"A Super Quantum Lie Day"

poster: Trace and Berezinian over \((\mathbb{Z}_2)^n\)-commutative algebras

speaker: Tiffany Covolo (Luxembourg)

abstract: A\((\mathbb{Z}_2)^n\)-commutative algebra (sometimes called color (super)algebra) is an associative \((\mathbb{Z}_2)^n\)-graded algebra in which multiplication satisfies the generalized sign rule \(ab = (-1)^{\langle\deg(a),\deg(b)\rangle} ba\) (for any pair of homogeneous elements \(a,b\)), where \(\langle.,.\rangle\) denotes the usual scalar product. Notably, these algebras include supercommutative algebras (\(n=1\)), and remarkable classical noncommutative algebras, such as quaternions and Clifford algebras. In this poster, I present three alternative approaches to obtain generalizations of notions of (super)determinant and trace in this higher graded setting: a direct approach via quasideterminants, a cohomological approach, and a categorical approach.

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