Questions tagged [lie-algebras]
Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie groups and differentiable manifolds.
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Are S(g) and U(g) isomorphic as g-modules for g Lie algebra over F_p ? Are S(g)^g and U(g)^g isomorphic as com.algs ?
If $g$ is Lie algebra over field char(k)=0,
then the following facts are well-known:
1) S(g) and U(g) are isomorphic as $g$-modules. (Symmetrization map S(g)->U(g) gives isomorphism).
2) S(g)^g and ...
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D-modules supported on the nilpotent cone
I am reading some papers which involve D-modules on a Lie algebra g, which are supported on the nilpotent cone n. They are equivariant for the action of G. (In particular, I consider g=sl_n).
It ...
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Tilting Objects in BGG Categories $\mathcal{O}$
Let $\mathcal{O}$ be the BGG category $\mathcal{O}$ with respect to a finite dimensional semisimple Lie algebra $\mathfrak{g}$ and its Borel subalgebra $\mathfrak{b}$ (as define in this book by ...
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Can one define quantized universal enveloping algebras in a basis-free way?
(For the background, I am learning about quantum groups — essentially in order to understand crystal/global/canonical bases in the context of this question — from the books by Jantzen and by Hong&...
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Torsion for Lie algebras and Lie groups
This question is about the relationship (rather, whether there is or ought to be a relationship) between torsion for the cohomology of certain Lie algebras over the integers, and torsion for ...
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Why is every deformation of the universal enveloping algebra of a complex semisimple Lie algebra trivial?
I have read in these lecture notes that every deformation $U_h(\mathfrak{g})$ of $U(\mathfrak{g})$ is trivial, i.e. isomorphic to $U(\mathfrak{g})[[h]]$ as associative $\mathbb{C}[[h]]$-algebras. Why ...
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Introducing division strategically in operads to accommodate formulas like the Baker-Campbell-Hausdorff formula
I am not as familiar with operad terminology as I'd like to be, so I might be missing some well-known term in the area. If so, I'd appreciate any pointers to the correct terms.
Consider the following ...
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What's the dimension of the Lie algebra generated by transpositions on $n$ objects?
Define a Lie bracket on the group algebra of the permutation group $S_n$ in the following way:
$$[\sigma, \tau] = \sigma\circ\tau - \tau\circ\sigma,$$
where $\sigma, \tau \in S_n$, and the ...
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Why the name O for category O?
What is the motivation behind naming the category O appearing in the theory of Lie algebras? Does O stand for something?
Here is a question Why the BGG category O? that further confuses me. It seems ...
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What are Harish-Chandra bimodules used for?
There are many recent papers on classification of Harish-Chandra bimodules for rational Cherednik algebras and, more generally, non-commutative algebras which are quantizations of symplectic ...
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Ado's theorem for metric Lie algebras?
Background
Ado's Theorem states that every finite-dimensional Lie algebra over a field of zero characteristic admits a faithful representation.
More precisely, if $\mathfrak{g}$ is a finite-...
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About enveloping algebras of direct sums
This question is imported from MSE. It is linked to this one in the case of semi-direct products.
My question Let us consider a Lie $R$-algebra ($R$ is a commutative ring) written as a (module) ...
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Free groups and free restricted Lie algebras
If $G$ is any group and $\gamma_k(G)$ denotes the $k$th term in the lower central series of $G$, then the commutator bracket on $G$ endows
$$\mathcal{L}(G) = \bigoplus_{k=1}^{\infty} \gamma_k(G) / \...
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Variety of nilpotent Lie algebras or $p$-groups
Here's a couple of analogous questions, one in terms of finite-dimensional complex Lie algebras and one in terms of finite $p$-groups; I'd be interested in an answer to either:
1) Let $\mathcal{L}$ ...
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Characteristic subgroup of nilpotent group that is not invariant under powering
I want an example of a nilpotent group $G$, a characteristic subgroup $H$, and a prime number $p$ such that:
$G$ is $p$-powered, i.e., every element of $G$ has a unique $p^{th}$ root in $G$.
$H$ is ...
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Significance of half-sum of positive roots belonging to root lattice?
Let $\Phi$ be a (crystallographic) root system and $\Phi^{+}$ a choice of positive roots, with $\Delta$ the corresponding choice of simple roots. So the root lattice of $\Phi$ is just $\mathbb{Z}\...
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Relationship between crystal root operators and usual $e_i, f_i$?
