

t ^∗ (η ^j ), e _i

= (−1) ^{he t,f h j i} (η ^j , t(e _i )) = (−1) ^{he t,f h j i} X

k

(η ^j , h _k )t ^k _i = (−1) ^{he t,f h j i} t ^j _i .

Onthe other hand,denoting

t ^∗ _i ^j

^the

(i, j)

^{-entry of the matrix}

^Γt T

^{, we have}

t ^∗ (η ^j ), e i

= X

k

(ε ^k t ^∗ _k ^j , e i ) = X

k

(−1) ^{h t f ∗ k j , e e i i} (ε ^k , e i )t _k ^{∗ j} = (−1) ^{h t f ∗ i j , e e i i} t ^∗ _i ^j .

Thus, for all

i, j

^,

t ^∗ _i ^j = (−1) ^{h t f ∗ i j , e e i i+he t,f h j i} t ^j _i

⁽¹⁰⁾

In partiular,

t f ^∗ _i ^j = t e ^j _i = e t + e e i + h e j

^{for all}

i, j

^{, so that the transpose morphism}

t ^∗

^{is also}

homogeneous ofsame degree

e t

^{. Hene, (10)rewrites as}

t ^∗ _i ^j = (−1) ^{he t+ e e i ,f h j + e e i i} t ^j _i

and the resultfollows.

In the super ase (i.e.

n = 1

^{), the graded transpose oinide with the well-known super}

transpose.

Itiseasilyveriedbystraightforwardomputations,thattheoperationofgraded-transposition

satisfythe followingfamiliar property.

Corollary 1. For anypair ofhomogeneous square graded matries

S, T ∈ M ₍ r ; A)

^{,of degrees}

s

^and

t

respetively, we have that

Γt (ST ) = (−1) ^{hs,ti Γt} T ^Γt S .

Consequently, we also have that

[ ^Γt S, ^Γt T ] = − ^Γt [S, T ] .

⁽¹¹⁾

2.3. GradedTrae. Asinthelassialontext,foranygraded

A

^-module

M

^{thereisanatural}

isomorphism of

A

^-modules

M ^∗ ⊗ A M ≃ End _{(M ) .}

(12)

It isgiven byreading atensor

α ⊗ m ∈ M ^∗ ⊗ A M

^asthe endomorphism

α ⊗ m : M ∋ m ^′ 7→ (−1) ^{h m, e m f ′ i} α(m ^′ )m ∈ M

In the ase where

M

^{is free (of rank}

r

), this endomorphism is represented by a matrix

T = (t ⁱ _j ) ∈ M ₍ r ; A)

^{, where the}

(i, j)

^{-th entry is}

t ⁱ _j = (−1) ^{h m,e e j i+h α f j , e e j + e e i i} α j m ⁱ

The above isomorphism permits to dene the graded trae of the matrix orresponding to

the endomorphism

α ⊗ m

^{asitsontration}

α(m)

^(asa

(1, 1)

^-tensor).

Denition1. Thegraded trae ofanhomogeneousmatrix

T = (T _uv ) ∈ M ^{e t} ₍ r ; A)

(onsidering hereits blokform(8)) isdened as

Γtr(T) := X

u

(−1) ^{hγ u +e t,γ u i} tr(T uu )

⁽¹³⁾

where

tr

^{denotes the lassi trae ofa matrix.}

Itis proved in [3℄that

Γtr : M ₍ r ; A) → A

^{isthe unique (upto}multipliation by a salarof degree

0

⁾ homomorphism of

A

^{-linear olored Lie algebras.}

2.4. Graded Berezinian. Fixinga standard basis

{e i } _i=1,...,r

^{of afree graded}

A

^-module

M

permits to represent degree-preserving automorphisms of this module as invertible

0

^-degree

matries, thegroup of whihwe denoteby

GL ⁰ ₍ r ; A)

^{. In[3℄we have introdued the notionof}

Graded Berezinian for this typeof matries. Letus reall the main resultof[3 ℄.

