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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Follow-up to Steinberg's problem (12) in his 1966 ICM talk?

Steinberg's lecture at the 1966 ICM in Moscow here surveyed his work on regular elements of semisimple algebraic groups, while also formulating a number of then-open questions as "problems" (...
Jim Humphreys's user avatar
18 votes
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977 views

"Special" meanders

One of the open problems in combinatorics is enumeration of meanders. Here on MO I only could find them under the heading not-especially-famous-long-open-problems-which-anyone-can-understand. Since my ...
მამუკა ჯიბლაძე's user avatar
17 votes
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Relations in a certain Lie algebra

Let ${\mathfrak g}$ be the (real) Lie algebra generated by infinitely many generators $D_i, E_i$ with $i=1,2,3,\dots$ subject to the following relations for any natural numbers $i,j$: \begin{gather*} [...
Terry Tao's user avatar
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Ado's theorem and the reduction to positive characteristic

The synopsis: proofs of Ado theorem in positive characteristic are simple, and in characteristic $0$ are difficult. Can one infer the characteristic $0$ case from the positive characteristic case? The ...
Dmitrii Korshunov's user avatar
17 votes
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546 views

Lie algebras vs. graph complexes

A ribbon graph is a graph in which every vertex has valence at least three and is equipped with a cyclic ordering of its adjacent half edges. The ribbon graph complex $\mathcal{G}_*$ is the chain ...
Manuel Rivera's user avatar
17 votes
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903 views

Combinatorial identity involving the Coxeter numbers of root systems

The setup is: $R$ = irreducible (reduced) root system; $D$ = connected Dynkin diagram of $R$, with nodes numbered $1,2,...,r$; $\hat D$ = extended Dynkin diagram, nodes numbered $0,1,2,...,r$; $\...
Jeffrey Adams's user avatar
16 votes
0 answers
739 views

Is there a "natural" proof of the equality $4^2=2^4$?

This question, or rather any answer that it might receive, would probably belong to the realm of Awfully sophisticated proof for simple facts. Still, I claim that I have quite serious motivation for ...
მამუკა ჯიბლაძე's user avatar
15 votes
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256 views

Lie theoretic meaning to $e^{\text{cycle}} = \text{permutation}$?

It is well known that exponentiating the EGF(exponential generating function) for cycles gives the EGF for permutations: link here. Usually summarized under the catchy slogan ...
Siddharth Bhat's user avatar
15 votes
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214 views

Geometry of Affine Kac-Moody Algebras

I recently asked this question on phys.SE and it was suggested to me to ask it here. One can reconstruct the unitary irreducible representations of compact Lie groups very beautifully in geometric ...
user356359's user avatar
14 votes
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355 views

Gel'fand and Fuks' "Globalizing" of cohomology of formal vector fields

I apologize for the length of this question. If anybody already spent some time with cohomology of (formal) vector fields and the results of Gel'fand and Fuks, I imagine a lot can be skipped. I do ...
Lukas Miaskiwskyi's user avatar
14 votes
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Local proof of Grothendieck-Riemann-Roch theorem

There is a theorem by Feigin and Tsygan(Theorem 1.3.3 here) which they call "Riemann-Roch" theorem. Given a smooth morphism $f:S\to N$ of relative dimension $n$ and a vector bundle $E/S$ of ...
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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 ...
ArB's user avatar
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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 ...
Jim Humphreys's user avatar
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208 views

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 ...
Jim Humphreys's user avatar
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503 views

Irreducible representations of Weyl group of F$_4$ on zero weight spaces?

This is a follow-up to a recent question here concerning the natural representation of a Weyl group $W$ on the zero weight space of an irreducible representation $L(\lambda)$ of highest weight $\...
Jim Humphreys's user avatar
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Quiver and relations for blocks of category $\mathcal{O}$

In Vybornov - Perverse sheaves, Koszul IC-modules, and the quiver for the category $\mathscr O$ an algorithm is presented to calculate quiver and relations for blocks of category $\mathcal{O}$ . ...
Mare's user avatar
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11 votes
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The $\frak{sl}_2$-representation on a symplectic manifold

Any symplectic manifold $(M,\omega)$ carries a representation of $\frak{sl}_2$: Define the maps $$ L: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~~~~ \Lambda: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~...
Pierre Dubois's user avatar
11 votes
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619 views

How to define (and compute) the Cartan-Killing form of the group of volume-preserving diffeomorphisms?

