HIGHER TRACE AND BEREZINIAN

OF

MATRICES OVER A CLIFFORD ALGEBRA

TIFFANY COVOLO

VALENTIN OVSIENKO

NORBERT PONCIN

Abstract.We deﬁne the notions of trace, determinant and, more generally, Berezinian of

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matrices over a (Z2) -gradedcommutative associative algebraA. Theapplications include a new

approach to the classical theory of matrices with coeﬃcients in a Cliﬀord algebra, in particular

of quaternionic matrices.In a special case, we recover the classical Dieudonn´ determinant of

quaternionic matrices, but in general our quaternionic determinant is diﬀerent.We show that

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the graded determinant of purely even (Z2) -gradedmatrices of degree 0 is polynomial in its

entries. Inthe case of the algebraA=Hof quaternions, we calculate the formula for the

Berezinian in terms of a product of quasiminors in the sense of Gelfand, Retakh, and Wilson.

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The graded trace is related to the graded Berezinian (and determinant) by a (Z2) -graded

version of Liouville’s formula.

Contents

1. Introduction

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2. (Z2) -GradedAlgebra

2.1. GeneralNotions

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2.2. (Z2and () -Z2on Cliﬀord Algebras) -Grading

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3. (Z2) -GradedTrace

3.1. FundamentalTheorem and Explicit Formula

3.2. Application:Lax Pairs

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4. (Z2Determinant of Purely Even Matrices of Degree 0) -Graded

4.1. Statementof the Fundamental Theorem

4.2. Preliminaries

4.3. ExplicitFormula in Terms of Quasideterminants

4.4. PolynomialStructure

arXiv:114.50.9.E5xa8m7pl7eSep 2011v1 [math.DG] 27

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5. (Z2) -GradedBerezinian of Invertible Graded Matrices of Degree 0

5.1. Statementof the Fundamental Theorem

5.2. ExplicitExpression

6. LiouvilleFormula

6.1. ClassicalLiouville Formulas

6.2. GradedLiouville Formula

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7. (Z2Determinant over Quaternions and Cliﬀord Algebras) -Graded

7.1. Relationto the Dieudonn´ Determinant

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2010Mathematics Subject Classiﬁcation.17A70, 58J52, 58A50, 15A66, 11R52.

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Key words and phrases.Cliﬀord linear algebra, quaternionic determinants, (Z2) -gradedcommutative algebra.

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TIFFANY COVOLO

VALENTIN OVSIENKO

NORBERT PONCIN

7.2. GradedDeterminant of Even Homogeneous Matrices of Arbitrary Degree

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8. Examplesof Quaternionic (Z2) -GradedDeterminants

8.1. QuaternionicMatrices of Degree Zero

8.2. HomogeneousQuaternionic Matrices of Nonzero Degrees

References

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1.Introduction

Linear algebra over quaternions is a classical subject.Initiated by Hamilton and Cayley, it

was further developed by Study [Stu20] and Dieudonn´ [Die71], see [Asl96] for a survey.The best

known version of quaternionic determinant is due to Dieudonn´, it is far of being elementary and

still attracts a considerable interest, see [GRW03].The Dieudonn´ determinant is not related to

any notion of trace.To the best of our knowledge, the concept of trace is missing in the existing

theories of quaternionic matrices.

The main diﬃculty of any theory of matrices over quaternions, and more generally over

Cliﬀord algebras, is related to the fact that these algebras are not commutative.It turns out

however, that the classical algebraHof quaternions can be understood as a graded-commutative

algebra. Itwas shown in [Lyc95], [AM99], [AM02] thatHis a graded commutative algebra over

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the Abelian group (Z2) =Z2×Z2(or over the even part of (Z2see [MGO09]).Quite similarly,) ,

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every Cliﬀord algebra withngenerators is (Z2) -gradedcommutative [AM02] (furthermore, a

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Cliﬀord algebra is understood as even (Z2Thiscommutative algebra in [MGO10]).) -graded

viewpoint suggests a natural approach to linear algebra over Cliﬀord algebras as generalized

Superalgebra.