There is a unique group homomorphism

ΓBer : GL ⁰ ₍ r ; A) → (A ⁰ ) ^×

suh that:

1) For every blok-diagonal matrix

X ∈ GL ⁰ ₍ r ; A)

^,

ΓBer(X) =

Y q u=1

det(X uu ) · Y N u=q+1

det ^-1 (X uu ) .

2) The image of any lower

(

^{resp., upper}

)

blok-unitriangular matrix in

GL ⁰ ₍ r ; A)

^equals

1 ∈ (A ⁰ ) ^×

^.

Here,

(A ⁰ ) ^×

^{denotestheinvertibleelementsofthe}

0

^{-degreepartof}

A

^,and

q := N/2 = 2 ⁿ⁻¹

^.

Notethat, similarlyto the lassial Berezinian,

ΓBer

^{isdened only for}

0

^{-degree invertible}

matries. In the partiular ase where the module

M

^{is graded by the even part}

( Z ₂ ) ⁿ _{¯ 0}

^{, the}

funtion

ΓBer

^{is a} polynomial. Moreover, if

A = H

then

ΓBer

^{oinides with the lassial}

Diedonné determinant (restritedto these type ofquaternioni matries), see[3℄.

3. Cohomologial Definition of the Graded Berezinian

Inthissetion,wedenetheGradedBerezinianmoduleanddesribeitsohomologialinter-

pretation. Weobtainthefuntion

ΓBer

^{fromtheationofthegroupof}degree-preservingauto-

3.1. Graded Berezinian Module. Given a free

A

^-module

M

^{, the} orresponding graded Berezinian module

ΓBer(M )

^{is the free}

A

^{-module of total rank}

1

^{built up from formal basis}

elements

B({e i })

^{for eah standard basis}

{e i } _i=1,...,r

^of

M

^{. The} transformation law indued by ahange ofbasis

e ^′ _i = e j t ^j _i

ⁱⁿ

M

^{(of transitionmatrix}

T = (t ^j _i )

^{)is given by}

B({e ^′ _i }) = B({e i }) ΓBer(T ).

⁽¹⁴⁾

Hene, intuitively,

ΓBer(M)

^{is the free}

A

^{-module of total rank}

1

^{whih is funtorial with}

respetto

0

-degree automorphisms of

A

^{-modules and if}

M

^{is onentrated in only one even}

degree (i.e. is just a lassial module) it oinides with the lassial determinant module

Det(M) := V top M

^.

Theabovedesriptionofthe module

ΓBer(M )

^{isquiteabstrat. Inthetextsetion, wewill}

present anexpliit ohomologialonstrution ofthis module.

3.2. Cohomologial Constrution. Consider the graded-ommutative algebra

S _A ^• (ΠM ⊕ M ^∗ )

^{, where}

M

^{isa (free) graded}

A

^{-moduleof rank}

r

.

S _A ^• (ΠM ⊕ M ^∗ )

^isthe

( Z ₂ ) ⁿ

-graded-ommutative algebra ofpolynomials in thegradedvari- ables

Πe i

^and

ε ⁱ

^{with oeients in}

A

^{. We dene the operator}

d

^{to be left}multipliation by the following element of

S _A ² (ΠM ⊕ M ^∗ )

^:

d = X

i

Πe i ε ⁱ ,

⁽¹⁵⁾

thatwe also denoteby

d

^{, byabuseof notation. Thishoie of}

d

^{is naturalsine}

d| M ≡ Π

^.

Thisdenesa ohain omplex

K ^• := S _A ^• (ΠM ⊕ M ^∗ ), d

,thanksto the following result.

Proposition 2. The operator

d

^is independent of the hoie of the basis and

d ² = 0

^.

Proof. Letusonsideratransformationmatrix

T = (t ⁱ _j ) ∈ GL ⁰ ₍ r ; A)

^fromthebasis

{e i } i=1,...,r

to the basis

{e ^′ _i } _i=1,...,r

^{(both standard,}

e e ^′ i = e e i ∀i

^{), i.e.}

e ^′ _i = P

j e j t ^j _i .

^The transformation matrix between the indued basis

{Πe _i } _i=1,...,r

^and

{Πe ^′ _i } _i=1,...,r

^of

ΠM

^{(resp. between the}

dualbasis

{ε ⁱ } _i=1,...,r

^and

{ε ^′i } _i=1,...,r

^of

M ^∗

^{)is then}

T

^(resp.

