On some gauge invariants of Hopf algebras Siu-Hung Ng Iowa State - PowerPoint PPT Presentation

On some gauge invariants of Hopf algebras Siu-Hung Ng Iowa State University, USA Non-commutative algebraic geometry 2011 Shanghai Workshop September 12-16, 2011 On some gauge invariants of Hopf algebras

Question: Let Q 8 , D 8 be the quaternion and dihedral groups. Complex irreducible representations of the two groups: degree 1: χ 1 , χ 2 , χ 3 , χ 4 degree 2: ψ . C D 8 ∼ = C Q 8 as C -algebras but not isomorphic as Hopf algebras C Q 8 - mod ∼ = C D 8 - mod as C -linear categories. They have the same fusion rules or Grothendieck ring : { χ 1 , χ 2 , χ 3 , χ 4 } ∼ = Z 2 × Z 2 , χ i ψ = ψ = ψχ i , ψψ = χ 1 + χ 2 + χ 3 + χ 4 . Are C D 8 - mod and C Q 8 - mod equivalent as tensor categories? On some gauge invariants of Hopf algebras

Question: Let Q 8 , D 8 be the quaternion and dihedral groups. Complex irreducible representations of the two groups: degree 1: χ 1 , χ 2 , χ 3 , χ 4 degree 2: ψ . C D 8 ∼ = C Q 8 as C -algebras but not isomorphic as Hopf algebras C Q 8 - mod ∼ = C D 8 - mod as C -linear categories. They have the same fusion rules or Grothendieck ring : { χ 1 , χ 2 , χ 3 , χ 4 } ∼ = Z 2 × Z 2 , χ i ψ = ψ = ψχ i , ψψ = χ 1 + χ 2 + χ 3 + χ 4 . Are C D 8 - mod and C Q 8 - mod equivalent as tensor categories? On some gauge invariants of Hopf algebras

Question: Let Q 8 , D 8 be the quaternion and dihedral groups. Complex irreducible representations of the two groups: degree 1: χ 1 , χ 2 , χ 3 , χ 4 degree 2: ψ . C D 8 ∼ = C Q 8 as C -algebras but not isomorphic as Hopf algebras C Q 8 - mod ∼ = C D 8 - mod as C -linear categories. They have the same fusion rules or Grothendieck ring : { χ 1 , χ 2 , χ 3 , χ 4 } ∼ = Z 2 × Z 2 , χ i ψ = ψ = ψχ i , ψψ = χ 1 + χ 2 + χ 3 + χ 4 . Are C D 8 - mod and C Q 8 - mod equivalent as tensor categories? On some gauge invariants of Hopf algebras

Question: Let Q 8 , D 8 be the quaternion and dihedral groups. Complex irreducible representations of the two groups: degree 1: χ 1 , χ 2 , χ 3 , χ 4 degree 2: ψ . C D 8 ∼ = C Q 8 as C -algebras but not isomorphic as Hopf algebras C Q 8 - mod ∼ = C D 8 - mod as C -linear categories. They have the same fusion rules or Grothendieck ring : { χ 1 , χ 2 , χ 3 , χ 4 } ∼ = Z 2 × Z 2 , χ i ψ = ψ = ψχ i , ψψ = χ 1 + χ 2 + χ 3 + χ 4 . Are C D 8 - mod and C Q 8 - mod equivalent as tensor categories? On some gauge invariants of Hopf algebras

Question: Let Q 8 , D 8 be the quaternion and dihedral groups. Complex irreducible representations of the two groups: degree 1: χ 1 , χ 2 , χ 3 , χ 4 degree 2: ψ . C D 8 ∼ = C Q 8 as C -algebras but not isomorphic as Hopf algebras C Q 8 - mod ∼ = C D 8 - mod as C -linear categories. They have the same fusion rules or Grothendieck ring : { χ 1 , χ 2 , χ 3 , χ 4 } ∼ = Z 2 × Z 2 , χ i ψ = ψ = ψχ i , ψψ = χ 1 + χ 2 + χ 3 + χ 4 . Are C D 8 - mod and C Q 8 - mod equivalent as tensor categories? On some gauge invariants of Hopf algebras

