NDLI: THE HIDDEN SEMI-INVARIANTS GENERATORS OF AN ALMOST-FROBENIUS BIPARABOLIC

Content Provider	Springer Nature Link
Author	JOSEPH, ANTHONY
Copyright Year	2014
Abstract	Let $$ \mathfrak{a} $$ be an algebraic Lie algebra and A its adjoint group. One calls $$ \mathfrak{a} $$ Frobenius if A has a dense orbit in $$ \mathfrak{a} $$ . In this case the algebra Sy( $$ \mathfrak{a} $$ ) spanned by the A semi-invariant polynomials on $$ \mathfrak{a} $$ is not reduced to scalars and is polynomial. Again d := dim a is even and after Ooms [27] the generators all appear as factors of the coe_cient of ^d $$ \mathfrak{a} $$ in the d = 2-th power of the Poisson bivector of $$ \mathfrak{a} $$ .Though appealing, this result does not give the weights of the generators, nor their degrees. In the present work we compute these weights and degrees in the case when $$ \mathfrak{a} $$ is a Frobenius biparabolic subalgebra $$ \mathfrak{q} $$ of a simple Lie algebra $$ \mathfrak{g} $$ . These results are relatively easy consequences of our earlier work on biparabolic subalgebras [7], [16], [18], except for the question of the existence of \hidden generators". The latter have squares which can be obtained by a general construction using the Hopf dual U( $$ \mathfrak{q} $$ )★ of U( $$ \mathfrak{q} $$ ), yet are themselves not easily proven to exist. These \hidden generators" only occur outside types A and C.To construct these hidden generators, the notion of a more general almost-Frobenius biparabolic subalgebra is introduced. It is shown that Sy( $$ \mathfrak{a} $$ ) is polynomial if q is an almost-Frobenius biparabolic and may itself have \hidden generators" outside types A and C. However, it is possible to construct these hidden generators through a rather precise description of the generators of an almost-Frobenius biparabolic subalgebra of a simple Lie algebra of type A.
Starting Page	735
Ending Page	778
Page Count	44
File Format	PDF
ISSN	10834362
Journal	Transformation Groups
Volume Number	19
Issue Number	3
e-ISSN	1531586X
Language	English
Publisher	Springer US
Publisher Date	2014-05-16
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Topological Groups, Lie Groups Algebra
Content Type	Text
Resource Type	Article
Subject	Algebra and Number Theory Geometry and Topology

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