^Γt (T ⁻¹ )

^{), i.e.}

Πe ^′ _i = X

j

Πe j t ^j _i

^and

ε ^′i = X

k

ε ^k t ^∗ _k ⁱ = X

k

ε ^k (−1) ^{h e e k + e e i , e e i i} t ⁱ _k

(16)

where

t ⁱ _k

denotes the

(i, k)

^{-th entryof}

T ⁻¹

^.

Hene, we have

X

i

Πe ^′ _i ε ^′i = X

i,j,k

Πe j t ^j _i ε ^k (−1) ^{h e e i + e e k , e e k i} t ⁱ _k

= X

j,k

Πe j ε ^k X

i

(−1) ^{h e e i + e e k , e e k i+h e e j + e e i , e e k i} t ^j _i t ⁱ _k

!

= X

j,k

Πe j ε ^k (−1) ^{h e e k , e e k + e e j i} X

i

t ^j _i t ⁱ _k

!

= X

j,k

Πe j ε ^k (−1) ^{h e e k , e e k + e e j i} δ ^j _k = X

j

Πe j ε ^j

sothat

d

^is well-dened.

The fatthat

d

^{squares to zero is easily heked bydiret} omputation, using the graded-

ommutativity.

To lighten the notation, we will denote by

x i

^{the even elements in}

{Πe i } ∪ {ε ⁱ }

^{and by}

ξ i

the odd ones (up to sign). Morepreisely, with

r ^′ := P q

i=1 r i

^{indiating the number of basis}

elementsof

M

^{of even degree, we set}

x _i :=

( ε ⁱ

^if

1 ≤ i ≤ r ^′ Πe i

^if

r ^′ + 1 ≤ i ≤ r

and

ξ _i :=

( Πe _i

^if

1 ≤ i ≤ r ^′

−(−1) ^{h e e i ,πi} ε ⁱ

^if

r ^′ + 1 ≤ i ≤ r

(17)

By onstrution,we still havethat

ξ e i = e x i + π

^forall

i

^.

With this notation, the dierential

d

^{orresponds to left} multipliation by

P

i ξ i x i

^{, and}

S _A ^• (ΠM ⊕ M ^∗ )

^{isnowviewed asthe}

( Z ₂ ) ⁿ

-ommutative algebra

A[x, ξ]

^ofpolynomials in the

( Z ₂ ) ⁿ

^{-graded variables}

x

^{-s and}

ξ

^{-s. Let us stress the fatthat the produt of} polynomials is herethe one whih isnaturally indued bythe graded signrule (2).

Theelementsoftheohainomplex

K ^•

^at

k

^-thlevel(

k ≥ 0

^{)aredenedasthe}polynomials in

A[x, ξ]

^with

k

^{-thtotal powerdegreein}

ξ

^{. Theelement}

ξ ₁ · · · ξ r

^{isaoyle. Indeed,wehave}

that

d(ξ ₁ · · · ξ r ) = X

i

(−1) h ^{ξ e} i , e x i + P

k<i ξ e k i x i ξ ₁ · · · ξ _i−1 ξ _i ² ξ _i+1 · · · ξ r = 0

sine

ξ i

^{are odd and hene squares to}

0

^{. By this same} observation,

K ^k = 0

^{for all}

k > r

^.

Hene, this oyle willplay the analogous roleof the lassial topelement.

Proposition 3.

H ^k (K ^• ) =

( 0

^if

k 6= r [ξ 1 · · ·ξ r ] · A

^if

k = r

Proof. Let us onsider the operator

ρ = P

i ∂

∂x i

∂

∂ξ i

, where

∂

∂x i , _∂ξ ^∂

i

are graded homogeneous

partialderivations, i.e.

∂

∂x i x j = δ _j ⁱ , _∂x ^∂

i ξ j = 0

∂

∂ξ i x j = 0 , _∂ξ ^∂

i ξ j = δ _j ⁱ .

for allindies

i, j

^{. Notethat the respetivedegrees are}

g ∂

∂x _i = x e i

^and

g ∂

∂ξ _i = ξ e i = x e i + π .

