36 pages
English

HIGHER TRACE AND BEREZINIAN

-

Obtenez un accès à la bibliothèque pour le consulter en ligne
En savoir plus

Description

ar X iv :1 10 9. 58 77 v1 [ ma th. DG ] 27 Se p 2 01 1 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 matrices over a (Z2)n-graded commutative associative algebra A. The applications 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 Dieudonne determinant of quaternionic matrices, but in general our quaternionic determinant is different. We show that the graded determinant of purely even (Z2)n-graded matrices of degree 0 is polynomial in its entries. In the case of the algebra A = H of quaternions, we calculate the formula for the Berezinian in terms of a product of quasiminors in the sense of Gelfand, Retakh, and Wilson. The graded trace is related to the graded Berezinian (and determinant) by a (Z2)n-graded version of Liouville's formula. Contents 1. Introduction 2 2. (Z2)n-Graded Algebra 4 2.1. General Notions 4 2.2. (Z2)n- and (Z2)n+1-Grading on Clifford Algebras 7 3.

  • invertible graded

  • graded vector

  • graded commutative

  • matrices over

  • ?? ? ?

  • structure can

  • associative algebra

  • quaternions


Sujets

Informations

Publié par
Nombre de lectures 29
Langue English

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
n
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
n
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.
n
The graded trace is related to the graded Berezinian (and determinant) by a (Z2) -graded
version of Liouville’s formula.

Contents
1. Introduction
n
2. (Z2) -GradedAlgebra
2.1. GeneralNotions
n n+1
2.2. (Z2and () -Z2on Clifford Algebras) -Grading
n
3. (Z2) -GradedTrace
3.1. FundamentalTheorem and Explicit Formula
3.2. Application:Lax Pairs
n
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
n
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
n
7. (Z2Determinant over Quaternions and Clifford Algebras) -Graded
7.1. Relationto the Dieudonn´ Determinant

2
4
4
7
8
8
9
10
10
11
15
18
23
24
24
25
27
27
28
30
30

2010Mathematics Subject Classification.17A70, 58J52, 58A50, 15A66, 11R52.
n
Key words and phrases.Clifford linear algebra, quaternionic determinants, (Z2) -gradedcommutative algebra.

2

TIFFANY COVOLO

VALENTIN OVSIENKO

NORBERT PONCIN

7.2. GradedDeterminant of Even Homogeneous Matrices of Arbitrary Degree
n
8. Examplesof Quaternionic (Z2) -GradedDeterminants
8.1. QuaternionicMatrices of Degree Zero
8.2. HomogeneousQuaternionic Matrices of Nonzero Degrees
References

31
32
32
34
35

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
2 3
the Abelian group (Z2) =Z2×Z2(or over the even part of (Z2see [MGO09]).Quite similarly,) ,
n
every Clifford algebra withngenerators is (Z2) -gradedcommutative [AM02] (furthermore, a
n+1
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.
n
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
n
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
n
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

n
(Z2TRACE AND BEREZINIAN) -GRADED

3

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).
n
•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