Question: Let Q 8 , D 8 be the quaternion and dihedral groups. Complex irreducible representations of the two groups: degree 1: χ 1 , χ 2 , χ 3 , χ 4 degree 2: ψ . C D 8 ∼ = C Q 8 as C -algebras but not isomorphic as Hopf algebras C Q 8 - mod ∼ = C D 8 - mod as C -linear categories. They have the same fusion rules or Grothendieck ring : { χ 1 , χ 2 , χ 3 , χ 4 } ∼ = Z 2 × Z 2 , χ i ψ = ψ = ψχ i , ψψ = χ 1 + χ 2 + χ 3 + χ 4 . Are C D 8 - mod and C Q 8 - mod equivalent as tensor categories? On some gauge invariants of Hopf algebras

Question: Let Q 8 , D 8 be the quaternion and dihedral groups. Complex irreducible representations of the two groups: degree 1: χ 1 , χ 2 , χ 3 , χ 4 degree 2: ψ . C D 8 ∼ = C Q 8 as C -algebras but not isomorphic as Hopf algebras C Q 8 - mod ∼ = C D 8 - mod as C -linear categories. They have the same fusion rules or Grothendieck ring : { χ 1 , χ 2 , χ 3 , χ 4 } ∼ = Z 2 × Z 2 , χ i ψ = ψ = ψχ i , ψψ = χ 1 + χ 2 + χ 3 + χ 4 . Are C D 8 - mod and C Q 8 - mod equivalent as tensor categories? On some gauge invariants of Hopf algebras

Question: Let Q 8 , D 8 be the quaternion and dihedral groups. Complex irreducible representations of the two groups: degree 1: χ 1 , χ 2 , χ 3 , χ 4 degree 2: ψ . C D 8 ∼ = C Q 8 as C -algebras but not isomorphic as Hopf algebras C Q 8 - mod ∼ = C D 8 - mod as C -linear categories. They have the same fusion rules or Grothendieck ring : { χ 1 , χ 2 , χ 3 , χ 4 } ∼ = Z 2 × Z 2 , χ i ψ = ψ = ψχ i , ψψ = χ 1 + χ 2 + χ 3 + χ 4 . Are C D 8 - mod and C Q 8 - mod equivalent as tensor categories? On some gauge invariants of Hopf algebras

Question: Let Q 8 , D 8 be the quaternion and dihedral groups. Complex irreducible representations of the two groups: degree 1: χ 1 , χ 2 , χ 3 , χ 4 degree 2: ψ . C D 8 ∼ = C Q 8 as C -algebras but not isomorphic as Hopf algebras C Q 8 - mod ∼ = C D 8 - mod as C -linear categories. They have the same fusion rules or Grothendieck ring : { χ 1 , χ 2 , χ 3 , χ 4 } ∼ = Z 2 × Z 2 , χ i ψ = ψ = ψχ i , ψψ = χ 1 + χ 2 + χ 3 + χ 4 . Are C D 8 - mod and C Q 8 - mod equivalent as tensor categories? On some gauge invariants of Hopf algebras

Question: Let Q 8 , D 8 be the quaternion and dihedral groups. Complex irreducible representations of the two groups: degree 1: χ 1 , χ 2 , χ 3 , χ 4 degree 2: ψ . C D 8 ∼ = C Q 8 as C -algebras but not isomorphic as Hopf algebras C Q 8 - mod ∼ = C D 8 - mod as C -linear categories. They have the same fusion rules or Grothendieck ring : { χ 1 , χ 2 , χ 3 , χ 4 } ∼ = Z 2 × Z 2 , χ i ψ = ψ = ψχ i , ψψ = χ 1 + χ 2 + χ 3 + χ 4 . Are C D 8 - mod and C Q 8 - mod equivalent as tensor categories? On some gauge invariants of Hopf algebras

