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On the freeness problem for truncated current algebras
Abstract
Molev (in: Doebner, Scherer, Nattermann (eds) Group 21, physical applications and mathematical aspects of geometry, groups, and algebras, World Scientific, Singapore, vol 1, pp 172–176, 1997) constructed generators of the center of the universal enveloping algebra \(U(\mathfrak{g}_m(n))\) for the truncated current Lie algebra \(\mathfrak{g}_m(n) = \mathfrak{gl}_n \otimes \mathbb{C}[x]/ (x^m)\). Such generators allow to define algebraic varieties associated to the center and the Gelfand–Tsetlin subalgebra. In this paper we prove that the Gelfand–Tsetlin variety is equidimensional of dimension \(mn(n-1)/2\) if and only if \(n=1,2\), implying that \(U(\mathfrak{g}_m(2))\) is free over the Gelfand–Tsetlin subalgebra.
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A famous result of Kostant states that the universal enveloping algebra of a semisimple complex Lie algebra is a free module over its center. We prove an analogue of this result for a class of filtered algebras and apply it to show the freeness over its center of the restricted Yangian and the universal enveloping algebra of the restricted current algebra associated with the general linear Lie algebra.
Sergei Ovsienko proved that the Gelfand–Tsetlin variety for [Formula: see text] is equidimensional and the dimension of all irreducible components equals [Formula: see text]. This implies in particular the equidimensionality of the nilfiber of the (partial) Kostant–Wallach map. We generalize this result for the [Formula: see text]-partial Kostant–Wallach map and prove that all its fibers are equidimensional of dimension [Formula: see text]. Also, we study certain subvarieties of the Gelfand–Tsetlin variety and show their equidimensionality which gives a new proof of Ovsienko’s theorem for [Formula: see text] and [Formula: see text].
In addition to being an interesting and profound subject in its own right, commutative ring theory is important as a foundation for algebraic geometry and complex analytical geometry. Matsumura covers the basic material, including dimension theory, depth, Cohen-Macaulay rings, Gorenstein rings, Krull rings and valuation rings. More advanced topics such as Ratliff's theorems on chains of prime ideals are also explored. The work is essentially self-contained, the only prerequisite being a sound knowledge of modern algebra, yet the reader is taken to the frontiers of the subject. Exercises are provided at the end of each section and solutions or hints to some of them are given at the end of the book.
Let Xn be the variety of n×n matrices, which k×k submatrices, formed by the first k rows and columns, are nilpotent for any k=1,…,n. We show, that Xn is a complete intersection of dimension (n−1)n/2 and deduce from it, that every character of the Gelfand–Zetlin subalgebra in U(gln) extends to an irreducible representation of U(gln).
We prove that if X is a locally complete intersection variety, then X has all the jet schemes irreducible if and only if X has canonical singularities. After embedding X in a smooth variety Y, we use motivic integration to express the condition that X has irreducible jet schemes in terms of data coming from an embedded resolution of X in Y. We show that this condition is equivalent with having canonical singularities. In the appendix, this result is used to prove a generalization of Kostant's freeness theorem to the setting of jet schemes. Comment: With an appendix by David Eisenbud and Edward Frenkel. Final version, to appear in Inventiones Mathematicae
We address two problems with the structure and representation theory of finite W-algebras associated with general linear Lie algebras. Finite W-algebras can be defined using either Kostant's Whittaker modules or a quantum Hamiltonian reduction. Our first main result is a proof of the Gelfand–Kirillov conjecture for the skew fields of fractions of finite W-algebras. The second main result is a parameterization of finite families of irreducible Gelfand–Tsetlin modules using Gelfand–Tsetlin subalgebra. As a corollary, we obtain a complete classification of generic irreducible Gelfand–Tsetlin modules for finite W-algebras.
Käre Bernt Lindström,Gratulerar på Din sextioårs dag. Vi hoppas att denna kommer falla Dig i smaken.Vi börjar med en kort studie i algebraiska matroider, och fortsätter med att bevisa relationen mellan Jacobianen av en mängd algebraiska funktioner och deras algebraiska oberoende. Med detta resulat bevisar vi de två huvudsatserna, som behandlar kanoniska former. Dessa satser reducerar frågan om en form är kanonisk för homogena polynom i q variabler och av grad p till att undersöka om ett homogent linjärt ekvations system har bara den triviala lösningen. Genom att använda apolaritet kan detta linjära system enkelt beskrivas. Till sist ger vi en mångfald av exempel av kanoniska former för homogena polynom.
We consider the polynomial current Lie algebra 9 [(t)Ix] corresponding to the general linear Lie algebra [(r,), and its factor-algebra ., by the ideal Ek>., [(r*) x'k We construct two families of algebraically independent generators of the center of the universal enveloping algebra U(.,) by using the quantum determinant and the quantum contraction for the Yangian of level m.
Contents §0. Introduction §1. The Yangian §2. The quantum determinant and the centre of §3. The twisted Yangian §4. The Sklyanin determinant and the centre of §5. The quantum contraction and the quantum Liouville formula for the Yangian §6. The quantum contraction and the quantum Liouville formula for the twisted Yangian §7. The quantum determinant and the Sklyanin determinant of block matrices
Bibliography
Une propriété des algèbres de Takiff
- F Geoffriau