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Weierstrass sections for some truncated parabolic subalgebras
Representation Theory · Submitted 26 Jul 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Extending the notion of a Weierstrass section introduced by Popov to the coadjoint action of the truncated parabolic subalgebra associated with $\mathfrak p$, we construct explicitly Weierstrass sections, which give the polynomiality (when it was not yet known) for the algebra generated by semi-invariant polynomial functions on the dual space $\mathfrak p^*$ of $\mathfrak p$ and which allow to linearize the semi-invariant generators.
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The abstract, sentence by sentence
In this paper, using Bourbaki's convention, we consider a simple Lie algebra $\mathfrak g\subset\mathfrak g\mathfrak l_m$ of type B, C or D and a parabolic subalgebra $\mathfrak p$ of $\mathfrak g$ associated with a Levi factor composed essentially, on each side of the second diagonal, by successive blocks of size two, except possibly for the first and the last ones. Extending the notion of a Weierstrass section introduced by Popov to the coadjoint action of the truncated parabolic subalgebra associated with $\mathfrak p$, we construct explicitly Weierstrass sections, which give the polynomiality (when it was not yet known) for the algebra generated by semi-invariant polynomial functions on the dual space $\mathfrak p^*$ of $\mathfrak p$ and which allow to linearize the semi-invariant generators. Our Weierstrass sections require the construction of an adapted pair, which is the analogue of a principal $\mathfrak s\mathfrak l_2$-triple in the non reductive case.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper, using Bourbaki's convention, we consider a simple Lie algebra $\mathfrak g\subset\mathfrak g\mathfrak l_m$ of type B, C or D and a parabolic subalgebra $\mathfrak p$ of $\mathfrak g$ associated with a Levi factor composed essentially, on each side of the second diagonal, by successive blocks of size two, except possibly for the first and the last ones. | background 0.92 | 0.49 | 0.03 | asserted 0.93 |
| s1 | Extending the notion of a Weierstrass section introduced by Popov to the coadjoint action of the truncated parabolic subalgebra associated with $\mathfrak p$, we construct explicitly Weierstrass sections, which give the polynomiality (when it was not yet known) for the algebra generated by semi-invariant polynomial functions on the dual space $\mathfrak p^*$ of $\mathfrak p$ and which allow to linearize the semi-invariant generators. | contribution 0.91 | 0.96 | 0.84 | asserted 1.00 |
| s2 | Our Weierstrass sections require the construction of an adapted pair, which is the analogue of a principal $\mathfrak s\mathfrak l_2$-triple in the non reductive case. | method 0.93 | 0.70 | 0.07 | asserted 1.00 |
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Original abstract
In this paper, using Bourbaki's convention, we consider a simple Lie algebra $\mathfrak g\subset\mathfrak g\mathfrak l_m$ of type B, C or D and a parabolic subalgebra $\mathfrak p$ of $\mathfrak g$ associated with a Levi factor composed essentially, on each side of the second diagonal, by successive blocks of size two, except possibly for the first and the last ones. Extending the notion of a Weierstrass section introduced by Popov to the coadjoint action of the truncated parabolic subalgebra associated with $\mathfrak p$, we construct explicitly Weierstrass sections, which give the polynomiality (when it was not yet known) for the algebra generated by semi-invariant polynomial functions on the dual space $\mathfrak p^*$ of $\mathfrak p$ and which allow to linearize the semi-invariant generators. Our Weierstrass sections require the construction of an adapted pair, which is the analogue of a principal $\mathfrak s\mathfrak l_2$-triple in the non reductive case.
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Then if there exists a Weierstrass section $y+V⊂^*$ for coadjoint action of $$, one has by [ref] that $y+V$ is an affine slice to the coadjoint action of $$.
Adapted pairs and Weierstrass sections · Affine slice. · 0.97
Cites as background 1 paper
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A Weierstrass section for coadjoint action of $$ (see [ref] ) is an affine subspace $y+V$ of $^*$ (with $y∈^*$ and $V$ a vector subspace of $^*$) such that restriction of functions of $S()=[^*]$ to $y+V$ induces an algebra isomorphism between $Y()$ and the algebra of polynomial functions $[y+V]$ on $y+V$.
Adapted pairs and Weierstrass sections · Weierstrass sections. · 0.91
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