Belavin-Drinfeld Structures: Generalized Cluster Sequel

Gone Fishing 2024 Generalized cluster structures on Generalized cluster structures on ??? (a sequel to Misha Gekhtman s talk) Dmitriy Voloshyn Institute for Basic Science Center for Geometry and Physics

Belavin-Drinfeld data (reminder) Setup. ? is a simple complex Lie algebra, a set of simple roots, ? the Cartan subalgebra, < ,> symmetric invariant nondegenerate form on ?. Def. A Belavin-Drinfeld triple (a BD triple) is ? 1, 2,? where 1, 2 and ?: 1 2 is a nilpotent isometry. (that is, ? 1 ? > 0 ??? 1) A Belavin-Drinfeld quadruple (a BD quadruple) is 1, 2,?,?0 where ?0: ? ? is a linear map that satisfies = id ?0+ ?0 ?01 ? ? = ?, ? 1 Compatible cluster structures don t depend on ?0, but the Poisson brackets do. Def. A Belavin-Drinfeld pair (a BD pair) (??,??) is a pair of any BD triples, which we denote (for historical reasons) as ?? ( 1?, 2 ?, 2 ?,??). ?,??) and ?? ( 1

Poisson varieties that we study Setup. ? is a connected simple complex Lie group. (Misha s yesterday talk) In the GSV program, we study the following objects parameterized by BD pairs ??,??: ?, , the group itself ??,?? ? ?(?), , ? ? ? ? as a group; it is the Drinfeld double of ?. Belavin & Drinfeld classification (1982) (??,??) ??,?? The following maps are Poisson: ? ?, , ?? A realization of the dual Poisson bracket in ? ?, , ? ?,? ?(?), , ; ??,?? ??,?? ?, ? 1 (?, , ?? ). ? ?(?), , (at least they show relations between the three objects) ??,?? Notation. ?std ( , , ) the trivial BD triple; Standard Poisson brackets: ? ? ? , std , , , std , ?std . , std , (?std ), ? ?,?std ?,?std ? ?std

The GSV program For every BD pair (??,??), construct compatible generalized cluster structures ??(??,??), ???(??,??) and ?? (??) and (?, , ?c ), respectively. ? on ?, , ??,?? , ?(?), , ??,?? That is, find an initial collection of function ?1,?2, ,?? such that ??,?? = ???????, ??? where , is the chosen Poisson bracket. The adjacency matrix ?? of the initial quiver is required to satisfy the compatibility equation: ?? = [? 0] ? ?,? ? ? where ????,?=1 .

The current approach ?); ?,?std (the same applies for ???(??,??) and ?? (??) as well) 1) Construct ??(?std 2) Construct a Poisson birational map ?: ?, , std (?, , ??,?? ); 3) Define the initial extended cluster for ??(??,??) via applying ? 1 to the initial extended cluster of ??(?std ? is expected to be a quasi-isomorphism. (for ?? (??) , the map is denoted as ?) (regularize the variables if necessary) ?). ?,?std (roughly, ? is a monomial transformation of cluster and frozen variables that is equivariant with respect to mutations)

The Poisson brackets , (??,??) and , ?? The trace form ?,?(??,??)? = ??? ?? ? , ?? ? ??? ? ?? ,? ?? (? GL?( )) Matrix of partial derivatives of ? ??c:??? ??? a linear map that depends on the BD data ?,??? ? = ??? ??,? , ??,? (at least one can say that conjugation plays a certain role in ?? (?) ) ??,? , ???

?) for (GL? , , std) ?,?std Recall ??(?std The initial extended cluster consists of two sets of flag minors. Flag minors with last columns: ?23 ?33 ?24 ?34, etc. ?14, det Flag minors with last rows: ?31 ?41 ?32 ?42, etc. ?42, det ?11 ?21 ?31 ?41 ?12 ?22 ?32 ?42 ?13 ?23 ?33 ?43 ?14 ?24 ?34 ?44 ? = GL4

The initial extended cluster for ??(?std) in type ? Select only one set of flag minors. (hence, two versions of ?? ? ) ?-functions ?-functions ? ? ?? ?-functions ??= ? ? ? ? (characters; also, Casimirs of , ?? ) -functions ? ?0 1??0 ? ? ?-functions ?-functions ??? (?std) ?? (?std) a vertex with a generalized mutation ?11? = det[?? ?11? = det[?1 (related to the highest weight theory) ?? 1??] ?? 1?1] ??? ??1 ??(?std,?std)

Poisson birational quasi-isomorphism ? (-convention) Setup. ? is a reductive complex Lie group. Decompose a generic element ? ? as ? = ?+?0? (Gauss decomposition). The map ?: ) (?, , ? ), ?:(?, ,std ? ?? . ? ? ? ? ? 1? ??(?), ? 1 The inverse: ? 1 ? ? ? 1 where the birational map :? ? satisfies the equation ? = ? ? (the equation can be solved via a recursive procedure) ? and , ? coincide, then ? is Poisson; Theorem. If the ?0 parts of ,std If ? SL? ,GL? , then ? is a quasi-isomorphism.

More features Marked variables: Frozen on the left but not frozen on the right 1, , ? marked in ?? (?std); 1, , ? marked in ?? ? ; ) (?, , ? ), For ?:(?, ,std the pullback 1, , ? 1 ? ? [ 1 1, , ? 1] ? :? ? 1 is an isomorphism of rings. ?? (?std) ?? (?) 2 1 3 ?: 3 2 1

Frozen variables Frozen variables generate Poisson prime ideals(?): ? ,? (?) for any frozen ? and any ? ?(?); Equivalently, ? ? ? ? = 0} is a union of symplectic leaves. Question: Why do frozen variables always have these special properties?

Status of the GSV program Setup. ? is a connected simply connected simple complex Lie group. (for non-simply connected groups, we only know what happens with PSL?( ), but it s not on arXiv) Trivial BD data: ???(?std,?std) ?? (?std) Only type ? Case of ?? ??(?std,?std) Only type ? What we know All Lie types Problem: what to replace the ?-functions with? Def. A BD pair (??,??) is aperiodic if ???0?? 1?0 Nontrivial BD data: (?0 is the longest Weyl group element) 1 is nilpotent. ??(??,??) ???(??,??) ?? (??) Case of ?? (for ??(??,??), only type ? is published) Only type ?, all ?? Only type ?, aperiodic (??,??) What we know All Lie types, aperiodic (??,??) Problem: the quasi-isomorphism is not rational when (??,??) is not aperiodic.

Thank you!