Letus ompute

[ρ , d]

^where

[ . , . ]

^{is the}

( Z ₂ ) ⁿ

^{-ommutator (6).}

[ρ , d] = X

i , j

∂

∂x _i

∂

∂ξ _i , ξ j x j

= X

i , j

∂

∂x i

∂

∂ξ i

, ξ j x j

+ X

i , j

(−1) ^{h e x i +π , πi} ∂

∂x i

ξ j x j

∂

∂ξ i

= X

i , j

∂

∂x i

∂

∂ξ i

, ξ _j

x _j + X

i , j

(−1) ^{h e x i +π , x e j +πi} ∂

∂x i

ξ _j ∂

∂ξ i

, x _j

− X

i , j

(−1) ^{h e x i , πi} ∂

∂x i

, ξ j

x j

∂

∂ξ i

− X

i , j

(−1) ^{h e x i , πi+h x e i , e x j +πi} ξ j

∂

∂x i

, x j

∂

∂ξ i

But, byonstrution,

h ∂

∂x i , x j

i = δ ⁱ _j , h

∂

∂x i , ξ j

i = 0 ,

h ∂

∂ξ _i , x j

i = 0 , h

∂

∂ξ _i , ξ j

i = δ _j ⁱ ,

sothatwe have

[ρ , d] = X

i

∂

∂x i

x _i − X

i

(−1) ^{h x e i , x e i i} ξ _i ∂

∂ξ i

= X

i

id +(−1) ^{h e x i , e x i i} x i

∂

∂x i

− X

i

(−1) ^{h x e i , x e i i} ξ i

∂

∂ξ i

= r id + X

i

x i

∂

∂x i

− X

i

ξ i

∂

∂ξ i

sine eah

x i

^{have even degree, i.e.}

h x e i , e x i i = 0

^{for all}

i

^.

Now,if

P

^isa homogeneousmonomial in

K ^k

^{, i.e.}

P = ξ ^α x ^β a αβ

with

α ∈ {0, 1} ^r

^{suh that}

|α| := P

i α i = k

^,

β ∈ N ^r

and

a _αβ

^a homogeneous element of

A

^,

we havefor every

i

^that

x i

∂

∂x i

(P ) =

( β i P

^if

β i 6= 0 0

^if

β _i = 0

and

ξ i

∂

∂ξ i

(P ) =

( P

^if

α _i = 1 0

^if

α i = 0

sothat

[ρ , d] (P) = (r + |β| − k) id(P ) .

Infat,we onlyhaveto onsider

0 ≤ k ≤ r

^{. Hene,}

c := r + |β| − k

^{iszeroifand onlyif}

k = r

and

β = 0

^{. It follows that, for}

k 6= r

^{, we have a ohain homotopy between the identity}

id

and the zeromap. Itis givenby

ρ/c

^{on monomialsof the same formof}

P

^{. Weonlude that}

H ^k (K ^• ) = 0

^forall

k 6= r

^.

Itremains to onsiderthe ase when

k = r

^{. Bydenition,}

H ^r (K ^• ) = ker(d : K ^r → K ^r+1 )/ im(d : K ^r−1 → K ^r )

where

K ^r = {ξ ₁ · · ·ξ r Q | Q ∈ A [x]}

^and

K ^r+1 = 0

^{. Hene,}

ker(d : K ^r → K ^r+1 ) = K ^r

^{. On the}

otherhand,bydiretomputation(e.g. apply

d

^{onanelementoftheform}

ξ _{α 1} · · · ξ _{α r− 1} Q _α

^with

Q α ∈ A[x]

^{a homogenous monomial), we obtain}

im(d : K ^r−1 → K ^r ) = ξ ₁ · · ·ξ r ( P

i A [x] · x i )

^.

Inonlusion,

H ^r (K ^• ) = ξ ₁ · · ·ξ r · A [x] /ξ 1 · · ·ξ r ( P

i A [x] · x i ) ≃ ξ ₁ · · ·ξ r ·A .

