Let ζ be a complex ℓ th root of unity for an odd integer ℓ>1 . For any complex simple Lie algebra g , let u ζ =u ζ (g) be the associated "small" quantum enveloping algebra. This algebra is a finite dimensional Hopf algebra which can be realised as a subalgebra of the Lusztig (divided power) quantum enveloping algebra U ζ and as a quotient algebra of the De Concini-Kac quantum enveloping algebra U ζ . It plays an important role in the representation theories of both U ζ and U ζ in a way analogous to that played by the restricted enveloping algebra u of a reductive group G in positive characteristic p with respect to its distribution and enveloping algebras. In general, little is known about the representation theory of quantum groups (resp., algebraic groups) when l (resp., p ) is smaller than the Coxeter number h of the underlying root system. For example, Lusztig's conjecture concerning the characters of the rational irreducible G -modules stipulates that p≥h . The main result in this paper provides a surprisingly uniform answer for the cohomology algebra H ∙ (u ζ ,C) of the small quantum group.

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**Christopher P. Bendel**, University of Wisconsin-Stout, Menomonie, Wisconsin.**Daniel K. Nakano**, University of Georgia, Athens.**Georgia, Brian J. Parshall**, University of Virginia, Charlottesville, Virginia.**Cornelius Pillen**, University of South Alabama, Mobile, Alabama.

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**Book Description **American Mathematical Society, United States, 2014. Paperback. Condition: New. Language: English . Brand New Book. Let Î be a complex â th root of unity for an odd integer â >1 . For any complex simple Lie algebra g , let u Î =u Î (g) be the associated small quantum enveloping algebra. This algebra is a finite dimensional Hopf algebra which can be realised as a subalgebra of the Lusztig (divided power) quantum enveloping algebra U Î and as a quotient algebra of the De Concini-Kac quantum enveloping algebra U Î . It plays an important role in the representation theories of both U Î and U Î in a way analogous to that played by the restricted enveloping algebra u of a reductive group G in positive characteristic p with respect to its distribution and enveloping algebras. In general, little is known about the representation theory of quantum groups (resp., algebraic groups) when l (resp., p ) is smaller than the Coxeter number h of the underlying root system. For example, Lusztig s conjecture concerning the characters of the rational irreducible G -modules stipulates that pâ ¥h . The main result in this paper provides a surprisingly uniform answer for the cohomology algebra H â (u Î ,C) of the small quantum group. Seller Inventory # AAN9780821891759

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**Book Description **American Mathematical Society, United States, 2014. Paperback. Condition: New. Language: English. Brand new Book. Let be a complex th root of unity for an odd integer >1 . For any complex simple Lie algebra g , let u =u (g) be the associated "small" quantum enveloping algebra. This algebra is a finite dimensional Hopf algebra which can be realised as a subalgebra of the Lusztig (divided power) quantum enveloping algebra U and as a quotient algebra of the De Concini-Kac quantum enveloping algebra U . It plays an important role in the representation theories of both U and U in a way analogous to that played by the restricted enveloping algebra u of a reductive group G in positive characteristic p with respect to its distribution and enveloping algebras. In general, little is known about the representation theory of quantum groups (resp., algebraic groups) when l (resp., p ) is smaller than the Coxeter number h of the underlying root system. For example, Lusztig's conjecture concerning the characters of the rational irreducible G -modules stipulates that p h . The main result in this paper provides a surprisingly uniform answer for the cohomology algebra H (u ,C) of the small quantum group. Seller Inventory # AAN9780821891759

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