Nilpotent subspaces of maximal dimension in semi-simple Lie algebras

Draisma, Jan; Kraft, Hanspeter; Kuttler, Jochen

doi:10.1112/s0010437x05001855

Cited by 18 publications

(24 citation statements)

References 18 publications

(10 reference statements)

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“…Littelmann and Procesi in [2] show that Z is isomorphic to the wonderful compactification of P GL(3)/P SO (3). In this part, we find equations defining the wonderful compactification in G(3, sl (3)).…”

Section: The Case Sl(3)mentioning

confidence: 79%

Equations of some wonderful compactifications

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2011

Ann. inst. Fourier

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De Concini and Procesi have defined in [1] the wonderful compactification X of a symmetric space X = G/G σ where G is a semisimple adjoint group and G σ the subgroup of fixed points of G by an involution σ. It is a closed subvariety of a grassmannian of the Lie algebra g of G. In this paper, we prove that, when the rank of X is equal to the rank of G, the variety is defined by linear equations. The set of equations expresses the fact that the invariant alternate trilinear form w on g vanishes on the −1-eigenspace of σ.

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Section: The Case Sl(3)mentioning

confidence: 79%

Equations of some wonderful compactifications

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2011

Ann. inst. Fourier

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“…It has been shown by Gerstenhaber [Ge58] that a linear subspace L of the nilpotent matrices N in M n of maximal possible dimension n 2 (see Proposition 3) is conjugate to the nilpotent upper triangular matrices, hence annihilated by a 1-PSG. Jointly with Jan Draisma and Jochen Kuttler we have generalized this result to arbitrary semisimple Lie algebras, see [DKK06].…”

Section: Some Examplesmentioning

confidence: 92%

Polarizations and Nullcone of Representations of Reductive Groups

Kraft

Wallach

2009

Progress in Mathematics

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Abstract. The paper starts with the following simple observation. Let V be a representation of a reductive group G, and let f 1 , f 2 . . . , fn be homogeneous invariant functions. Then the polarizations of f 1 , f 2 . . . , fn define the nullcone of k ≤ m copies of V if and only if every linear subspace L of the nullcone of V of dimension ≤ m is annhilated by a one-parameter subgroup (shortly a 1-PSG). This means that there is a group homomorphism λ :This is then applied to many examples. A surprising result is about the group SL 2 where almost all representations V have the property that all linear subspaces of the nullcone are annihilated. Again, this has interesting applications to the invariants on several copies.Another result concerns the n-qubits which appear in quantum computing. This is the representation of a product of n copies of SL 2 on the n-fold tensor product C 2 ⊗ C 2 ⊗ · · · ⊗ C 2 . Here we show just the opposite, namely that the polarizations never define the nullcone of several copies if n ≥ 3.(An earlier version of this paper, distributed in 2002, was split into two parts; the first part with the title "On the nullcone of representations of reductive group" is published in Pacific J. Math. 224 (2006), 119-140.) Linear subspaces of the nullconeIn this paper we study finite dimensional complex representations of a reductive algebraic group G. It is a well-known and classical fact that the nullcone N V of such a representation V plays a fundamental rôle in the geometry of the representation. Recall that N V is defined to be the union of all G-orbits in V containing the origin 0 in their closure. Equivalently, N V is the zero set of all non-constant homogeneous G-invariant functions on V .In a previous paper [KrW06] we have seen that certain linear subspaces of the nullcone play a central role for understanding its irreducible components. In this paper we will discuss arbitrary linear subspaces of the nullcone N V of a representation V of a reductive group G and show how they relate to questions about system of generators and systems of parameter for the invariants.We first recall the definition of a polarization of a regular function f ∈ O(V ). For k ≥ 1 and arbitrary parameters t 1 , . . . , t k we write

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“…Lemma 5.2 also follows from the main theorem in [6]. The above proof, which uses only elementary methods, is included in order to be self-contained.…”

Section: Automorphisms Of Moduli Spaces Of Symplectic Bundlesmentioning

confidence: 98%

Automorphisms of Moduli Spaces of Symplectic Bundles

2012

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Let X be an irreducible smooth complex projective curve of genus g ≥ 4. Fix a line bundle L on X. Let M Sp (L) be the moduli space of semistable symplectic bundles (E, ϕ : E ⊗ E → L) on X, with the symplectic form taking values in L. We show that the automorphism group of M Sp (L) is generated by the automorphisms of the form E → E ⊗ M , where M 2 ∼ = O X , together with the automorphisms induced by automorphisms of X.

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Nilpotent subspaces of maximal dimension in semi-simple Lie algebras

Cited by 18 publications

References 18 publications

Equations of some wonderful compactifications

Equations of some wonderful compactifications

Polarizations and Nullcone of Representations of Reductive Groups

Automorphisms of Moduli Spaces of Symplectic Bundles

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