1984/01/01 by M. Cohen, Miriam Cohen, S. Montgomery +1 · 17 citations
Mathematics · #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Advanced Operator Algebra Research
paper · pdf · doi:10.1090/s0002-9947-1984-0728711-4
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebra graded by a finite group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">A1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the component for the identity element of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We consider such a grading as a “coaction” by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , in that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k left-bracket upper G right-bracket Superscript asterisk"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">[</mml:mo> <mml:mi>G</mml:mi> <mml:msup> <mml:mo stretchy="false">]</mml:mo> <mml:mo> ∗ </mml:mo> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">k[G]^ ∗ </mml:annotation> </mml:semantics> </mml:math> </inline-formula> -module algebra. We then study the smash product <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A number-sign k left-bracket upper G right-bracket Superscript asterisk"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi mathvariant="normal"> # </mml:mi> <mml:mi>k</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">[</mml:mo> <mml:mi>G</mml:mi> <mml:msup> <mml:mo stretchy="false">]</mml:mo> <mml:mo> ∗ </mml:mo> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">A# k[G]^ ∗ </mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; it plays a role similar to that played by the skew group ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R asterisk upper G"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mspace width="thinmathspace"/> <mml:mo> ∗ </mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">R ∗ G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the case of group actions, and enables us to obtain results relating the modules over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A comma upper A 1"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">A, A1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A number-sign k left-bracket upper G right-bracket Superscript asterisk"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi mathvariant="normal"> # </mml:mi> <mml:mi>k</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">[</mml:mo> <mml:mi>G</mml:mi> <mml:msup> <mml:mo stretchy="false">]</mml:mo> <mml:mo> ∗ </mml:mo> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">A# k[G]^ ∗ </mml:annotation> </mml:semantics> </mml:math> </inline-formula> . After giving algebraic versions of the Duality Theorems for Actions and Coactions (results coming from von Neumann algebras), we apply them to study the prime ideals of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics>