All Questions
6,486 questions
1
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On reflexive bialgebras
Let $A$ be a bialgebra. We can consider $A$ as a relfexive algebra (i.e. $A\cong A^{o*}$) or relfexive coalgebra (i.e. $A\cong A^{*o}$ where in each case $o$ denotes what is sometimes called ...
5
votes
0
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435
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How to define $\mathbb{R}^\frac{1}{2}$?
The Cayley-Dickson construction generates higher-dimensional hyper complex numbers from lower-dimensional ones, producing algebras of dimension $2^n$.
I want to generate an algebra of dimension $2^{-1}...
- 75
4
votes
1
answer
197
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Lie algebra cohomology and Lie groups
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\U{U}\DeclareMathOperator\Sp{Sp}$Let $G$ be a complex Lie group and $\mathfrak{g}$ its Lie algebra (which is over $\mathbb{C}$). My first question is:
(...
- 2,348
1
vote
1
answer
115
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Block-diagonal embedding of $U(n)$ into $U(mn)$
What is known about the subgroup $U(n)\subset U(mn)$ for $m,n\in\mathbb{N}$ given by the diagonal embedding
$$ \alpha\mapsto \text{diag}(\alpha,\cdots, \alpha),$$
for $\alpha$ appearing $m$ times?
For ...
- 3,054
2
votes
1
answer
122
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Slowly increasing smooth mappings with values in a Lie group?
Let $G$ be $l$-dimensional compact Lie group and consider any smooth $F : \mathbb{R}^n \to G$.
Then, the first-order derivative of $F$ at each $x \in \mathbb{R}^n$ can be regarded as a linear mapping $...
- 286
22
votes
6
answers
6k
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What are hypergroups and hyperrings good for?
I came across the concept of a hyperring in two recent papers by Connes and Consani (From monoids to hyperstructures: in search of an absolute arithmetic and The hyperring of adèle classes). It's a ...
- 1,228
4
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0
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79
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Closed character formula for the module $L(a\omega_i)$
Let $\mathfrak{g}$ be a complex finite-dimensional simple Lie algebra with a fixed Cartan subalgebra $\mathfrak{h}$. Assume that $\omega_1, \omega_2, \dots, \omega_n\in\mathfrak{h}^{*}$ is the ...
- 41
14
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2
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741
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Is a "separable" algebra over a field finite-dimensional?
Let $k$ be a field and $A$ a unital associative (possibly non-commutative) $k$-algebra, and let $A^e$ denote the enveloping algebra of $A$, namely, $A^e = A \otimes_k A^{op}$.
It seems that there are ...
- 11
0
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0
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69
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A weakening of the definition of positive roots for a root system
Given a root system $\Delta$ a choice of positive/negative roots is a decomposition of the elements of $\Delta$ into two subsets $\Delta^+$ and $\Delta^-$ satisfying
$$\Delta^+ = - \Delta^-\tag{$*$}\...
- 12.9k
2
votes
1
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123
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Stabilizer of a lattice $\Gamma \subset G$ in the group of automorphisms $\operatorname{Aut}(G)$ is always discrete?
$\DeclareMathOperator\Aut{Aut}$Let $G$ be a (simply) connected Lie group whose semisimple part has no compact factors, and let $\Gamma $ be a lattice (uniform?) in $G$.
Is it true that the stabilizer $...
- 309
0
votes
1
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304
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A Lie group whose Lie algebra is equal to (the Lie algebra? of )all functions with fibrewise polynomial growth
Let $M$ be a Riemannian manifold. We denote by $\mathfrak{g}$ the space of all smooth function $f:TM\to \mathbb{R}$ with fibre wise polynomial growth. Is it a Lie algebra wrt the Poisson bracket ...
- 8,098
10
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0
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225
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Third homology of simply connected Chevalley–Demazure group schemes
I’ve been studying the group of $\mathbb{Z}$-points of the simply connected Chevalley–Demazure group scheme of type $E_7$, denoted $G_{\text{sc}}(E_7,\mathbb{Z})$; see Vavilov and Plotkin - Chevalley ...
- 545
21
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4
answers
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Why should we study derivations of algebras?
