All Questions
931 questions with no upvoted or accepted answers
4
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150
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Differential graded Lie algebra over an ordinary Lie algebra
Given a dg (differential graded) Lie algebra $L$ and an ordinary Lie algebra $\mathfrak{g}$,
is there written somewhere a formal definition of $L$ as a dg Lie algebra over $\mathfrak{g}$?
- 3,880
4
votes
0
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155
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Is one of the hyperplane partitions of a irreducible root system always generate the whole Weyl group?
Let $\Delta$ be a irreducible root system and $\Delta^+$ be its positive roots.
We say a subset $\Delta^{\prime}\subset \Delta^+$ can generate the Weyl group if reflections of roots in $\Delta^{\...
- 9,019
4
votes
0
answers
300
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Number of connected components of the centre of a Levi subgroup
Let $G$ be a connected complex semisimple algebraic group and $T\subset B\subset G$ a choice of maximal torus and Borel subgroup. Let $\Phi$ be the root system and $\Pi\subset\Phi$ the set of simple ...
- 41
4
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0
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64
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Constructing $\delta$ to show Chevalley-Eilenberg complex $\Lambda \mathfrak g \otimes \mathcal U\mathfrak g \rightarrow k$ is null homotopic
Let $\mathfrak g$ be a finite dimensional Lie algebra over a field $k$ and $(\Lambda^\bullet \mathfrak g \otimes \mathcal U\mathfrak g, d)$ be the Chevalley-Eilenberg chain complex.
I would like to ...
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4
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0
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133
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Non-cyclic Galois groups over the field of formal Laurent series in positive characteristic
This should be an easy question, but I am unfortunately not able to give an answer, so I am sorry if this is not the appropriate level for the site.
Let $C$ be an algebraically closed field of ...
- 41
4
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0
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234
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$L_\infty$-quasi inverse for the contravariant Cartan model on principal bundles
First of all I want to apologize for the much too long post.
A Lie group $G$ is acting on a smooth manifold $M$, then we define
\begin{align*}
T^k_G(M)=
(S^\bullet \mathfrak{g}\otimes T^k_\mathrm{...
4
votes
0
answers
396
views
Lie algebras with unique invariant bilinear symmetric form
A complex, finite dimensional Lie algebra admitting a (up to multiplication by a non-zero scalar) unique, invariant, symmetric, non-degenerate bilinear form is 1-dimensional or simple. Cf 'Lie ...
- 427
4
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0
answers
169
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Fibered surfaces degenerating to Frobenius
Let $R$ be a DVR with algebraically closed, positive characteristic residue field $k$. Let $X\rightarrow Spec(R)$ and $C\rightarrow Spec(R)$ be smooth projective morphisms of relative dimension 2 and ...
- 599
4
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0
answers
108
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Mysterious identity in cosimplicial $R$-module with Lie brackets
I have a cosimplicial $R$-module $\mathscr{A}:=(A_n)_{n\ge 0}$ and on each $A_n$ a Lie bracket $[-,-]_n:A_n\otimes A_n\to A_n$. Denote the cofaces by $d_i:A_n\to A_{n+1}$ and the codegeneracies by $...
- 1,623
4
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279
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Are vertex operator algebras ever conspiratorial?
I have a vertex operator algebra (VOA) $V$ with all niceness properties (unitary, rational, CFT type, etc). Its Lie algebra $\mathfrak{g} = V_1$ of spin-$1$ fields is large, and I understand how the ...
- 54.6k
4
votes
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132
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Panyushev's conjectured duality for root poset antichains
In his 2004 paper "ad-nilpotent ideals of a Borel subalgebra: generators and duality" (https://www.sciencedirect.com/science/article/pii/S0021869303006380), Panyushev conjectured (Conjecture 6.1) the ...