This question was raised a while ago in a blog post by Terry Tao on the Euler-Arnold equation and he called it "quite tricky". Has anyone in the meantime tried to formulate this question precisely, ...
S.Surace's user avatar
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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 ...
Jim Humphreys's user avatar
11 votes
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410 views

Connection between Gelfand-Tsetlin basis and SSYT basis in Schur module

Consider an $n$-dimensional complex vector space $V$ with a chosen basis $e_1,\ldots,e_n$. This basis defines a Cartan decompostion of $GL(V)\cong GL_n$ and for an (integral dominant) highest weight $\...
Igor Makhlin's user avatar
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10 votes
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Each simple real Lie algebra as a representation of its maximal compact subalgebra

I am interested in a detailed description of the Cartan decomposition of each type of simple, real, finite-dimensional Lie algebra. (This is essentially a question about the classification of simple, ...
Greg Kuperberg's user avatar
10 votes
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96 views

Non-linear version of the Chevalley–Eilenberg complex

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra. In small degrees, the differentials of the Chevalley–Eilenberg complex $C^\bullet(\mathfrak{g}, \mathfrak{g})$ with values in the adjoint ...
Tobias Diez's user avatar
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285 views

Did anybody come across this Lie algebra representation?

Let $\mathfrak{g}$ be – for the sake of simplicity – a complex Lie Algebra. Then we define the antisymmetric linear transformations as $$ \mathfrak{A(g)} =\big\{\left.\,\alpha \in \mathfrak{gl(g)}\,\...
Marius S.L.'s user avatar
10 votes
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397 views

Gram matrix determinant in dimension 4 and $E_8$

Consider a determinant of a Gram matrix in dimension $4$. $$\begin{vmatrix} 1 & -\cos(\alpha_1) & -\cos(\alpha_2) & -\cos(\alpha_3)\\ -\cos(\alpha_1) & 1 & -\cos(\alpha_6)& -\...
Daniil Rudenko's user avatar
10 votes
0 answers
1k views

Relative Lie algebra cohomology

Forgive me if this question is too elementary however, I haven't found an answer. If $\mathfrak{g}$ is a Lie algebra one can define its Lie algebra cohomology: the definition is quite similar to the ...
truebaran's user avatar
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10 votes
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479 views

What is the importance of the number $k+h^{∨}$ (level+dual Coxeter number)?

The number $k+h^{∨}$ appears at many places in the representation theory of affine Lie algebras (and probably elsewhere). Here $h^{∨}$ is the dual Coxeter number of the root system, and $k$ is the ...
Adam Gyenge's user avatar
10 votes
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266 views

differentiating positive energy LG reps

Background:Let $G$ be a cscsc¹ Lie group, and let $\widetilde{LG}$ be the universal central extension (center = $S^1$) of $LG:=Map_{C^\infty}(S^1,G)$, with the topology inherited from the $C^\infty$ ...
André Henriques's user avatar
10 votes
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517 views

Explicit change of basis for the Schur-Weyl basis

The Schur-Weyl duality states that $\bigotimes_{m=1}^n \mathbb{C}^k$ can be decomposed as a direct sum over the tensor product of irreductible representations of $SU(k)$ and of the symmetric group $\...
Damien S.'s user avatar
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345 views

How large must the characteristic of $k$ be, for the cohomology of the Lie algebra $\mathfrak{sl}_n(k)$ to be exterior as in characteristic zero?

$\DeclareMathOperator\SU{SU}$In this question, all Lie algebra cohomology is of the form $H^*(\mathfrak{g}; k)$, with $k$ the trivial one-dimensional representation of $\mathfrak{g}$. All Lie algebra ...
user509184's user avatar
9 votes
0 answers
461 views

What is wrong with $A^{(2)}_{2n}$?

When dealing with affine Kac-Moody groups, especially geometrically (e.g. by examining their affine flag varieties or affine Grassmannians) I've been taught that time and time again, issues arise in ...
Marc Besson's user avatar
9 votes
0 answers
189 views

Octonionic Stiefel manifolds

The Stiefel manifolds are presented in this Wikipedia article over the division algebras $\mathbb{R,C,H}$. In fact, they are presented as homogeneous spaces, respectively for the $A,B,C$,and $D$ ...
Fofi Konstantopoulou's user avatar
9 votes
0 answers
159 views

Can semisimple orbits be written $\exp(\mathfrak{g})\cdot x$?

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $G$ be its adjoint group. If $x\in\mathfrak{g}^{rs}$ is a regular semisimple element, is its orbit $$G\cdot x=\{\mathrm{Ad}_gx:g\in G\}$$ ...
Simon Parker's user avatar
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450 views

Branching rules for compact Lie groups

Let $G$ be a compact connected Lie group, and let $H\subset G$ be a closed subgroup. For an irreducible representation $\pi:G\to\mathrm{End}_\mathbb{C}(V)$ of $G$ ($\dim\pi<\infty$) I want to know ...
Bedovlat's user avatar
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9 votes
0 answers
242 views

Homology of an interesting Lie algebra

Let $U$ and $V$ be finite dimensional complex vector spaces (or perhaps graded vector spaces). Let $E(U)$ be the "square zero extension" $\mathbb C \oplus U$, made into a commutative ring in such a ...
Dan Petersen's user avatar
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9 votes
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187 views

