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Letter from Mrs. Dorothy L. Feigin to John Sloan, May 26, 1950
No full text1 leaf (single sided)Letter from Mrs. Dorothy L. Feigin to John Sloan, May 26, 195
Letter from Mrs. Dorothy L. Feigin to John Sloan, May 26, 1950
No full text1 leaf (single sided)Letter from Mrs. Dorothy L. Feigin to John Sloan, May 26, 195
THE FEIGIN TETRAHEDRON
No full textThe first goal of this note is to extend the well-known Feigin homomorphisms taking quantum groups to quantum polynomial algebras. More precisely, we define generalized Feigin homomorphisms from a quantum shuffle algebra to quantum polynomial algebras which extend the classical Feigin homomorphisms along the embedding of the quantum group into said quantum shuffle algebra. In a recent work of Berenstein and the author, analogous extensions of Feigin homomorphisms from the dual Hall-Ringel algebra of a valued quiver to quantum polynomial algebras were defined. To relate these constructions, we establish a homomorphism, dubbed the quantum shuffle character, from the dual Hall-Ringel algebra to the quantum shuffle algebra which relates the generalized Feigin homomorphisms. These constructions can be compactly described by a commuting tetrahedron of maps beginning with the quantum group and terminating in a quantum polynomial algebra. The second goal in this project is to better understand the dual canonical basis conjecture for skew-symmetrizable quantum cluster algebras. In the symmetrizable types it is known that dual canonical basis elements need not have positive multiplicative structure constants, while this is still suspected to hold for skew-symmetrizable quantum cluster algebras. We propose an alternate conjecture for the symmetrizable types: the cluster monomials should correspond to irreducible characters of a KLR algebra. Indeed, the main conjecture of this note would establish this ''KLR conjecture'' for acyclic skew-symmetrizable quantum cluster algebras: that is, we conjecture that the images of rigid representations under the quantum shuffle character give irreducible characters for KLR algebras. We sketch a proof in the symmetric case giving an alternative to the proof of Kimura-Qin that all non-initial cluster variables in an acyclic skew-symmetric quantum cluster algebra are contained in the dual canonical basis. With these results in mind we interpret the cluster mutations directly in terms of the representation theory of the KLR algebra.This paper is a contribution to the Special Issue on New Directions in Lie Theory. The full collection is
available at http://www.emis.de/journals/SIGMA/LieTheory2014.html.
The author would like to thank Sasha Kleshchev for introducing him to KLR algebras and for
leading him in this direction of research. The author would also like to thank Arkady Berenstein
for introducing him to Hall algebras and their beautiful properties. Finally, special thanks need
to be given to the anonymous referees for helping to solidify the proof of Theorem 8.4
Feigin, Harry
No full textBody cremated. Lillian Feigin - wifehttps://stars.library.ucf.edu/cfm-ch-memoranda-1940/1217/thumbnail.jp
Quiver Grassmannians and degenerate flag varieties
No full textQuiver Grassmannians are varieties parametrizing subrepresentations of a quiver
representation. It is observed that certain quiver Grassmannians for type A quivers
are isomorphic to the degenerate flag varieties investigated earlier by Feigin. This
leads to the consideration of a class of Grassmannians of subrepresentations of
the direct sum of a projective and an injective representation of a Dynkin quiver.
It is proved that these are (typically singular) irreducible normal local complete
intersection varieties, which admit a group action with finitely many orbits and
a cellular decomposition. For type A quivers, explicit formulas for the Euler
characteristic (the median Genocchi numbers) and the Poincaré polynomials are
derived
Bosonization of Feigin-Odesskii Poisson varieties
No full textThe derived moduli stack of complexes of vector bundles on a Gorenstein Calabi-Yau curve admits a 0-shifted Poisson structure. Projective spaces with Feigin-Odesskii Poisson brackets are examples of such moduli spaces over complex elliptic curves [6,7]. By generalizing several results in our previous work [10–12] we construct a collection of auxiliary Poisson varieties equipped with Poisson morphisms to Feigin-Odesskii varieties. We call them bosonizations of Feigin-Odesskii varieties. These spaces appear as special cases of the moduli spaces of chains, which we introduce. We show that the moduli space of chains admits a shifted Poisson structure when the base is a Calabi-Yau variety of an arbitrary dimension. Using bosonization spaces mapping to the zero loci of the Feigin-Odesskii varieties, we show that the Feigin-Odesskii Poisson brackets on projective spaces (associated with stable bundles of arbitrary rank on elliptic curves) admit no infinitesimal symmetries. We also derive explicit formulas for the Poisson brackets on the bosonizations of the Feigin-Odesskii varieties associated with line bundles in a simplest nontrivial case
On Conormal Lie Algebras of Feigin-Odesskii Poisson Structures
No full textInternational audienceThe main result of the paper is a description of conormal Lie algebras of Feigin-Odesskii Poisson structures. In order to obtain it, we introduce a new variant of a definition of a Feigin-Odesskii Poisson structure: we define it using a differential on the second page of a certain spectral sequence. In the general case, this spectral sequence computes morphisms and higher Ext ′ s between filtered objects in an Abelian category. Moreover, we use our definition to give another proof of the description of symplectic leaves of Feigin-Odesskii Poisson structures
On a conjecture of Feigin, Wang and Yoshinaga
No full textWe settled a conjecture of Feigin, Wang and Yoshinaga, appeared in the
preprint "Integral expressions for derivations of multiarrangements" (arXiv:
2309.01287v2)
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