Suppose I am working in a symmetrizable Kac–Moody Lie algebra $\mathfrak{g}$. Let $e_1,\dotsc,e_n,f_1,\dotsc,f_n$ denote the usual Chevalley generators of $\mathfrak{g}$. Let $V$ be a highest weight ...
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Conceptual meaning of (g, K)-cohomology
Let $G$ be a reductive Lie group (over $\bf C$, say), $K$ a maximal compact subgroup and $\frak g$, $\frak k$ their Lie algebras.
It is standard that Lie algebra cohomology $H^n({\frak g}, V)$ of ...
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Source of a formula for tensor product multiplicities?
This is a follow-up to a recent question by Allen Knutson here, involving a special type of tensor product multiplicity for a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ (or other ...
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Elliptic Curves, Lattices, Lie Algebras
I've recently started to look at elliptic curves and have three basic questions:
Is it correct to say that elliptic curves $E$ in the projective plane are in bijective correspondence with lattices $...
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Geodesics on $SU(4)$
Are the geodesics of the following metrics on $SU(4)$ known or easy (in a way not known to me!) to find?
In the adjoint representation, one can express the Killing form as a matrix and consider it as ...
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Representation Theory of Lie Groups: Reference Request
I am looking for a reference that describes the correspondence between the (finite-dimensional) representations of (real) Lie groups and the representations of their Lie algebras. More precisely, ...
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Weight lattice and the fundamental group
Let $G$ be a compact connected Lie group and let $T$ be a maximal torus of $G$, with Lie algebras $\frak{g}$ and $\frak{t}$ respectively. Then, $\frak{t}$ can be considered as a Cartan subalgebra of $...
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Semisimplicity of Lie algebra in positive characteristic
Let $F$ be a field of characteristic $p > 0$. Let $\mathfrak{g}$ be a linear Lie algebra, that is $\mathfrak{g}\subset M_n(F)$ for some natural number $n$. Does there exist a condition involving $n$...
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Breaking up the free Lie algebra into GL irreps
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\L{\mathfrak{L}}$The free Lie algebra $\L(V)$ generated by an $r$-dimensional vector space $V$ is, in the
language of https://en.wikipedia.org/wiki/...
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$A \wedge A \wedge A$ in Chern-Simons
I am confused with the wedging operations of Lie algebra valued differential forms. Especially, for instance, I have some problems with the Chern-Simons 3-form
$$A \wedge dA + \frac{2}{3}A \wedge A \...
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Type of 26-dimensional representation of different real forms of the complex simple Lie algebra $F_4$
The exceptional complex simple Lie algebra $F_4$ has an irreducible 26-dimensional representation $V$ with Dynkin label [0,0,0,1] in the usual ordering of the simple roots one can find, say, in ...
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calculating Littlewood-Richardson coefficients
It is known that if $\alpha,\beta,\gamma$ are three partitions then the Littlewood-Richardson coefficient $c_{\alpha \beta}^{\gamma}$ is positive when the triple ($\alpha,\beta,\gamma$
) occurs as ...
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Real and quaternionic representations according to weights
According to this question, it is easy to know whether a (complex, finite-dimensional) representation is self-dual or not: just check if the weight distribution in space is symmetric about the origin.
...
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Which nilpotent Lie algebras appear as nilradicals of parabolic subalgabras?
I am interested to identify (ideally classify) nilpotent Lie algebras that occur as nilradicals of parabolic subalgebras in (say) reductive Lie algebras.
For example, all Heisenberg Lie algebras ...
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Lie algebras with abelian Cartan subalgebras
The Cartan subalgebras of a reductive Lie algebra are abelian.
Are there non-reductive Lie algebras with abelian Cartan subalgebras?
In fact, the elements of a Cartan subalgebra of a reductive ...
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How are Sheffer polynomials related to Lie theory?
Sheffer polynomials $\{P_n(x)\}$ have generating function $P(x,t) = \sum_{n=0}^{\infty}P_n(x)t^n=A(t)e^{xu(t)}$.
This form reminds me of the Lie group–Lie algebra correspondence. Is there any ...
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Non-conjugate subgroups that are conjugate in complexification
In trying to come up with a counter-example in my line of research, I would like to find an example as follows:
$G$ is a semisimple Lie group with complexification $G^{\mathbb{C}}$. $H_1, H_2 \...
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Summary of Lie-Algebra integration tactics
If this is in the scope of MO, I would like to gather here the known tactics of
Lie algebra integration, since it appear surprisingly hard to find such a
compendium, library or any other kind of ...
