Noncommutative Algebraic Geometry and Representations of Quantized Algebras

1. Front Matter

   Pages i-xii

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2. Noncommutative Affine Schemes

   • Alexander L. Rosenberg

   Pages 1-47

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3. The Left Spectrum and Irreducible Representations of ‘Small’ Quantized and Classical Rings

   • Alexander L. Rosenberg

   Pages 48-109

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4. Noncommutative Local Algebra

   • Alexander L. Rosenberg

   Pages 110-141

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5. Noncommutative Local Algebra and Representations of certain rings of mathematical physics

   • Alexander L. Rosenberg

   Pages 142-187

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6. Skew PBW monads and representations

   • Alexander L. Rosenberg

   Pages 188-237

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7. Six spectra and two dimensions of an abelian category

   • Alexander L. Rosenberg

   Pages 238-275

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8. Noncommutative Projective Spectrum

   • Alexander L. Rosenberg

   Pages 276-305

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9. Back Matter

   Pages 306-322

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This book is based on lectures delivered at Harvard in the Spring of 1991 and at the University of Utah during the academic year 1992-93. Formally, the book assumes only general algebraic knowledge (rings, modules, groups, Lie algebras, functors etc.). It is helpful, however, to know some basics of algebraic geometry and representation theory. Each chapter begins with its own introduction, and most sections even have a short overview. The purpose of what follows is to explain the spirit of the book and how different parts are linked together without entering into details. The point of departure is the notion of the left spectrum of an associative ring, and the first natural steps of general theory of noncommutative affine, quasi-affine, and projective schemes. This material is presented in Chapter I. Further developments originated from the requirements of several important examples I tried to understand, to begin with the first Weyl algebra and the quantum plane. The book reflects these developments as I worked them out in reallife and in my lectures. In Chapter 11, we study the left spectrum and irreducible representations of a whole lot of rings which are of interest for modern mathematical physics. The dasses of rings we consider indude as special cases: quantum plane, algebra of q-differential operators, (quantum) Heisenberg and Weyl algebras, (quantum) enveloping algebra ofthe Lie algebra sl(2) , coordinate algebra of the quantum group SL(2), the twisted SL(2) of Woronowicz, so called dispin algebra and many others.

• Abelian category
• Algebraic K-theory
• Dimension
• Grad
• K-theory
• algebra
• algebraic geometry
• associative ring
• commutative algebra
• commutative ring
• representation theory

• Department of Mathematics, Indiana University, Bloomington, USA

  Alexander L. Rosenberg

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