- 24.2k
4
votes
0
answers
223
views
Deformation of the Hochschild-Kostant-Rosenberg isomorphism for universal enveloping algebra
Let $\mathfrak{g}$ be a Lie algebra over a char. $0$ field and let $\iota: U\mathfrak{g}\rightarrow S\mathfrak{g}$ be the Poincaré-Birkhoff-Witt (PBW) isomorphism, inverse to that natural map from the ...
user108998
4
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0
answers
391
views
Trace of the adjoint action is an eigenvalue in $\mathrm{U}(L)$?
Let $L$ be a finite-dimentional complex Lie algebra. $\forall x \in L$, one defines the adjoint action of $x$ on $L$ as the map
$\mathrm{ad}_x : L \to L, \text{ with } \mathrm{ad}_x(y) = [x,y]$
for ...
- 71
4
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62
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About isomorphism of Extension groups between Category $\mathcal{O}^\mathfrak{p}$ and $\mathcal{O}$
In the paper: Kostant modules in blocks of category $\mathcal{O}^\mathfrak{p}$
In section 8.2 (p.22), I use the notation in Humphrey's Category $\mathcal{O}$ book.
Then the passage said the ...
- 1,875
4
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275
views
Symmetric power contained in tensor power?
Let $V$ be an $R$-module. Traditionally the symmetric algebra $S(V)$ is defined as a quotient of the tensor algebra $T(V)$, by the ideal generated by all $a\otimes b-b\otimes a$.
Can $S^n(V)$ also be ...
- 2,519
4
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answers
91
views
Good range and fair range
Let $G$ be a noncompact simple Lie group with complexified Lie algebra $\mathfrak{g}$. Fix a Cartan involution $\theta$, which defines a maximal compact subgroup $K$ of $G$. Take a $\theta$-stable ...
- 951
4
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468
views
Quaternion algebras in characteristic 2
Let $k$ be a field and let $Q$ be a quaternion algebra over $k$.
It is well known that, if $\mathrm{char}\,k\neq 2$, one can define $Q$ as the $k$-algebra of dimension $4$ generated by elements $x,y$ ...
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4
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answers
256
views
Second symmetric square of the adjoint representation
I have just come across the following experimental fact. Let ${\mathfrak g}$ be a simple complex Lie algebra.
Fact: ${\mathfrak g}$ is a constituent of $S^2{\mathfrak g}$ if and only if ${\mathfrak ...
- 12.4k
4
votes
0
answers
139
views
"Signature Changing" Generalization of Lie Algebra?
I have in mind a mathematical structure I've never heard of before. Does anyone know what might be?
It is a manifold with vector fields whose Lie brackets have structure coefficients that are ...
4
votes
0
answers
112
views
Restricted universal extensions and lifting of derivations
Let $L$ be a perfect Lie algebra. Then it is well-known that $L$ has a universal central extension $\hat{L}$ and every derivation of $L$ can be lifted to a derivation of $\hat{L}$. (See e.g. Section 2 ...
- 5,878
4
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350
views
How should one understand shifted Lie algebras, like $T_X[-1]$?
Let $X$ be an algebraic variety, $at(E) \in Ext^1(E, E \otimes T)$ the Atiyah class of a complex $E \in D(Coh X)$ (see Markaryan, $\S$1.1-1.2).
Then $at(\Omega[1])$ gives a "shifted Lie algebra" on $...
- 1,980
4
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0
answers
202
views
Shifted Schur functions
Let's fix the ground field $\mathbb{C}$.
In the paper "Shifted Schur Functions" http://arxiv.org/abs/q-alg/9605042 Andrei Okounkov and Grigori Olshanski introduce a special basis for the center $Z(\...
- 501
4
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0
answers
302
views
Does the tensor algebra $T(V)$ of $V$ isomorphic to the symmetric algebra of the free Lie algebra over $V$?
Let $V$ be a finite dimensional vector space. Let $T(V)$ be the tensor algebra over $V$.
Do we have $T(V) \cong S(Lie(V))$ as a graded vector space? Here $S(Lie(V))$ is the symmetric algebra of the ...