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On the isomorphism problem of enveloping algebras
Let $\mathfrak{g}$ and $\mathfrak{g}'$ be Lie algebras. It is known that if $U(\mathfrak{g})\cong U(\mathfrak{g}')$ as associative algebras, then it is not necessarily true that $\mathfrak{g}\cong \...
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Is there a canonical Hopf structure on the center of a universal enveloping algebra?
Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\mathbb C$. Define $\mathcal Z(\mathfrak g)$ to be the center of the universal enveloping algebra $\mathcal U\mathfrak g$, and define $(\...
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Chiral homology for the Virasoro algebra and/or affine Lie algebra
I want to understand what concrete analytical objects are found in chiral homology of higher degree of a vertex algera (-module) $M$. More precisely: I can obtain conformal blocks on a surface $\Sigma$...
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Some questions about the Malcev completion
Let $G$ be an abstract group. The Malcev completion $\widehat{G}$ of $G$ (over $\mathbb{Q}$) is the set of group-like elements in the complete Hopf algebra $\widehat{\mathbb{Q}[G]} = \lim_n \mathbb{Q}...
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Bounding weight multiplicities by number of certain Coxeter elements
This question concerns lower bounds of certain weight multiplicities in finite dimensional representations of algebraic groups (or Lie groups, Lie algebras).
Let's say $G$ is a simple algebraic group ...
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Smallest dimension of nontrivial representation of a simple Lie algebra over `$\mathbb{C}$`
The question involved here is natural and very classical, but I'm unsure what has been formally stated and proved in the literature. The only approach I know involves assembling facts that apparently ...
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Comparing two similar procedures for quantizing a Casimir Lie algebra
My primary reference for this question is the very good book Quantum Groups and Knot Invariants by C. Kassel, M. Rosso, and V. Turaev. I'm also drawing from P. Etingof and O. Schiffmann, Lectures on ...
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What's the most simple proof of Kostant's version of Borel-Weil-Bott for Lie Algebra cohomology?
Besides Kostant's original proof (in http://www.math.tamu.edu/~jml/kostant61.pdf) of the above mentioned theorem (using the Lie Algebra Laplacian), there are a few other approaches:
Casselman-Osborne,...
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Lie's third theorem via differential graded algebras?
Dennis Sullivan, "Infinitesimal computations in topology", Publ. IHES: At the end of section 8, he writes, among other things, roughly the following.
Let $\mathfrak{g}$ be a (finite-dimensional, real)...
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Matrices into path algebras
I was thinking about quivers recently, and the following idea came to me.
Let ei,j denote the matrix unit in Mn for 1 ≤ i,j ≤ n. Let Γ denote the complete quiver on vertices {1, …,...
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Representations of $U_q(\mathfrak{sl}(2))$ as differential / difference operators
$\mathfrak{sl}(2)$ (over $\mathbb{C}$) with basis $E_\pm, H$ with commutation relations
$$
[H,E_{\pm}]=\pm 2 E_\pm,\quad [E_+,E_-]=H
$$
admits the well-known representation on $\mathbb{C}[x]$ with
$$
...
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Comparing a Chevalley basis with the canonical basis of the adjoint module?
First some background: Given a simple Lie algebra $\mathfrak{g}$ over an algebraically closed field of characteristic 0 such as $\mathbb{C}$, fix a Cartan decomposition $\mathfrak{g} = \mathfrak{h} \...
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What are the simple Lie superalgebras of type E?
Background
Simple finite dimensional Lie superalgebras over $\Bbb C$ have been classified. There are the Cartan type superalgebras (algebras of purely odd vector fields), two strange families P(n) ...
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Cartan involution for finite W-algebras
Does anybody know if there is an analog of the Cartan (anti)involution for W-algebra
associated to a nilpotent element e, which is principal in some Levi subalgebra
of semi-simple Lie algebra g? ...
- 6,957
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Is there a reasonable way to define "reductive Lie algebra" in prime characteristic?
Among the finite dimensional Lie algebras over a field of characteristic 0, there is a sensible definition of "reductive Lie algebra" going back at least to the 1960 first chapter of N. Bourbaki's ...
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Failure of surjectivity in Hotta-Springer specialization: examples for special unipotents?
Last weekend's workshop on Springer theory and its generalizations at UMass demonstrated how far the subject has expanded over four decades, but the original set-up for the Springer correspondence ...
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