Lectures on Supergeometry

E-print/Working paper (2012)

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 ▲]

Fiber bundles and connections

E-print/Working paper (2012)

Geometric structures encoded in the Lie structure of an Atiyah algebroid

in Transformation Groups (2011), 16(1), 137--160

Higher Loday Algebras

Scientific Conference (2011)

Geometry of Manifolds and Mathematical Physics

Report (2011)

Equivariant quantization of orbifolds

in Journal of Geometry and Physics (2010), 60(9), 1103--1111

Strongly r-matrix induced tensors, Koszul cohomology and arbitrary-dimensional quadratic Poisson cohomology

in Pacific Journal of Mathematics (2010), 245(1), 1--23

Equivariant quantization of orbifolds

Scientific Conference (2010)

Théorie des représentations du groupe symétrique

E-print/Working paper (2009)