By Lucy Gow
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Additional resources for Yangians of Lie Superalgebras
Example text
T[2] (u) = i,j=1 Then since ∆ is an algebra homomorphism and we must have that ∆ : T (u)−1 → T[2] (u)−1 T[1] (u)−1 , which gives explicitly: m+n tkj (u) ⊗ tik (u)(−1)(i+k)(j+k) . ∆(tij (u)) = k=1 It is easy to see that this coincides with ((ζ ⊗ ζ) ◦ τ ◦ ∆ ◦ ζ) (tij (u)). (r) Let ϕm|n : Y (glm|n ) → Y (glm+k|n ) be the inclusion which sends each tij in (r) Y (glm|n ) to the generator tk+i,k+j in Y (glm+k|n ); and let ψk : Y (glm|n ) → Y (glm+k|n ) be the injective homomorphism defined by ψk = ωm+k|n ◦ ϕm|n ◦ ωm|n .
The corresponding metric space topology on M is called the h-adic topology. The sets m + hn M, where m ∈ M, n ∈ N, form a basis for the h-adic topology on M [41]. Given two Cauchy sequences {xn }n∈N and {yn }n∈N in a k[[h]]-module M , we say that they are equivalent if the difference xn − yn tends to zero as n goes to ˆ of the module M is the set of equivalence classes of infinity. The completion M Cauchy sequences in M , with operations given pointwise: {xn } + {yn } = {xn + yn }, a{xn } = {axn }, for all a ∈ A.
Cm (u) := τ ∈Sm It is clear that Cm (u) is an element of the subalgebra of Y (glm|n ) generated by the (r) set {tij }1≤i,j≤m;r≥0 . This subalgebra is isomorphic to the Yangian Y (glm ) of the Lie (r) algebra glm by the inclusion Y (glm ) → Y (glm|n ) which sends each generator tij in Y (glm ) to the generator of the same name in Y (glm|n ). Moreover, Cm (u) is the image under this map of the quantum determinant of Y (glm ) [6, 38]. 32 in [39]) that we have: Cm (u) = d1 (u)d2 (u − 1) · · · dm (u − m + 1).
Yangians of Lie Superalgebras by Lucy Gow
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