Some authors have written that derivation of an algebra is an important tools for studying its structure. Could you give me a specific example of how a derivation gives insight into an algebra's ...
- 12.9k
5
votes
1
answer
148
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A general form of a maximal totally isotropic subspace in the split octonion algebra
$\DeclareMathOperator\Ker{Ker}$Let $\mathbb O'$ be the split octonion algebra over $\mathbb R$. For each nonzero divisor of zero $x\in \mathbb O'$ ($x \neq 0$, $N(x)=0$) the kernel of the left ...
- 12.9k
11
votes
1
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383
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Is there a comprehensive survey of the discrete series representation of a real reductive group?
Vague form of the question: where can one find a comprehensive and possible modern account of the discrete series representations of a real reductive group?
Motivation:
I am a master's student trying ...
- 869
2
votes
2
answers
336
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Orthosymplectic superalgebra
Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space over $\mathbb C$. Suppose $V$ has an even non-degenerate bilinear form $(-, -)$
which is symmetric on $V_0$, skew symmetric on $V_1$, and ...
- 12.9k
1
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1
answer
115
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Equivalent definition of Spin group in terms of automorphisms
Let $\mathrm{Cl}(\mathbb{R}^n)$ denote the (real) Clifford algebra on $\mathbb{R}^n$ with respect to the Euclidean inner product. Let $\mathrm{Cl}^0({\mathbb{R}^n})$ denote the even part of $\mathrm{...
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1
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473
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A subfield $R \subseteq \mathbb{C}(x,y)$ with 'many' generators $w$, $R(w)=\mathbb{C}(x,y)$
Let $R \subseteq \mathbb{C}(x,y)$ and assume that $R=\mathbb{C}(u,v)$, where $u,v \in \mathbb{C}[x,y]$ are algebraically independent over $\mathbb{C}$.
Here $\mathbb{N}$ includes $0$.
Assume that $R$ ...
- 2,837
0
votes
0
answers
61
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Representation and Laplacian on the Heisenberg group
Let $\pi_{\lambda}$ be the the Schrödinger representations of the Heisenberg group $H=\mathbb C\times \mathbb R$. For $\varphi\in L^2(\mathbb R)$, we have
$$\pi_{\lambda} (z,t)\varphi(\xi)=e^{i\...
- 650
1
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0
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35
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An algebraic characterization of dual translation projective planes
It is well-known that translation projective planes are coordinatized by quasifields. More precisely, a projective plane is translation if and only if it has a ternary-ring $R$ which is linear, the ...
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1
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73
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Relationship between units and primitive characters 2
This is a follow up to this question.
Let $(R,+,\cdot)$ be a finite ring.
Definition Given the dual group $\widehat{R}$ of $(R,+)$, a character $\chi\in\widehat{R}$ is said to be primitive with ...
- 12.9k
1
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1
answer
263
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Relationship between units of a ring and primitive characters of the ring under addition
Let $(R,+,\cdot)$ be a finite ring. Obviously, $(R,+)$ is an abelian group; however the unit (multiplicative) group need not be abelian.
My question is the following problem:
Given the dual group $\...
1
vote
1
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215
views
Reference about cancellation property for semigroups
Have the semigroups with the following cancellation property been studied?
Property: Let $S$ be a semigroup and $x,y\in S$ such that $xz=yz,$ for all $z\in S,$ then $x=y$.
- 468
1
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0
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146
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Which elements lie in a Cartan subalgebra?
Let $\mathfrak{g}$ be a Lie subalgebra of the real upper triangular matrices (i.e. $\mathfrak{g}$ is a completely solvable real Lie algebra). $\mathfrak{g}$ has an exponential radical $\mathfrak{r}$, ...
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1
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219
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Finding $\mathbb{C}(u,v)$ such that $\mathbb{C}(u,v,x^p+y^p)=\mathbb{C}(x,y)$, for every prime number $p$
Denote the set of prime numbers by $P$, $P=\{2,3,5,7,\ldots\}$.