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Geometric motivations to consider (Z2) -gradingscome from the study of higher vector

bundles [GR07].IfEdenotes a vector bundle with coordinates (x, ξ), a kind of universal Legendre

transform

∗ ∗∗

T E∋(x, ξ, y, η)↔(x, η, y,−ξ)∈T E

2∗

provides a natural and rich (Z2) -degree ((0,0),(1,0),(1,1),(0,1)) onT[1]E[1]. Multigraded

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vector bundles give prototypical examples of (Z2manifolds.) -graded

Quite a number of geometric structures can be encoded in supercommutative algebraic

structures, see e.g.[GKP11b], [GKP10a], [GKP10b], [GP04].On the other hand, supercommutative

algebras deﬁne supercommutative geometric spaces.It turns out, however, that the

classicalZ2∗

graded commutative algebras Sec(∧E) of vector bundle forms are far from being suﬃcient.For

instance, whereas Lie algebroids are in 1-to-1 correspondence with homological vector ﬁelds of

∗

split supermanifolds Sec(∧E), the supergeometric interpretation of Loday algebroids [GKP11a]

requires aZ2-graded commutative algebra of non-Grassmannian type, namely the shuﬄe algebra

D(EHowever, not only other types of algebras, but also) of speciﬁc multidiﬀerential operators.

more general grading groups must be considered.

Let us also mention that classical Supersymmetry and Supermathematics are not completely

suﬃcient for modern physics (i.e., the description of anyons, paraparticles).

All the aforementioned problems are parts of our incentive to investigate the basic notions of

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linear algebra over a (Z2commutative unital associative algebra) -gradedAconsider the. We

spaceM(r;A) of matrices with coeﬃcients inAand introduce the notions of graded trace and

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(Z2TRACE AND BEREZINIAN) -GRADED

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Berezinian (in the simplest case of purely even matrices we will talk of the determinant).We

prove an analog of the Liouville formula that connects both concepts.Although most of the

results are formulated and proved for arbitraryA, our main goal is to develop a new theory of

matrices over Cliﬀord algebras and, more particularly, over quaternions.

Our main results are as follows:

•There exists a unique homomorphism of gradedA-modules and graded Lie algebras

Γtr :M(r;A)→A ,

deﬁned for arbitrary matrices with coeﬃcients inA.

•There exists a unique map

0 0

Γdet :M(r0;A)→A ,

deﬁned on purely even homogeneous matrices of degree 0 with values in the

commuta0

tive subalgebraA⊂Aconsisting of elements of degree 0 and characterized by three

properties: a)Γdet is multiplicative, b) for a block-diagonal matrix Γdet is the product

of the determinants of the blocks, c) Γdet of a lower (upper) unitriangular matrix equals

1. Inthe caseA=H, the absolute value of Γdet coincides with the classical Dieudonn´

determinant.

•There exists a unique group homomorphism

0 0×

ΓBer :GL(r;A)→(A),

deﬁned on the group of invertible homogeneous matrices of degree 0 with values in the

0

group of invertible elements ofA, characterized by properties similar to a), b), c).

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•The graded Berezinian is connected with the graded trace by a (Z2version of) -graded

Liouville’s formula

ΓBer(exp(εX)) = exp(Γtr(εX)),

whereεis a nilpotent formal parameter of degree 0 andXa graded matrix.

•For the matrices with coeﬃcients in a Cliﬀord algebra, there exists a unique way to

extend the graded determinant to homogeneous matrices of degree diﬀerent from zero,

if and only if the total matrix dimension|r|satisﬁes the condition

|r|= 0,4)1 (mod.

In the case of matrices overH, this graded determinant diﬀers from that of Dieudonn´.

The reader who wishes to gain a quick and straightforward insight into some aspects of the

preceding results, might envisage having a look at Section 8 at the end of this paper, which can

be read independently.

Our main tools that provide most of the existence results and explicit formulæ of graded

determinants and graded Berezinians, are the concepts of quasideterminants and quasiminors,

see [GGRW05] and references therein.

Let us also mention that in the case of matrices over a Cliﬀord algebra, the restriction for

the dimension of theA-module,|r|= 0,1 (mod 4), provides new insight into the old

problem initiated by Arthur Cayley, who considered speciﬁcally two-dimensional linear algebra over

quaternions. Itfollows that Cayley’s problem has no solution, at least within the framework of