32,089 reputation
368142
bio website math.umass.edu/~jeh
location U. Massachusetts, Amherst
age 75
visits member for 5 years, 3 months
seen 9 hours ago
More-or-less retired professor at UMass Amherst. Basically an algebraist with interests in Lie theory, specifically representation theory and related algebraic geometry.

1d
revised Algebraic groups “generated” by a Lie algebra element
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1d
answered Algebraic groups “generated” by a Lie algebra element
May
21
comment Preprint by Wall on Sjogren's theorem
Wall's paper is short (and the volume exists in many libraries), but the standard Springer link online gives only the first two pages: link.springer.com/chapter/10.1007/BFb0100741#page-1
May
21
comment real representation of a product group
@Qiaochu: I was only commenting on the fact that finite groups seldom benefit from being viewed as compact Lie groups. Of course, they technically fall under this heading, but does anyone use Lie theory to study representations of arbitrary finite groups? Most of the essential Lie theory presupposes connectedness. (Of course it's true historically in the other direction that finite group ideas helped inspire some developments for compact groups.)
May
20
comment About supersolvable Lie algebras
I'm not sure where this definition is stated, but your version doesn't make sense. Can you be more exact about sources (and try to avoid anonymous sources)?
May
20
comment real representation of a product group
@Qiaochu: Yes, but finite groups are usually misleading when regarded as compact Lie groups. Maybe the definition allows for this, but books treating representations of compact Lie groups don't usually cover finite groups and their rich theory beyond perhaps noting that some ideas for the compact (usually connected) groups arise by analogy from the Frobenius-Schur theory. Fine points of the representations of compact Lie groups usually involve maximal tori, Weyl groups, roots, etc.
May
20
comment real representation of a product group
@Megan: I'm confused by your example, since there is no obvious structure of "compact Lie group" on $\mathbb{Z}_p$. How are you defining compact Lie groups?
May
16
comment Which p-adic algebraic groups are type I?
@Alain: Note that Joseph Bernstein's name has an extra "n", though there is also a mathematician named Berstein. (To add to the name confusion, early English translations of Russian papers co-authored by Joseph Bernstein gave his initials as I.N.)
May
16
comment Which p-adic algebraic groups are type I?
To add a further link for those wanting Bernstein's note in the original Russian: mathnet.ru/php/…
May
15
comment How bad can $\pi_1$ of a linear group orbit be?
@hic: While algebraic groups (as in your reference to McNinch and Sommers) come into play indirectly here, the component groups in the purely algebraic situation are always finite.
May
15
awarded  Notable Question
May
12
comment Motivation for a representation of the Witt algebra
These are useful but informal (and sometimes incomplete) lecture notes with minimal references, so it's better for you to consult published sources along with the notes. Aside from that, the variable $z$ here appears at the very beginning and differentiation obeys the usual rules. The notation comes from the way physicists use calculus, but here the set-up is more abstract and algebraic.
May
9
comment A table for irreducible integral representation of finite cyclic groups
Concerning the edit, it's worth emphasizing that there is a big difference between "irreducible" and "indecomposable" in this kind of representation theory.
May
7
revised Representation Theory of Lie Groups: Reference Request
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May
7
answered Representation Theory of Lie Groups: Reference Request
May
6
revised Reference for class of involutions containing longest element of finite Coxeter group?
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May
5
asked Reference for class of involutions containing longest element of finite Coxeter group?
Apr
29
comment What is the importance of the number $k+h^{∨}$ (level+dual Coxeter number)?
I can't provide an answer, but maybe it's worthwhile to point out that the question is indirectly related to the mysterious role of the "critical level" usually defined as $k = -h^\vee$: in this extreme case your number is 0 and can't be divided out. At the critical level the representations of an affine Lie algebra turn out to be closely related to characteristic $p$ representations.
Apr
27
awarded  Good Answer
Apr
24
comment Centralizer of a non-regular Lie algebra element
The formulation of the question isn't clear to me: is the Lie algebra here just the full $n \times n$ matrix algebra over $\mathbb{C}$? In general, a "regular" element of a semisimple (or reductive) Lie algebra is one whose centralizer has minimal dimension, necessarily equal to the rank. This makes sense even if the element isn't semisimple (= diagonalizable over algebraic closure). Moreover, the centralizer is then commutative (an abelian Lie algebra), but in general that's far from true. There is a lot of literature about this in the Lie algebra or the algebraic group setting.