- 6,211
4
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0
answers
628
views
Lusztig's definition of quantum groups
In his book Introduction to quantum groups, Lusztig gives a definition (Def 3.1.1) of the rational form $U^{\mathbb{Q}(q)}_q$ that is rather different from the usual approach (see [1,Ch.9.1] for ...
- 1,813
4
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324
views
Center of affine W-algebras
Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and $k$ a complex number. Denote by $\hat{\mathfrak{g}}$ the corresponding affine Lie algebra ($\hat{\mathfrak{g}}=\mathfrak{g}((t)...
- 3,357
4
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0
answers
72
views
Harmonicity on semisimple groups
I asked this on Math.SE and got no answer, so I'll try my luck here.
Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in ...
- 41
4
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0
answers
327
views
Lie algebras whose derivation algebra is nilpotent
Let $L$ be a Lie algebra and denote by $\mathrm{Der}(L)$ the derivation algebra of $L$. If $L$ is finite-dimensional, then a theorem of Leger and Togo (Duke
Math. J. $\bf{26}$ (1959), 623 – 628, DOI: ...
- 5,878
4
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318
views
Action of orthogonal group on the free Lie algebra
This question is somewhat related and inspired by this post of professor Montgomery.
The free Lie algebra $L(V)$ generated by an $r$-dimensional vector space $V$ is, in the language of https://en....
- 277
4
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0
answers
173
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Poincaré-Birkhoff-Witt theorem for Leibniz algebras
Leibniz algebras can be seen as a non-skew-symmetric generalization of Lie algebras. I have already taken a look at some papers related to Leibniz algebras and extending main results of Lie algebras ...
- 367
4
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0
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233
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Affine Steinberg groups vs Steinberg groups over Laurent polynomials
Let $R$ be a commutative ring and $\Phi$ be a finite (also called spherical) reduced irreducible root system of rank $\geq 2$. I will denote by $\mathrm{St}(\Phi,R)$ the Steinberg group of type $\Phi$ ...
- 493
4
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0
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643
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The existence of zero-divisors in the universal enveloping algebra of an infinite-dimensional Lie algebra
The intuition for this problem comes from $\S$17 Exercise 1 Humphreys' Introduction to Lie Algebras and Representation Theory which essentially asks us to use PBW in order to prove that if a Lie ...
4
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0
answers
285
views
Application of Frobenius splitting in characteristic $0$
In general, Frobenius splitting only defines on field of characteristic $p$ (algebraically closed) field.
I am reading Brion and Kumar's book and I can see that there are geometric results can be ...
- 849
4
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0
answers
557
views
Prove that a Verma module is projective only if its highest weight is dominant?
Let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over $\mathbb{C}$ with a fixed Cartan subalgebra $\mathfrak{h}$ and a fixed system of simple roots. It is stated in Exercise 3.11 of ...
- 671
4
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0
answers
220
views
What does "control of a deformation problem" mean?
Is the expression "control of a deformation problem' ever defined? There are of course many examples relating a dg-Lie or L-infty algebra to a deformation problem, and the phrase is evocative. Is it ...
- 3,880
4
votes
0
answers
85
views
Homological dimension of Joseph quotients
Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ not isomorphic to $sl(n)$.
Let $\mathcal O$ be the minimal nilpotent orbit in $\mathfrak g^*$. Joseph proved that there exists unique two-...
- 7,037
4
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0
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335
views
The geometry of the holomorph of a Lie group
Every Lie group $G$ is naturally contained in its holomorph Hol($G$) = $G \rtimes $ Aut($G$)
Is Hol$(G)$ always a Lie group?
If the answer is yes our main questions:
1.For a left invariant ...
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4
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0
answers
209
views
Partial differential Equation over characteristic p
I want some references on partial differential equations over characteristic $p$. If we have a first order partial differential equation, how can we check whether there exists polynomials or rational ...
4
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0
answers
226
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Explicit description of graded (counital) cofree cocommutative coalgebras
Let $k$ be a field of characteristic $p \neq 2$, and $V = \oplus V_{n}$ be a graded vector space over $k$.
Then, can one compute the graded (counital) cofree cocommutative coalgebra $C(V)$ ...
