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The adjoint representation in rings of functions [PDF]

Representation Theory of the American Mathematical Society, 1997
Let G G be a connected, simple Lie group of rank n n defined over the complex numbers. To a parabolic subgroup P P in G G of semisimple rank r r , one can associate n − r n-r positive integers coming from the theory of ...
Peter E. Trapa, Eric Sommers
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Adjoint Representations and the Derivative of exp [PDF]

, 2020
In this chapter, in preparation for defining the Lie bracket on the Lie algebra of a Lie group, we introduce the adjoint representations of the group \(\mathbf {GL}(n, {\mathbb {R}})\) and of the Lie algebra \({\mathfrak {gl}}(n, {\mathbb {R}})\). The map \(\mathrm {Ad}\colon \mathbf {GL}(n, {\mathbb {R}})\rightarrow \mathbf {GL}({\mathfrak {gl}}(n ...
Jocelyn Quaintance, Jean Gallier
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Exterior Powers of the Adjoint Representation

Canadian Journal of Mathematics, 1997
AbstractExterior powers of the adjoint representation of a complex semisimple Lie algebra are decomposed into irreducible representations, to varying degrees of satisfaction.
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Adjoint representations of exceptional Lie algebras

Theoretical and Mathematical Physics, 1987
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
M. A. Ol'shanetskii, V. B. K. Rogov
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The Adjoint Representation and the Adjoint Action

2002
The purpose of this article is to study in detail the actions of a semisimple Lie or algebraic group on its Lie algebra by the adjoint representation and on itself by the adjoint action. We will focus primarily on orbits through nilpotent elements in the Lie algebra; these are called nilpotent orbits for short.
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A homotopy construction of the adjoint representation for Lie groups [PDF]

Mathematical Proceedings of the Cambridge Philosophical Society, 2002
Let G be a compact, simply-connected, simple Lie group and T ⊂ G a maximal torus. The purpose of this paper is to study the connection between various fibrations over BG (where G is a compact, simply-connected, simple Lie group) associated to the adjoint representation and homotopy colimits over poset categories [Cscr ], hocolim[Cscr ]BGI where
Nitu Kitchloo, Natàlia Castellana
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LIE ALGEBRAS WITH AN ALGEBRAIC ADJOINT REPRESENTATION

Mathematics of the USSR-Sbornik, 1984
An algebra R over a field K satisfies the property P locally, if P holds for every finitely generated subalgebra of R. A famous result of A. I. Kostrikin claims that every Lie algebra G satisfying the Engel condition g(ad h)\({}^ n=0\) for any g,\(h\in G\), is locally nilpotent if char K\(=0\) or char K\(=p>n\).
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The deduction of the Lax representation for constrained flows from the adjoint representation

Journal of Physics A: Mathematical and General, 1993
Summary: For \(x\)- and \(t_ n\)-finite-dimensional Hamiltonian systems obtained from the decompositions of zero-curvature equations, it is shown that their Lax representations can be deduced directly from the adjoint representations of the auxiliary linear problems.
Yishen Li, Yunbo Zeng
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Adjoints, representables and limits

2014
We have approached the idea of universal property from three different angles, producing three different formalisms: adjointness, representability, and limits. In this final chapter, we work out the connections between them. In principle, anything that can be described in one of the three formalisms can also be described in the others.
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Adjoint Functors and Representation Dimensions

Acta Mathematica Sinica, English Series, 2006
We study the global dimensions of the coherent functors over two categories that are linked by a pair of adjoint functors. This idea is then exploited to compare the representation dimensions of two algebras. In particular, we show that if an Artin algebra is switched from the other, then they have the same representation dimension.
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