Littlewood-Richardson sequences and Littlewood-Richardson coefficients

I'm looking for a proof or a reference for the following statement, I give the definitions below: There exists a Littlewood-Richardson sequence of type $(\alpha, \beta, \lambda)$ if and only if $...
Maffred's user avatar
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9 votes
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181 views

Super-extensions of the Poincaré Lie algebra

For $(\mathfrak{g},[-,-])$ an ordinary Lie algebra let me say that a super-extension of it (maybe not the best terminology) is a super-Lie algebra $(\mathfrak{s}, [-,-]_{\mathfrak{s}})$ whose bosonic ...
Urs Schreiber's user avatar
9 votes
0 answers
654 views

Software for explicit computations in representations of classical Lie algebras

I'm pretty sure many a mathematician has longed for such a tool but I wasn't able to find such a question here, so here we go. Is there, by any chance, an existing package or program that allows one ...
Igor Makhlin's user avatar
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9 votes
0 answers
356 views

Finding $U,V$ in Thompson's Formula

Thompson's formula says, given $A,B \in \mathfrak{su}(n)$, there exists $U,V \in SU(n)$ such that: $e^{A}e^{B}=e^{UAU^{\dagger} + VBV^{\dagger}}$ Given $a,b \in \mathfrak{su}(4)$ defined by: $a=J_x ...
Benjamin's user avatar
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9 votes
0 answers
404 views

How to show the compatibility between Duflo isomorphism and Harish-Chandra isomorphism for semi-simple Lie algebras?

I was told that the Duflo isomorphism is compatible with the Harish-Chandra isomorphism when the Lie algebra $\mathfrak{g}$ is semi-simple. However I cannot see why this is true. All I can show is ...
Xuxu's user avatar
  • 663
8 votes
0 answers
217 views

What can we say about the homogeneous spaces $E_8/E_7$ and $E_7/E_6$?

For the three exceptional compact Lie groups $E_8, E_7, E_6$ we have the inclusions $$ E_6 \subseteq E_7 \subseteq E_8. $$ What can we say about the the homogeneous spaces $$ E_8/E_7, ~~~~ E_7/E_6? $$ ...
Alain Rochefort's user avatar
8 votes
0 answers
643 views

Which infinite-dimensional Lie algebras have realizations as algebras of global sections of vector bundles with special structure?

I would not be surprised by downvotes since this question is at the same time very naïve, very vague and might be asking about things well known for decades. Specifically vagueness comes from the word ...
მამუკა ჯიბლაძე's user avatar
8 votes
0 answers
186 views

Cohomology algebra of the maximal nilpotent subalgebra of a semisimple Lie algebra

The answer to this question may be well-known, but I failed to locate it in any obvious source. From the results of Bott (Ann. Math. 66 (1957), 203-248) and Kostant (Ann. Math. 74 (1961), 329-387), it ...
Vladimir Dotsenko's user avatar
8 votes
0 answers
154 views

Alternate proof of easiest special case of PBW theorem

Let $L$ be a Lie algebra over a field $k$ of characteristic $0$ (I'm happy for that field to be $\mathbb{C}$) and let $U(L)$ be its universal enveloping algebra. One of the standard consequences of ...
Roberta's user avatar
  • 153
8 votes
0 answers
176 views

Chevalley-Eilenberg cohomology of polynomial vector fields on $\mathbb{A}^2$

I have a question similar to one given here. What is the cohomology of the Lie algebra of polynomial vector fields on an affine space $\mathbb{A}^2$ over a field of characteristic $0?$ (I'm interested ...
Grisha Taroyan's user avatar
8 votes
0 answers
118 views

Invariant complex structures for simple Lie groups

For which simple Lie groups there exists a left-invariant complex structure?
Vít Tuček's user avatar
  • 8,159
8 votes
0 answers
368 views

Significance of half sum of non-simple positive roots

In representation theory, there are plenty of places that a $\rho$-shift makes an appearance, where $\rho$ is the half sum of positive roots. See, for instance, this post for some discussions of the ...
Dr. Evil's user avatar
  • 2,641
8 votes
0 answers
385 views

Which representations of the Lie algebra of a Lie group come from representations of the group itself?

I think this is very classic mathematics, but I can't find a complete answer in the literature. Let $G$ be a Lie group, $\mathfrak{g}$ the Lie algebra of $\mathfrak{g}$. Suppose $\rho : \mathfrak{g} \...
Rick Sternbach's user avatar
8 votes
0 answers
144 views

Semisimple Lie groups admitting a free algebra of invariants

Assume we work over an algebraically closed field of characteristic zero. I know that for a connected semisimple algebraic group there is an upper bound for the number of isomorphism classes of ...
svelaz's user avatar
  • 189
8 votes
0 answers
162 views

Generalizing a theorem of Kostant to arbitrary parabolics

Let $\mathfrak{g}$ be a simple complex Lie algebra and let $\Delta$ be a system of positive roots relative a choice of Cartan subalgebra and $\mathfrak{b}$ the corresponding Borel subalgebra. Let $B&...
Andy Sanders's user avatar
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