Question: Let Q 8 , D 8 be the quaternion and dihedral groups. Complex irreducible representations of the two groups: degree 1: χ 1 , χ 2 , χ 3 , χ 4 degree 2: ψ . C D 8 ∼ = C Q 8 as C -algebras but not isomorphic as Hopf algebras C Q 8 - mod ∼ = C D 8 - mod as C -linear categories. They have the same fusion rules or Grothendieck ring : { χ 1 , χ 2 , χ 3 , χ 4 } ∼ = Z 2 × Z 2 , χ i ψ = ψ = ψχ i , ψψ = χ 1 + χ 2 + χ 3 + χ 4 . Are C D 8 - mod and C Q 8 - mod equivalent as tensor categories? On some gauge invariants of Hopf algebras

Question: Let Q 8 , D 8 be the quaternion and dihedral groups. Complex irreducible representations of the two groups: degree 1: χ 1 , χ 2 , χ 3 , χ 4 degree 2: ψ . C D 8 ∼ = C Q 8 as C -algebras but not isomorphic as Hopf algebras C Q 8 - mod ∼ = C D 8 - mod as C -linear categories. They have the same fusion rules or Grothendieck ring : { χ 1 , χ 2 , χ 3 , χ 4 } ∼ = Z 2 × Z 2 , χ i ψ = ψ = ψχ i , ψψ = χ 1 + χ 2 + χ 3 + χ 4 . Are C D 8 - mod and C Q 8 - mod equivalent as tensor categories? On some gauge invariants of Hopf algebras

Question: Let Q 8 , D 8 be the quaternion and dihedral groups. Complex irreducible representations of the two groups: degree 1: χ 1 , χ 2 , χ 3 , χ 4 degree 2: ψ . C D 8 ∼ = C Q 8 as C -algebras but not isomorphic as Hopf algebras C Q 8 - mod ∼ = C D 8 - mod as C -linear categories. They have the same fusion rules or Grothendieck ring : { χ 1 , χ 2 , χ 3 , χ 4 } ∼ = Z 2 × Z 2 , χ i ψ = ψ = ψχ i , ψψ = χ 1 + χ 2 + χ 3 + χ 4 . Are C D 8 - mod and C Q 8 - mod equivalent as tensor categories? On some gauge invariants of Hopf algebras

FS-indicators for semisimple Hopf algebras Tambara-Yamagami [98] answered the question by considering the fusion categories with such fusion rules. For other groups or semisimple quasi-Hopf algebras, more general but computable invariants are needed to be discovered. Two Hopf algebras H , K are said to be gauge equivalent if H - mod and K - mod are equivalent as monoidal categories. Let C be a collection of Hopf algebras which is closed under gauge equivalence. A quantity f ( H ) defined for any Hopf algebra H in C is called a gauge invariant if f ( H ) = f ( K ) for all Hopf algebras K gauge equivalent to H . On some gauge invariants of Hopf algebras

FS-indicators for semisimple Hopf algebras Tambara-Yamagami [98] answered the question by considering the fusion categories with such fusion rules. For other groups or semisimple quasi-Hopf algebras, more general but computable invariants are needed to be discovered. Two Hopf algebras H , K are said to be gauge equivalent if H - mod and K - mod are equivalent as monoidal categories. Let C be a collection of Hopf algebras which is closed under gauge equivalence. A quantity f ( H ) defined for any Hopf algebra H in C is called a gauge invariant if f ( H ) = f ( K ) for all Hopf algebras K gauge equivalent to H . On some gauge invariants of Hopf algebras

FS-indicators for semisimple Hopf algebras Tambara-Yamagami [98] answered the question by considering the fusion categories with such fusion rules. For other groups or semisimple quasi-Hopf algebras, more general but computable invariants are needed to be discovered. Two Hopf algebras H , K are said to be gauge equivalent if H - mod and K - mod are equivalent as monoidal categories. Let C be a collection of Hopf algebras which is closed under gauge equivalence. A quantity f ( H ) defined for any Hopf algebra H in C is called a gauge invariant if f ( H ) = f ( K ) for all Hopf algebras K gauge equivalent to H . On some gauge invariants of Hopf algebras