Let $F \subseteq \mathbb{C}(x,y)$ be a subfield of $\mathbb{C}(x,y)$, and for $w \in \mathbb{C}[x,y]$ denote by $F(w)$ the subfield of $\...
- 148k
1
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0
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78
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The partial orders on the elements of a root system coming from the positive spans of the weights and the roots
Let $(\Delta,V)$ be a root system with a choice of positive roots $\Delta^+$. Denote the $\mathbb{N}_0$-span of the positive roots by $\mathcal{O}^+$, and the $\mathbb{N}_0$-span of the associated ...
1
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1
answer
335
views
How many rings exist when ring is subspace of finite dimensional vector space?
Suppose we have a ring $R[M]$ for a monoid $M$ over the real numbers $R$. The number of generators for the monoid is finite. Now suppose that every ring element $r$ has a decomposition in finite ...
- 1,507
9
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3
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790
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A manifold whose tangent space is a sum of line bundles and higher rank vector bundles
I am looking for an example of the following situation. Let $M$ be a connected (if possible compact) manifold such that its tangent bundle $T(M)$ admits a vector bundle decomposition
$$
T(M) = A \...
- 108k
3
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2
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468
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How fast does the number of "fixed" points grow compared to the size of the ball in the following group?
I have copied this question from Math.StackExchange, in the hope that some experts here can provide some relevant insight.
Let $M = \oplus_{i\in \mathbb Z} V^{(i)}$ where each $ V^{(i)} \cong \mathbb ...
- 1,819
5
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1
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209
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Reconstructing a Lie group from its Maurer-Cartan form (role of completeness)
Theorem III.8.7 in Sharpe and Chern's "Differential Geometry: Cartan's Generalization of Klein's Erlangen Program" states:
If $M$ is a simply connected manifold, $\mathfrak{g}$ is a Lie ...
- 26.3k
3
votes
1
answer
2k
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LUB and GLB on a lexicographically ordered complete lattice product
I am trying to define the LUB and GLB on a product of lattices that are partially ordered lexicographically.
Is there any papers or help that I could read up on? I would particularly like proofs on ...
- 4,219
2
votes
1
answer
245
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Characterization of a certain class of modules-broader than Noetherian
Let $R$ be a commutative ring with $1$.
An $R$-module $K$ has the 'S' property if $K/T \simeq K$ (i.e. isomorphic) implies that the submodule $T$ is trivial.
By Fitting's lemma any Noetherian module ...
- 63.9k
4
votes
1
answer
523
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Is automorphism on a compact group necessarily homeomorphism? How about N-dimensional torus？ [closed]
Is automorphism on a compact group necessarily homeomorphism? I don't think so，but I think it is possible on the N-dimensional torus.
- 12.9k
1
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1
answer
114
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Source for highest weight vectors for $\text{SL}_n(\mathbb{C})$ representations
I have a list of a lot of irreducible $\text{SL}_n(\mathbb{C})$ representations and need to know what all of their highest weight vectors are. For example, using the notation from Fulton and Harris, I ...
- 550
1
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0
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58
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Can a maximal rank subgroup of a simply connected Lie group have simply connected factors?
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\rank{rank}$Take a simply connected Lie group $G$ such as $\SU(N)$ and a maximal rank subgroup $H$, i.e. $\rank(G) = \rank(H)$. Assume that $H$ takes ...
- 12.9k
2
votes
1
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88
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Number of real forms of a (not semisimple, solvable) Lie algebra
Consider the Lie algebra $\mathfrak{r}_2=\mathfrak{aff}(\mathbb{C})$ of the group of affine maps of $\mathbb{C}$, and let $\mathfrak{g}=\mathfrak{r}_2 \oplus \mathfrak{r}_2$.
I am interested in ...
- 12.9k
8
votes
1
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340
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Density of extended Mersenne numbers?
Consider the subset of odd positive integers defined and constructed as follows by these rules :
A) $1$ is in the set.
B) if $x$ is in the set , then $2x + 1$ is in the set.
C) if $x$ and $y$ are in ...
