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The geometric reductivity of the quantum group S L q ( 2 )

Michał Kępa, Andrzej Tyc (2011)

Colloquium Mathematicae

We introduce the concept of geometrically reductive quantum group which is a generalization of the Mumford definition of geometrically reductive algebraic group. We prove that if G is a geometrically reductive quantum group and acts rationally on a commutative and finitely generated algebra A, then the algebra of invariants A G is finitely generated. We also prove that in characteristic 0 a quantum group G is geometrically reductive if and only if every rational G-module is semisimple, and that in...

The groups of automorphisms of the Witt W n and Virasoro Lie algebras

Vladimir V. Bavula (2016)

Czechoslovak Mathematical Journal

Let L n = K [ x 1 ± 1 , ... , x n ± 1 ] be a Laurent polynomial algebra over a field K of characteristic zero, W n : = Der K ( L n ) the Lie algebra of K -derivations of the algebra L n , the so-called Witt Lie algebra, and let Vir be the Virasoro Lie algebra which is a 1 -dimensional central extension of the Witt Lie algebra. The Lie algebras W n and Vir are infinite dimensional Lie algebras. We prove that the following isomorphisms of the groups of Lie algebra automorphisms hold: Aut Lie ( Vir ) ≃ Aut Lie ( W 1 ) ≃ { ± 1 } ⋉ K * , and give a short proof that Aut Lie ( W n ) ≃ Aut K - alg ( L n ) ≃ GL n ( ℤ ) ⋉ K * n .

The Harish-Chandra homomorphism for a quantized classical hermitian symmetric pair

Welleda Baldoni, Pierluigi Möseneder Frajria (1999)

Annales de l'institut Fourier

Let G / K a noncompact symmetric space with Iwasawa decomposition K A N . The Harish-Chandra homomorphism is an explicit homomorphism between the algebra of invariant differential operators on G / K and the algebra of polynomials on A that are invariant under the Weyl group action of the pair ( G , A ) . The main result of this paper is a generalization to the quantum setting of the Harish-Chandra homomorphism in the case of G / K being an hermitian (classical) symmetric space

The Hughes subgroup

Robert Bryce (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let G be a group and p a prime. The subgroup generated by the elements of order different from p is called the Hughes subgroup for exponent p . Hughes [3] made the following conjecture: if H p ⁢ G is non-trivial, its index in G is at most p . There are many articles that treat this problem. In the present Note we examine those of Strauss and Szekeres [9], which treats the case p = 3 and G arbitrary, and that of Hogan and Kappe [2] concerning the case when G is metabelian, and p arbitrary. A common proof is...

The Hurwitz determinants and the signatures of irreducible representations of simple real Lie algebras

Alexander Rudy (2005)

Open Mathematics

The paper deals with the real classical Lie algebras and their finite dimensional irreducible representations. Signature formulae for Hermitian forms invariant relative to these representations are considered. It is possible to associate with the irreducible representation a Hurwitz matrix of special kind. So the calculation of the signatures is reduced to the calculation of Hurwitz determinants. Hence it is possible to use the Routh algorithm for the calculation.

The Kuranishi space of complex parallelisable nilmanifolds

Sönke Rollenske (2011)

Journal of the European Mathematical Society

We show that the deformation space of complex parallelisable nilmanifolds can be described by polynomial equations but is almost never smooth. This is remarkable since these manifolds have trivial canonical bundle and are holomorphic symplectic in even dimension. We describe the Kuranishi space in detail in several examples and also analyse when small deformations remain complex parallelisable.

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