A tale of three homotopies

E-print/Working paper (2012)

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 ▲]

Higher categorified algebras versus bounded homotopy algebras

in Theory and Applications of Categories (2011), 25

Geometric structures encoded in the Lie structure of an Atiyah algebroid

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

Lectures on Algebraic Operads

E-print/Working paper (2011)

Coalgebraic approach to the Loday infinity category, stem differential for 2n-ary graded and homotopy algebras

in Annales de l'Institut Fourier (2010), 60(1), 355--387

The Lie superalgebra of a supermanifold

in Journal of Lie Theory (2010), 20(4), 739--749

Higher categorified algebras versus truncated homotopy algebras

Presentation (2010)

A first approximation for quantization of singular spaces

in Journal of Geometry and Physics (2009), 59(4), 503--518