- 763
5
votes
1
answer
246
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Infinite-dimensional, non-unital Frobenius algebras
A Frobenius algebra is a tuple $(A,\mu,\delta,\eta,\varepsilon)$, where $A$ is a vector space over some field, $(A,\mu,\eta)$ a unital associative algebra, and $(A,\delta,\varepsilon)$ a counital ...
- 985
6
votes
1
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284
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Infinite dimensional finitely generated algebraic division algebra
Is there a division algebra $D$ with center $K$ that satisfies the
following 3 conditions?
1) $D$ is of infinite dimension over $K$;
2) every element of $D$ is algebraic over $K$;
3) $D$ is ...
- 35.2k
2
votes
0
answers
119
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gcrd and associates of an element of the quaternion algebra over a totally real number field $K$
Let $K$ be a totally real number field of class number 1, and $Q$ the quaternion algebra over the ring of integers of $K$ with basis
$\{1,i,j,k\}$ such that $i^2 = j^2 = k^2 = -1$ and $ij = -ji, ik = ...
1
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0
answers
108
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Schur Weyl Duality for Maximal Torus
I wanted to know if there's some version of Schur Weyl Duality for the maximal torus $T \subset \operatorname{GL}(V)$? Is there also some combinatorial object which might be useful for the same?
2
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1
answer
244
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Decomposition of an $\text{SL}_n(\mathbb{C})$ representation
Let $W = V \oplus V^*$, where $V$ is the standard $\text{SL}_n(\mathbb{C})$ rep and $V^*$ is its dual. I'm ultimately trying to decompose the space $(W \otimes \bigwedge^2 W) / {\bigwedge^3 W}$.
This ...
- 954
2
votes
1
answer
123
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Signed measures on algebras (fields) and their boundedness properties
I asked this question here on math.StackEchange, but it might be too technical so I re-post it here.
Let $X$ be a compact Hausdorff second countable topological space. Let $\mathcal{B}$ a countable ...
- 2,830
0
votes
1
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171
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Prime index subgroups of $\langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle$ that is invariant under matrix $Q$
Let $Q $ be a matrix in $ \operatorname{GL}(2, \mathbb{Q}) $ and consider the group
$G = \langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle := \langle Q^{i}(v) \mid i \in \mathbb Z, v \in \...
- 63.9k
1
vote
0
answers
28
views
Most general filtered algebras with Hilbert polynomials and multiplicities
Let $k$ be any base field and $A$ an affine infinite dimensional $k$-algebra.
Let $\mathcal{F}= \{ A_i \}_{i \geq 0}$ be a finite dimensional filtration for $A$: that is, $k \subset A_0$ and each $A_i$...
- 3,318
3
votes
0
answers
152
views
My category is rigid: what this implies for representation theory?
I am studying a subcategory $\mathcal{C}$ of modules for an associative noncommutative algebra $A$ (which is in fact also a Hopf algebra).
It is clear from our definition of $\mathcal{C}$ that it is ...
- 3,318
1
vote
0
answers
125
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Ring $R$ has IBN $\iff$ $R^n \ncong R^{n+1}$ for all $n \in \mathbb{N}$?
Let $R$ be a ring with $1$. Is it true that $R$ has IBN $\iff$ $R^n \ncong R^{n+1}$ for all $n \in \mathbb{N}$?
Per definition $R$ has IBN (invariant basis number) if $R^{m} \cong R^{n}$ as left $R$-...
- 261
3
votes
2
answers
214
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Characterize rings $R$, such that the countable product $P=R^N$ has the property that every finitely generated submodule of $P$ is free
What are the rings whose countable power has the property that every finitely generated submodule is free?
- 5,031
1
vote
0
answers
181
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Examples of semirings where the additive neutral element is not absorbing for multiplication
In the case of a non unital ring, the additive 0 must be absorbing for the multiplication because we have a⋅0 = a⋅(a − a) = a⋅a − a⋅a = 0 and similarly on the other side.
In the case of a unital ...
- 22.3k
6
votes
0
answers
184
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Reference request: fusion rules for unitary dual of SL(3,R)?
By the fusion rules, I mean: given two unitary irreps of the group, what unitary irreps occur in their tensor product and with what "multiplicity"? (I am guessing that direct integrals ...
- 25.8k