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A presentation by generators and relations of Nichols algebras of diagonal type and convex orders on root systems

Iván Ezequiel Angiono (2015)

Journal of the European Mathematical Society

We obtain a presentation by generators and relations of any Nichols algebra of diagonal type with finite root system. We prove that the defining ideal is finitely generated. The proof is based on Kharchenko’s theory of PBW bases of Lyndon words. We prove that the lexicographic order on Lyndon words is convex for PBW generators and so the PBW basis is orthogonal with respect to the canonical non-degenerate form associated to the Nichols algebra.

Abelian ideals of a Borel subalgebra and root systems

Dmitri I. Panyushev (2014)

Journal of the European Mathematical Society

Let 𝔤 be a simple Lie algebra and 𝔄𝔟 o the poset of non-trivial abelian ideals of a fixed Borel subalgebra of 𝔤 . In [8], we constructed a partition 𝔄𝔟 o = μ 𝔄𝔟 μ parameterised by the long positive roots of 𝔤 and studied the subposets 𝔄𝔟 μ . In this note, we show that this partition is compatible with intersections, relate it to the Kostant-Peterson parameterisation and to the centralisers of abelian ideals. We also prove that the poset of positive roots of 𝔤 is a join-semilattice.

Invariants of the half-liberated orthogonal group

Teodor Banica, Roland Vergnioux (2010)

Annales de l’institut Fourier

The half-liberated orthogonal group O n * appears as intermediate quantum group between the orthogonal group O n , and its free version O n + . We discuss here its basic algebraic properties, and we classify its irreducible representations. The classification of representations is done by using a certain twisting-type relation between O n * and U n , a non abelian discrete group playing the role of weight lattice, and a number of methods inspired from the theory of Lie algebras. We use these results for showing that...

Quantization of Drinfeld Zastava in type A

Michael Finkelberg, Leonid Rybnikov (2014)

Journal of the European Mathematical Society

Drinfeld Zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of the affine Lie algebra 𝔰𝔩 ^ n . We introduce an affine, reduced, irreducible, normal quiver variety Z which maps to the Zastava space bijectively at the level of complex points. The natural Poisson structure on the Zastava space can be described on Z in terms of Hamiltonian reduction of a certain Poisson subvariety of the dual space of a (nonsemisimple) Lie algebra. The quantum Hamiltonian...

Restriction to Levi subalgebras and generalization of the category 𝒪

Guillaume Tomasini (2013)

Annales de l’institut Fourier

The category of all modules over a reductive complex Lie algebra is wild, and therefore it is useful to study full subcategories. For instance, Bernstein, Gelfand and Gelfand introduced a category of modules which provides a natural setting for highest weight modules. In this paper, we define a family of categories which generalizes the BGG category, and we classify the simple modules for a subfamily. As a consequence, we show that some of the obtained categories are semisimple.

Riesz transforms for Dunkl transform

Bechir Amri, Mohamed Sifi (2012)

Annales mathématiques Blaise Pascal

In this paper we obtain the L p -boundedness of Riesz transforms for the Dunkl transform for all 1 < p < .

The freeness of ideal subarrangements of Weyl arrangements

Takuro Abe, Mohamed Barakat, Michael Cuntz, Torsten Hoge, Hiroaki Terao (2016)

Journal of the European Mathematical Society

A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers–Tymoczko. In particular, when an ideal subarrangement is equal to the entireWeyl arrangement, our...

The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications

Guo-Niu Han (2010)

Annales de l’institut Fourier

The paper is devoted to the derivation of the expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov in their study of the Seiberg-Witten Theory. We provide a refinement based on a new property of t -cores, and give an elementary proof by using the Macdonald identities. We also obtain an extension by adding two more parameters, which appears to be a discrete interpolation between the Macdonald identities and the generating function...

The structure of split regular Hom-Poisson algebras

María J. Aragón Periñán, Antonio J. Calderón Martín (2016)

Colloquium Mathematicae

We introduce the class of split regular Hom-Poisson algebras formed by those Hom-Poisson algebras whose underlying Hom-Lie algebras are split and regular. This class is the natural extension of the ones of split Hom-Lie algebras and of split Poisson algebras. We show that the structure theorems for split Poisson algebras can be extended to the more general setting of split regular Hom-Poisson algebras. That is, we prove that an arbitrary split regular Hom-Poisson algebra is of the form = U + j I j with U...

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