Introduction to Axiomatic 33 HIRSCH. Differential Topology. Set Theory. 2nd ed. 34 SPITZER. Principles of Random Walk. 2nd ed. 2 OXTOBY. Measure and Category. 2nd ed. 35 WERMER. Banach Algebras and Several 3 SCHAEFFER. Topological Vector Spaces. Complex Variables. 2nd ed. 4 HILTON/STAMMBACH. A Course in 36 KELLEy/NAMIOKA et al. Linear Topological Homological Algebra. Spaces. 5 MAC LANE. Categories for the Working 37 MONK. Mathematical Logic. Mathematician. 38 GRAUERT/FRITZSCHE. Several Complex 6 HUGHES/PIPER. Projective Planes. Variables. 7 SERRE. A Course in Arithmetic. 39 ARVESON. An Invitation to C*-Algebras. 8 TAKEUTI/ZARING. Axiomatic Set Theory. 40 KEMENy/SNELL/KNAPP. Denumerable Markov 9 HUMPHREYS. Introduction to Lie Algebra~ Chains. 2nd ed. and Representation Theory. 41 ApOSTOL. Modular Functions and Dirichlet COHEN. A Course in Simple Homotopy Series in Number Theory. 2nd ed. Theory. 42 SERRE. Linear Representations of Finite CONWAY. Functions of One Complex Groups. Variable. 2nd ed. 43 GILLMAN/JERISON. Rings of Continuous BEALS. Advanced Mathematical Analysis. Functions. I3 ANDERSON/FULLER. Rings and Categories of 44 KENDIG. Elementary Algebraic Geometry. Modules. 2nd ed. 45 LoEVE. Probability Theory I. 4th ed. GOLUBITSKy/GUILEMIN. Stable Mappings and 46 LoEvE. Probability Theory II. 4th ed. Their Singularities. 47 MOISE. Geometric Topology in Dimensions 2 BERBERIAN. Lectures in Functional Analysis and 3. and Operator Theory. 48 SACHSlWu. General Relativity for WINTER. The Structure of Fields. Mathematicians. ROSENBLATT. Random Processes. 2nd ed. 49 GRUENBERGIWEIR. Linear Geometry. 2nd ed. HALMos. Measure Theory. 50 EDWARDS. Fermat's Last Theorem. HALMos. A Hilbert Space Problem Book. 51 KLINGENBERG. A Course in Differential 2nd ed. Geometry. HUSEMOLLER. Fibre Bundles. 3rd ed. 52 HARTSHORNE. Algebraic Geometry. HUMPHREYS. Linear Algebraic Groups. 53 MANIN. A Course in Mathematical Logic. BARNES/MACK. An Algebraic Introduction to 54 GRAVERIW ATKINS. Combinatorics with Mathematical Logic. Emphasis on the Theory of Graphs. GREUB. Linear Algebra. 4th ed. 55 BROWN/PEARCY. Introduction to Operator HOLMES. Geometric Functional Analysis and Theory I: Elements of Functional Analysis. Its Applications.

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To any cleft Hopf Galois object, i.e., any algebra H[t] obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle t, we attach two "universal algebras" A(H,t) and U(H,t). The algebra A(H,t) is obtained by twisting the multiplication of H with the most general two-cocycle u formally cohomologous to t. The cocycle u takes values in the field of rational functions on H. By construction, A(H,t) is a cleft H-Galois extension of a "big" commutative algebra B(H,t). Any "form" of H[t] can be obtained from A(H,t) by a specialization of B(H,t) and vice versa. If the algebra H[t] is simple, then A(H,t) is an Azumaya algebra with center B(H,t). The algebra U(H,t) is constructed using a general theory of polynomial identities that we set up for arbitrary comodule algebras; it is the universal comodule algebra in which all comodule algebra identities of H[t] are satisfied. We construct an embedding of U(H,t) into A(H,t); this embedding maps the center Z(H,t) of U(H,t) into B(H,t) when the algebra H[t] is simple. In this case, under an additional assumption, A(H,t) is isomorphic to B(H,t) \otimes_{Z(H,t)} U(H,t), thus turning A(H,t) into a central localization of U(H,t). We work out these constructions in full detail for the four-dimensional Sweedler algebra.Comment: 39 page

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