Quantum Deformations of Algebras and Their Representations

1-dimensional A-modules abelian action affine algebra map algebraic group automorphism BD infinitesimals bialgebra bijective bijective antipode canonical Cartan matrix coalgebra cochain cocycle cohomology commutative consider construction cyclic cyclic homology D₁ defined denote dense orbit differential dihedral dimension elements filtration finite dimensional follows formula graded group algebra Hall algebra Hecke symmetry hence highest weight vector homomorphism Hopf algebra idempotents invariant isomorphic LEMMA linear Math monomial morphism multiplication noetherian non-zero object obtain polynomial proof Proposition quadratic form quantized quantized enveloping algebras quantum groups quantum transposition quiver quotient relations representations resp ring root of unity satisfies semisimple Lie algebra simple Lie algebra simple modules structure subalgebra subspace summand Suppose tensor product theorem tilting modules Uq(g V₁ vector space W₁ Weyl group Yang-Baxter equation Z/pZ zero Σ Σ ΣΣ