Remark 2. a) Proposition 3 implies that

H(K ^• ) = H ^r (K ^• ) ≃ [ξ ₁ · · ·ξ r ] · A

^{, hene is a free}

A

^{-moduleof total rank}

1

^{. Thedegree of the basiselement is}

ξ ^ ₁ · · ·ξ r = X q i=1

(γ i + π)r i + X N i=q+1

γ i r i = r ^′ π + X N

i=1

γ i r i .

b) Theresultis independent ofthe hosen parity funtor

Π

^{in the ohainomplex}

K ^•

^.

A degree-preserving automorphism of

M

^,

ϕ ∈ Aut ⁰ _{(M )}

(represented by a matrix

T ∈ GL ⁰ ₍ r ; A)

^{), naturally indues two}automorphisms

ϕ ^Π : ΠM → ΠM

^and

(ϕ ⁻¹ ) ^∗ : M ^∗ → M ^∗ ,

and heneanautomorphism (of

A

^-algebras)

φ : S ^• _A (ΠM ⊕ M ^∗ ) → S ^• _A (ΠM ⊕ M ^∗ ) .

on the totalspae

K := L

k K ^k = S _A ^• (ΠM ⊕ M ^∗ )

^{of the} orresponding omplex. Expliitly, itis given by

φ(x i ) =

( (ϕ ^∗ ) ⁻¹ (x _i ) 1 ≤ i ≤ r ^′

ϕ ^Π (x i ) r ^′ < i ≤ r

^and

φ(ξ i ) =

( ϕ ^Π (ξ _i ) 1 ≤ i ≤ r ^′

(ϕ ^∗ ) ⁻¹ (ξ i ) r ^′ < i ≤ r

⁽¹⁸⁾

i.e. orrespondsto matrix multipliation by

T

^on

ΠM

^{and by}

^Γt (T ⁻¹ )

^on

M ^∗

^.

Thedierential

d : K → K

^{isinvariantunderthis}transformation(seeProposition2). Hene, we obtain anautomorphism on

ker d/ im d

^{. Thislattermodule, is equalto}

ker d/ im d = L

k ker d| _K ^k / L

k im d| _K k−1

= M

k

ker d| _K ^k / im d| _K k−1

= M

k

H ^k (K ^• )

and heneisjust

H ^r (K ^• )

^{, thanksto} Proposition 3.

By means of a graded matrix

T ∈ GL ⁰ ₍ r ; A)

representing

ϕ

^{, the obtained map rewrites as}

a group ationof

GL ⁰ ₍ r ; A)

^on

H ^r (K ^• )

^{. In otherwords, wehave agroup morphism}

Φ : GL ⁰ ₍ r ; A) → Aut ⁰ _(H ^r _(K ^• _{)) ≃ (A} ⁰ ₎ ^×

(19) given by

Φ(T ) ([ξ ₁ · · · ξ r ]) = [φ(ξ ₁ · · · ξ r )] .

We will now provethat thismorphism oinides with the Graded Berezinian.

Proposition4. Forall

T ∈ GL ⁰ ₍ r ; A)

^,

Φ(T)

^{istheoperatorofright}multipliationby

ΓBer(T )

^.

Proof. Wewillrstexpliitindetailthe proofforthe superase(i.e. withgrading group

Z ₂

), followingthedesriptiongivenin[6℄. Hene,inthisasethegradedBerezinianandthegraded

trae redueto the lassial Berezinianandthe supertrae.

Letus onsidertwo partiular typesof transformations

ϕ

^.

1) Let

ϕ

^{be a diagonal} transformation, i.e. the orresponding graded matrix is blok- diagonal.

T =

A 0 0 B

∈ GL ⁰ r = (r ^′ , r ^′′ ); A

Thematrix orresponding to the inverse dualisalso blok-diagonal

st (T ⁻¹ ) = _t

A ⁻¹ 0 0 ^t B ⁻¹

∈ GL ⁰ ₍ r ; A)

Letus denote

a ⁱ _j

^{the entries of}

A

^and

b _j ⁱ

the entriesof

t B ⁻¹

^.

