There is a well-known correspondence between the objects of algebra and
geometry: a space gives rise to a function algebra; a vector bundle over
the space corresponds to a projective module over this algebra;
cohomology can be read off the de Rham complex; and so on. In this book
Yuri Manin addresses a variety of instances in which the application of
commutative algebra cannot be used to describe geometric objects,
emphasizing the recent upsurge of activity in studying noncommutative
rings as if they were function rings on "noncommutative spaces." Manin
begins by summarizing and giving examples of some of the ideas that led
to the new concepts of noncommutative geometry, such as Connes'
noncommutative de Rham complex, supergeometry, and quantum groups. He
then discusses supersymmetric algebraic curves that arose in connection
with superstring theory; examines superhomogeneous spaces, their
Schubert cells, and superanalogues of Weyl groups; and provides an
introduction to quantum groups. This book is intended for mathematicians
and physicists with some background in Lie groups and complex geometry.
Originally published in 1991.
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