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Associative rings and algebras Definition

Quasitriangular Hopf algebra

Builds on Hopf algebraTensor product of vector spaces

Definition

Let kk be a field and let HH be a Hopf algebra over kk with coproduct Δ\Delta. Let τ ⁣:H⊗H→H⊗H\tau \colon H \otimes H \to H \otimes H be the flip a⊗b↦b⊗aa \otimes b \mapsto b \otimes a, and let Δop=τ∘Δ\Delta^{\mathrm{op}} = \tau \circ \Delta. For R=∑iai⊗biR = \sum_i a_i \otimes b_i in the tensor product H⊗HH \otimes H, put R12=∑iai⊗bi⊗1R_{12} = \sum_i a_i \otimes b_i \otimes 1, R13=∑iai⊗1⊗biR_{13} = \sum_i a_i \otimes 1 \otimes b_i and R23=∑i1⊗ai⊗biR_{23} = \sum_i 1 \otimes a_i \otimes b_i. The pair (H,R)(H, R) is a quasitriangular Hopf algebra if RR is invertible in H⊗HH \otimes H and:

  1. R Δ(x) R−1=Δop(x)R \, \Delta(x) \, R^{-1} = \Delta^{\mathrm{op}}(x) for all x∈Hx \in H.
  2. (Δ⊗id⁡)(R)=R13R23(\Delta \otimes \id)(R) = R_{13} R_{23}.
  3. (id⁡⊗Δ)(R)=R13R12(\id \otimes \Delta)(R) = R_{13} R_{12}.

The element RR is the universal R-matrix.

Remarks

Example: let H=k[G]H = k[G] be the group algebra of a group GG, with Δ(g)=g⊗g\Delta(g) = g \otimes g, and let R=1⊗1R = 1 \otimes 1. Then Δop=Δ\Delta^{\mathrm{op}} = \Delta, so clause 1 holds, and both sides of clauses 2 and 3 equal 1⊗1⊗11 \otimes 1 \otimes 1. Non-example: let GG be a finite non-abelian group and let H=kGH = k^G be the dual Hopf algebra of functions on GG. Since kGk^G is commutative, R Δ(x) R−1=Δ(x)R \, \Delta(x) \, R^{-1} = \Delta(x) for every invertible RR, so clause 1 would force Δ=Δop\Delta = \Delta^{\mathrm{op}}. This fails because kGk^G is not cocommutative, so no RR makes kGk^G quasitriangular. The axioms imply (ε⊗id⁡)(R)=1=(id⁡⊗ε)(R)(\varepsilon \otimes \id)(R) = 1 = (\id \otimes \varepsilon)(R) for the counit ε\varepsilon. They also imply the Yang–Baxter equation R12R13R23=R23R13R12R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}. A triangular Hopf algebra is a quasitriangular one with R21R=1R_{21} R = 1, where R21=τ(R)R_{21} = \tau(R).

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Further reading

  1. Web https://en.wikipedia.org/wiki/Quasitriangular_Hopf_algebra
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MSC16T05, 16T25, 81R50
DefinesQuasitriangular Hopf algebra
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