Graded differential geometry of graded matrix algebras

Abstract

We study the graded derivation-based noncommutative differential geometry of the Z 2 -graded algebra M(n͉m) of complex (nϩm)ϫ(nϩm)-matrices with the ''usual block matrix grading'' ͑for n m). Beside the ͑infinite-dimensional͒ algebra of graded forms, the graded Cartan calculus, graded symplectic structure, graded vector bundles, graded connections and curvature are introduced and investigated. In particular we prove the universality of the graded derivation-based first-order differential calculus and show that M(n͉m) is a ''noncommutative graded manifold'' in a stricter sense: There is a natural body map and the cohomologies of M(n͉m) and its body coincide ͑as in the case of ordinary graded manifolds͒.

Noncommutative $\varepsilon$-graded connections

“…This is a key difference with the non-graded case (see Example 3.11). The cohomology of (3.23) has been computed in [31] and is given by:…”

“…The derivation-based differential calculus has been introduced in [5,6,21]. It has been studied for various algebras in [8,9,22,23,10,11,24,25,26] (see also [27,7,28,29] for reviews) and some propositions to generalize this construction to graded algebras have been presented for instance in [30,31].…”

Lectures on Graded Differential Algebras and Noncommutative Geometry

“…The above models were the first ones of classical Yang-Mills-Higgs models based on noncommutative geometry. They certainly admit a natural supersymmetric extension since there is a natural extension of the derivation-based calculus to graded matrix algebras [42]. There is also another extension of the above calculus where C ∞ (R s+1 ) ⊗M n (C) is replaced by the algebra ΓEnd(E) of smooth sections of the endomorphisms bundle of a (nontrivial) smooth vector bundle E (of rank n) admitting a volume over a smooth ((s + 1)dimensional) manifold [33].…”

On the structure of graded Leibniz triple systems

“…The study of gradings on Lie algebras begins in the 1933 seminal Jordan's work, with the purpose of formalizing Quantum Mechanics [3]. Since then, the interest on gradings on different classes of algebras has been remarkable in the recent years, motivated in part by their application in physics and geometry [4][5][6][7]. Recently, in [7][8][9][10][11][12], the structure of arbitrary graded Lie algebras, graded Lie superalgebras, graded commutative algebras, graded Leibniz algebras and graded Lie triple systems have been determined by the techniques of connections of roots.…”