Twisted Real Structures in Noncommutative Geometry

Twisted real structures are a generalisation of real structures for spectral triples which are motivated as a way to implement the conformal transformation of a Dirac operator without needing to twist the noncommutative 1-forms. Taking inspiration from this example, in this thesis, we study further applications of twisted real structures, in particular those pertaining to commutative or almost-commutative geometries. We investigate how a reality operator can implement a noncommutative Clifford algebra Morita equivalence bimodule and find that the corresponding real structure on a commutative spectral triple must be twisted. We also investigate how the presence of a twisted real structure affects the implementation of the C*-algebra self-Morita equivalence bimodule which gives the gauge transformations of a spectral triple and find that the twist operators must be tightly constrained to yield meaningful physical action functionals. The form of the resulting action functionals suggests that the twist operator may implement a Krein structure, which often appears in pseudo-Riemannian generalisations of spectral triples. Thus we further investigate if twisted real structures can implement Wick rotations, and though we do not find a fully satisfactory construction, our preliminary attempts are encouraging and suggest that the possibility cannot yet be ruled out. Lastly we identify from the literature that the twisted spectral triple for kappa-Minkowski space admits a reality operator which gives a twisted real structure. This indicates that twisted real structures are compatible with twisted spectral triples as had been previously conjectured, opening up a whole new range of potential applications

On ℤ2-graded identities of UT 2 (E) and their growth

Let F be an infinite field of characteristic different from two and E be the infinite dimensional Grassmann algebra over F. We consider the upper triangular matrix algebra UT2(E) with entries in E endowed with the ℤ2-grading inherited by the natural Z2-grading of E and we study its ideal of ℤ2-graded polynomial identities (Tℤ2-ideal) and its relatively free algebra. In particular we show that the set of ℤ2-graded polynomial identities of UT2(E) does not depend on the characteristic of the field. Moreover we compute the ℤZ2-graded Hilbert series of UT2(E) and its ℤZ2-graded Gelfand-Kirillov dimension.Let F be an infinite field of characteristic different from two and E be the infinite dimensional Grassmann algebra over F. We consider the upper triangular matrix algebra UT2(E) with entries in E endowed with the ℤ2-grading inherited by the natural Z2-gr471469499FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO2013/06752-4305339/2013-3Aljadeff, E., Kanel-Belov, A., Representability and Specht problem for G-graded algebras (2010) Adv. Math., 225 (5), pp. 2391-2428Bahturin, Y., Drensky, V., Graded polynomial identities of matrices (2002) Linear Algebra Appl., 357, pp. 15-34Bahturin, Y., Giambruno, A., Riley, D.M., Group graded algebras with polynomial identity (1998) Israel J. Math., 104, pp. 145-155Belov, A., Counterexamples to the Specht problem (2000) Sb. Math., 191 (3-4), pp. 329-340Centrone, L., On some recent results about the graded Gelfand-Kirillov dimension of graded PI-algebras (2012) Serdica Math. 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