- 41
4
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0
answers
144
views
When does finite presentability of the associated graded Lie algebra of a group imply the group is finitely presented?
Let $G$ be a finitely generated group; let $L(G)$ denote the graded Lie algebra (over $\mathbb{Q}$) associated to the lower central series of $G$. I would like to know conditions for when the finite ...
- 373
4
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0
answers
169
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Automorphisms of Nilmanifolds
Let $\mathfrak{g}$ be an n-dimensional, rational, nilpotent Lie algebra with simply connected that lie group $G$. It is stated in some papers that if $A$ is an automorphism of $\mathfrak{g}$ which is ...
- 419
4
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0
answers
304
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Lie algebra cohomology
Let $\mathfrak{g}$ be a simple complex Lie algebra of $rank(\mathfrak{g})\geq2$ and dimension $d$. Fix a (non-zero) invariant bilinear form $(\cdot,\cdot)$ on $\mathfrak{g}$ and let $\{x_i\}_{1\leq i\...
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4
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185
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Are these subspaces of $\mathbb{Z}/3[[x]]$ stable under the shallow Hecke algebra?
This is a characteristic $3$ analog of part of my earlier question, "Are these two subspaces of $\mathbb{Z}/2[[x]]$ the same?"
Notation
Fix a prime $N$ other than $3$. Let $F,G \in \mathbb{Z}/3[[x]]$...
- 5,422
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1k
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How to find the unitary matrices in this exponential matrix representation
In the following post
Representing a product of matrix exponentials as the exponential of a sum
there is a statement regarding the result of the multiplication of two matrix exponentials:
if $A$ and $...
- 149
4
votes
0
answers
164
views
Is there an analogue of distributions in characteristic p?
Some motivation: When working over $\mathbb{C}$, distributions (in the sense of generalized functions) act as natural generators for $D$-modules (in the sense that any regular holonomic $D$-module is ...
- 1,488
4
votes
0
answers
189
views
Does this space of mod 2 modular forms admit a (Z/8)* degree decomposition?
Fix an odd N>0. Let M consist of all odd elements of Z/2[[x]] that are the mod 2 reductions of elements of Z[[x]] arising as the Fourier expansions of modular forms for (Gamma_0)(N); it's easy to see ...
- 5,422
4
votes
0
answers
814
views
Adjunction Formula for Weil Divisors on a Normal Variety X
Let $X$ be a normal variety over an algebraically closed field $k$ of characteristic $p>0$ and $S$ be a prime Weil divisor on $X$ which is normal too. Now if $K_X+S$ is NOT $\mathbb{Q}$-Cartier, ...
- 165
4
votes
0
answers
136
views
A subring of the Serre Swinnerton -Dyer ring of level N modular power series
Suppose ell is prime and (N,ell)=1. Consider those power series over Z that are expansions at infinity of modular forms for gamma_0 (N) of weight a multiple of ell-1. I'll say that an element of (Z/...
- 5,422
4
votes
0
answers
214
views
Harmonic analysis and non-symmetric Macdonald polynomials?
I have recently been reading a lot about Macdonald polynomials, the symmetric and the non-symmetric ones. One thing that strikes me is that the symmetric Macdonald polynomials admit a positive theory, ...
- 41
4
votes
0
answers
741
views
The normalizer a maximal compact subgroup of a semi-simple Lie group
Let $G$ be a semi-simple real Lie group such that $|\pi_0(G)|<\infty$ and let $K$ be a maximal compact subgroup of $G$.
Q1: How does one prove that $N_G(K)=K$?
So I know a nice (and low-tech) ...
- 7,609
4
votes
0
answers
174
views
Number of submodules in $\wedge^2 V$ and $S^2V$ isomorphic to $\mathfrak{g}$
Let $\mathfrak{g}$ be a simple complex Lie algebra. Let
$\mathfrak{g}\subset\mathfrak{so}(V)$ be an orthogonal
irreducible representation. It can be shown that the number of
$\mathfrak{g}$-...
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