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See detailOn noncommutative deformations, cohomology of color-commutative algebras and formal smoothness
Gohr, Aron Samuel UL

Doctoral thesis (2009)

The main topic under study in the present work is the deformation theory of color algebras. Color algebras are generalized analogues of associative superalgebras, where the underlying grading can be over ... [more ▼]

The main topic under study in the present work is the deformation theory of color algebras. Color algebras are generalized analogues of associative superalgebras, where the underlying grading can be over an arbitrary abelian group and the Koszul sign is replaced by a bicharacter from the group into the base ring. A special case of particular interest are color-commutative algebras, which satisfy a commutation identity similar to (but much more general than) supercommutative algebras. Examples of color-commutative algebras include commutative and supercommutative superalgebras, the quaternions and para-quaternions, full matrix algebras over suitable base rings, Clifford algebras, and group rings over certain nonabelian groups. In the present work, Gerstenhaber-type formal deformations of these algebras are studied. In doing so, we extend previous work by Scheunert and provide a different approach to noncommutative deformation theory as introduced by Pinczon and Nadaud. In preparation of developing deformation theory for color algebras, we adapt a number of tools from ungraded Hochschild theory to our setting: among them, we derive an adapted Ext-functor, a color Gerstenhaber bracket, twisted graded versions of pre-Lie-algebras and pre-Lie-systems and colored analogs of the classical results linking infinitesimal deformations and obstructions to extension of deformations to second and third Hochschild cohomology. Additionally, we discuss the impact of some decisions in the construction of the trivial deformation object (color power series rings of given degree) on the resulting deformation theory. Finally, color-commutative deformations of color-commutative algebras are discussed and a suitable version of Harrison cohomology is developed. Also, the problem of classifying the color-commutative structures compatible with a given ungraded algebra is discussed and one nontrivial example is studied in detail. In support of all of these efforts, a number of structure theorems about color-commutative algebras are shown. [less ▲]

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