Decomposing moduli of representations of finite-dimensional algebras

Chindris, Calin; Kinser, Ryan

doi:10.1007/s00208-018-1687-7

Cited by 5 publications

(8 citation statements)

References 53 publications

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“…It is shown in [CK18,Prop. 3] that any GL(d)-stable, irreducible, closed subvariety of rep A (d) which has as least one θ-semistable point admits a θ-stable decomposition.…”

Section: Moduli Spacesmentioning

confidence: 99%

“…Every irreducible component of M(d) ss θ is of the form M(C) ss θ where C is an irreducible component of rep A (d), so this covers all of them. Here we have combined the three parts of the main theorem of [CK18] for simplicity; this is enough for our application. We also note that the map of this theorem is quite simplistic on the set-theoretical level, sending a list of representations to their direct sum.…”

Section: Moduli Spacesmentioning

confidence: 99%

“…Such results on the singularities of irreducible components, specifically their normality, can be combined with the main theorem of [CK18] to study the geometry of decompositions of moduli spaces of semistable representations (in sense of Geometric Invariant Theory). We discuss such applications and related semistability results in Section 5.…”

mentioning

confidence: 99%

“…Theorem 5.3 [CK18]. Let A be a finite-dimensional algebra and let C ⊆ rep A (d) ss θ be an irreducible component such thatC ss θ = ∅.…”

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confidence: 99%

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Representation varieties of algebras with nodes

Kinser,

Lőrincz

2018

Preprint

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We study the behavior of representation varieties of quivers with relations under the operation of node splitting. Working in the "relative setting" (splitting one node at a time) allows us to combinatorially enumerate irreducible components of representation varieties, and show they have rational singularities, for a wide class of algebras. This class contains all radical square zero algebras but also many others, as illustrated by examples throughout the paper. We also give applications to generic decomposition within irreducible components and decomposition of moduli spaces of semistable representations of certain algebras.

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“…It is shown in [CK18,Prop. 3] that any GL(d)-stable, irreducible, closed subvariety of rep A (d) which has as least one θ-semistable point admits a θ-stable decomposition.…”

Section: Moduli Spacesmentioning

confidence: 99%

Section: Moduli Spacesmentioning

confidence: 99%

mentioning

confidence: 99%

“…Theorem 5.3 [CK18]. Let A be a finite-dimensional algebra and let C ⊆ rep A (d) ss θ be an irreducible component such thatC ss θ = ∅.…”

mentioning

confidence: 99%

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Representation varieties of algebras with nodes

Kinser,

Lőrincz

2018

Preprint

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“…Then any irreducible component of a moduli space M(A, d) ss θ is isomorphic to a product of projective spaces. The isomorphism of the theorem results from a general decomposition theorem for moduli spaces proved in [CK17]. The key geometric condition needed to apply this theorem is that certain representation varieties are normal.…”

Section: Introductionmentioning

confidence: 99%

Moduli Spaces of Representations of Special Biserial Algebras

Carroll

Chindris

Kinser

et al. 2018

International Mathematics Research Notices

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We show that the irreducible components of any moduli space of semistable representations of a special biserial algebra are always isomorphic to products of projective spaces of various dimensions. This is done by showing that irreducible components of varieties of representations of special biserial algebras are isomorphic to irreducible components of products of varieties of circular complexes, and therefore normal, allowing us to apply recent results of the second and third authors on moduli spaces.

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Algorithms for representations of quiver Yangian algebras

Galakhov,

Gavshin,

Morozov

et al. 2024

J. High Energ. Phys.

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In this note, we aim to review algorithms for constructing crystal representations of quiver Yangians in detail. Quiver Yangians are believed to describe an action of the BPS algebra on BPS states in systems of D-branes wrapping toric Calabi-Yau three-folds. Crystal modules of these algebras originate from molten crystal models for Donaldson-Thomas invariants of respective three-folds. Despite the fact that this subject was originally at the crossroads of algebraic geometry with effective supersymmetric field theories, equivariant toric action simplifies applied calculations drastically. So the sole pre-requisite for this algorithm’s implementation is linear algebra. It can be easily taught to a machine with the help of any symbolic calculation system. Moreover, these algorithms may be generalized to toroidal and elliptic algebras and exploited in various numerical experiments with those algebras. We illustrate the work of the algorithms in applications to simple cases of $$ \textrm{Y}\left({\mathfrak{sl}}_2\right) $$ Y sl 2 , $$ \textrm{Y}\left({\hat{\mathfrak{gl}}}_1\right) $$ Y gl ̂ 1 and $$ \textrm{Y}\left({\hat{\mathfrak{gl}}}_{\left.1\right|1}\right) $$ Y gl ̂ 1 1 .

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Decomposing moduli of representations of finite-dimensional algebras

Cited by 5 publications

References 53 publications

Representation varieties of algebras with nodes

Representation varieties of algebras with nodes

Moduli Spaces of Representations of Special Biserial Algebras

Algorithms for representations of quiver Yangian algebras

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