Determinants over graded-commutative algebras, a categorical viewpoint

E-print/Working paper (2016)

We generalize linear superalgebra to higher gradings and commutation factors, given by arbitrary abelian groups and bicharacters. Our central tool is an extension, to monoidal categories of modules, of ... [more ▼]

We generalize linear superalgebra to higher gradings and commutation factors, given by arbitrary abelian groups and bicharacters. Our central tool is an extension, to monoidal categories of modules, of the Nekludova-Scheunert faithful functor between the categories of graded-commutative and supercommutative algebras. As a result we generalize (super-)trace, determinant and Berezinian to graded matrices over graded-commutative algebras. For instance, on homogeneous quaternionic matrices, we obtain a lift of the Dieudonné determinant to the skew-field of quaternions. [less ▲]

(Z_2)^n-Superalgebra and (Z_2)^n-Supergeometry

Doctoral thesis (2014)

The present thesis deals with a development of linear algebra, geometry and analysis based on (Z_2)^n-superalgebras: associative unital algebras which are (Z_2)^n-graded (for some natural number n) and ... [more ▼]

The present thesis deals with a development of linear algebra, geometry and analysis based on (Z_2)^n-superalgebras: associative unital algebras which are (Z_2)^n-graded (for some natural number n) and graded-commutative, i.e., satisfying ab = (-1)^<deg(a),deg(b)> ba, for all homogeneous elements a, b of respective degrees deg(a), deg(b) in (Z_2)^n, and < . , . > denoting the usual scalar product. This generalization widens the range of applications of supergeometry to many mathematical structures -- the algebra of quaternions H and more generally Clifford algebras, Deligne algebra of superdifferential forms, higher vector bundles ... -- and appears also in physics -- for describing anyons, paraparticles -- proving its worth and relevance. In this dissertation, we first focus on (Z_2)^n-superalgebra theory: we define and characterize the notions of trace, determinant and Berezinian of matrices over graded-commutative algebras. Special attention is given to the case of Clifford algebras, where our study gives a new approach to treat the classical problem of finding a “good” determinant for matrices with non-commuting (quaternionic) entries. Further, we undertake the study of (Z_2)^n-graded differenital geometry. Privileging the ringed space approach, we define (smooth) (Z_2)^n-supermanifolds modeling their algebras of functions on the (Z_2)^n-commutative algebra of formal power series in graded variables, and develop the theory along the lines of supergeometry. Notable results are: the graded Berezinian and its cohomological interpretation (essential to establish integration theory); the theorem of morphism, which states that a morphism of (Z_2)^n-supermanifolds can be recovered from its coordinate expression; an analogous of Batchelor-Gawedzki theorem for (Z_2)^n-supermanifolds. [less ▲]

Z_2^n-Supergeometry II: Batchelor-Gawedzki Theorem

E-print/Working paper (2014)

Graded Algebra and Geometry

Report (2012)

Abstracts and list of participants of the Workshop on "Graded Algebra and Geometry" organized in December 2012 at the University of Luxembourg by Tiffany Covolo, Valentin Ovsienko, and Norbert Poncin

Higher trace and Berezinian of matrices over a Clifford algebra

in Journal of Geometry and Physics (2012), 62(11), 22942319

We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z_2)^n-graded commutative associative algebra A. The applications include a new approach to the classical ... [more ▼]

We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z_2)^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 Dieudonné determinant of quaternionic matrices, but in general our quaternionic determinant is different. We show that the graded determinant of purely even (Z_2)^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 (Z_2)^n-graded version of Liouville’s formula. [less ▲]