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The extended Green and repository Drinfeld–Jimbo coproduct conventions agree on simple generators

Statement

For a vertex simple $S_{i}$ , the DJ-compatible extended Green coproduct has $Δ_{-} ([S_{i}]) = [S_{i}] \otimes k_{i}^{- 1} + 1 \otimes [S_{i}],$ which is exactly the repository Drinfeld–Jimbo coproduct formula on $E_{i}$ . The alternative Hall convention $Δ_{+} ([S_{i}]) = [S_{i}] \otimes 1 + k_{i} \otimes [S_{i}]$ is distinct and is not silently identified with $Δ_{-}$ . The positive Borel coproduct formulas are well-defined at $ν = \sqrt{q} > 1$ .

Facts & Assumptions

Given: A finite loopless quiver, its positive Borel presentation, and the extended Green coproduct of The extended Green coproduct is an algebra map.

[F1]

The repository Drinfeld–Jimbo coproduct is $Δ (E_{i}) = E_{i} \otimes K_{i}^{- 1} + 1 \otimes E_{i}$ and $Δ (K_{i}) = K_{i} \otimes K_{i}$ (The Drinfeld–Jimbo formulas define a Hopf algebra).

[F2]

The positive Borel coproduct assignments are the same formulas in the presentation of The Hall-normalized extended positive Borel of a loopless quiver.

[F3]

The positive Serre ideal is a coideal for these coproducts in the toral action quotient (The coproduct preserves the positive and negative quantum Serre ideals). Its identities are Laurent-polynomial identities in $q_{i}$ ; at $q_{i} = ν > 1$ all quantum factorial denominators are nonzero, so those identities specialize.

[F4]

For a simple $S_{i}$ , the extended coproduct is $Δ_{-} ([R]) = \sum ν^{⟨ M, N ⟩} P_{M, N}^{R} a_{R}^{- 1} [M] \otimes k_{- [M]} [N]$ (The extended Green coproduct is an algebra map).

Proof

technique · direct comparison of generator formulas

1.1F4givenalgebra

In the sum [F4] for $R = S_{i}$ , the only endpoint pairs are $(S_{i}, 0)$ and $(0, S_{i})$ . Each endpoint-labelled exact-sequence count is $a_{S_{i}}$ , and the Euler exponents involving zero are zero. Their terms are respectively $[S_{i}] \otimes k_{i}^{- 1}$ and $1 \otimes [S_{i}]$ .

2.1F1F2F3step 1.1algebra∎

The formula in step 1.1 agrees term by term with [F1] under $E_{i} \mapsto [S_{i}]$ and $K_{i} \mapsto k_{i}$ . The group-like torus formulas agree as well. Formula [F2] is well-defined on the positive Serre quotient by [F3] and specialization at $ν > 1$ . Schiffmann’s $Δ_{+}$ has different simple terms, so no equality between those two coproduct conventions is asserted.

Depends on

Used by

The Ringel assignment is a bialgebra homomorphism onto the extended composition subalgebra Proposition

Dependency tree · two levels

31 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Olivier Schiffmann, Lectures on Hall Algebras, arXiv:math/0611617v2 (standard reference, not scraped)
Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390 (standard reference, not scraped)