We havethat

φ(ξ i ) =

 



ϕ ^Π (ξ _i ) = X

1≤j≤r ^′ ξ _j a ^j _i

^for

1 ≤ i ≤ r ^′ (ϕ ^∗ ) ⁻¹ (ξ i ) = X

r ^′ <j≤r ξ j b ^{i−r ′}

j−r ^′

^for

r ^′ < i ≤ r

sothat

φ(ξ ₁ · · · ξ r ) = ϕ ^Π (ξ ₁ ) · · · ϕ ^Π (ξ r ^′ ) · (ϕ ^∗ ) ⁻¹ (ξ r ^′ +1 ) · · · (ϕ ^∗ ) ⁻¹ (ξ r )

= ξ ₁ · · · ξ r

X

σ∈ S _{r ′}

sign σ a ^σ(1) ₁ · · ·a ^σ(r _r ′ ^{′ )}

! X

σ∈ S _{r−r ′}

sign σ b ¹

σ(1) · · · b ^{r−r ′}

σ(r−r ^′ )

!

= ξ ₁ · · · ξ r det(A) det(B ⁻¹ )

= ξ ₁ · · · ξ _r Ber(T ) .

Here,signsappearsbeausetheelementsofthesubset

{ξ i } _1≤i≤r ^′

(respetively,

{ξ i } r ^′ <i≤r

⁾

areof the sameodddegree, heneantiommute.

2) Let

ϕ

^beaunitriangulartransformation,i.e. theorrespondinggradedmatrixisblok- unitriangular. Thevalueof itsBerezinian thenequals

1

^{.Wewill onsider onlythe ase}

ofanupperunitriangularmatrix,theaseofalower unitriangularmatrixbeingsimilar.

Let

T =

I C 0 I

∈ GL ⁰ ₍ r ; A).

Thenthe orrespondingdualinverse isalso blok-unitriangular, more preisely

st (T ⁻¹ ) =

I 0

t C I

.

We havein thisase

φ(ξ i ) =

( ϕ ^Π (ξ i ) = ξ i

^for

1 ≤ i ≤ r ^′ (ϕ ^∗ ) ⁻¹ (ξ i ) = ξ i

^for

r ^′ < i ≤ r

sothat

φ(ξ ₁ · · · ξ r ) = ξ ₁ · · · ξ r = ξ ₁ · · · ξ r Ber(T ).

Hene, we have proved that

Φ

^{oinide with left} multipliation by

Ber

^{on blok diagonal}

and blok unitriangular matries. By the uniqueness result onerning the Berezinian, this

suesto onlude.

Thisstrategyofproofgeneralizestotheaseofgradinggroup

( Z ₂ ) ⁿ

^forhigher

n ∈ N

,thanks to the analogous uniqueness resultof the graded Berezinian(see setion 2.4). We heneonly

haveto verify1. and 2. in this multigraded ase.

Let

M

^{be, as usual, a free module of rank}

r = (r ₁ , r ₂ , . . . , r _N )

^{. The odd elements}

ξ _i

^are

byonstrution ordered bydegree, sothatin eahsubset

{ξ i } ^P

α<γ r α <i≤ P

α≤γ r α

^{the elements}

are of the same odd degree, hene they antiommute. This impliesthat in the rst step the

expetedsigns(andhenethedeterminants)appears,asinthe superase. Intheseondstep,

by denition of the graded transpose (see setion 2), we still have thatif

T

^{is a blokupper}

(resp. lower)unitriangularmatrixthen

Γt (T ⁻¹ )

^{islower(resp. upper)blok}unitriangular. Let us onsider, for simpliity,

n = 2

^and

r = (1, 1, 1, 1)

^{. We then have only four odd}

ξ _i

^{, of two}

dierent degrees

(0, 1)

^and

(1, 0)

^{. For a gradedmatrix}

T =



 



1 a ⋆ ⋆ 1 ⋆ ⋆ 1 b 1



 

 ∈ GL ⁰ _{((1, 1, 1, 1); A)}

we have

Γt (T ⁻¹ ) =



 

 1

−a 1

⋆ ^′ ⋆ ^′ 1

⋆ ^′ ⋆ ^′ b 1



 

