Unanswered Questions
1,115 questions with no upvoted or accepted answers
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What did Gelfand mean by suggesting to study "Heredity Principle" structures instead of categories?
Israel Gelfand wrote in his remarkable talk "Mathematics as an adequate language (a few remarks)", given at "The Unity of Mathematics" Conference in honor of his 90th birthday, the ...
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Groups whose complex irreducible representations are finite dimensional
By a complex irreducible representation of a group $G$, I mean a simple $\mathbb CG$-module. So my representations need not be unitary and we are working in the purely algebraic setting.
It is easy ...
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Greatly expanded new edition of a Bourbaki chapter on algebra?
Recently I discovered by accident that Bourbaki issued in 2012 a radically expanded version of their 1958 Chapter 8 Modules et anneaux semi-simples (like other chapters, initially in French) within ...
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Is there a field $F$ which is isomorphic to $F(X,Y)$ but not to $F(X)$?
Is there a field $F$ such that $F \cong F(X,Y)$ as fields, but $F \not \cong F(X)$ as fields?
I know only an example of a field $F$ such that $F$ isomorphic to $F(x,y)$ : this is something like $F=k(...
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What algebraic structure characterizes all natural operations between differential operators and differential forms?
On a smooth manifold $M$ one can define various algebraic structures, natural with respect to diffeomorphisms:
the differential graded-commutative algebra $\Omega(M)$ of differential forms on $M$;
...
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23
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Topological loops vs. algebro-geometric suspension in Hochschild homology
Let $k$ be a base commutative ring, and let $A$ be a (unital but not necessarily commutative) $k$-algebra. The cone on $A$ is the ring $CA$ of infinite matrices $(a_{ij})_{i,j \geq 1}$ that are ...
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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*}
[...
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When is the determinant an $8$-th power?
I am working over $\mathbb{R}$ (though most of the story goes over any field). I am looking for linear spaces of matrices such that the restriction of the determinant to this spaces can be written (...
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Are $0, 1, 4, 7, 8$ the only dimensions in which a bivector-valued cross product exists?
It is a well-known mathematical curiosity that ordinary (vector-valued) cross products over $\mathbb{R}$ exist only in dimensions $0, 1, 3$ and $7$ (this fact is related to Hurwitz's theorem that real ...
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How to explain the picturesque patterns in François Brunault's matrix?
How to explain the patterns in the matrix defined in François Brunault's
answer to the question Freeness of a Z[x] module depicted below? --
Choosing colors according to the highest power of 2 which ...
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Is "being a full ring of quotients" a Morita invariant property?
Definition and context:
An (associative, unital, not necessarily commutative) ring $R$ is called classical if every regular element of $R$ is a unit. Equivalently, $R$ is its own classical ring of ...
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14
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Rings that fail to satisfy the strong rank condition
In T.Y. Lam's book Lectures on Modules and Rings, a ring $R$ is said to satisfy the strong rank condition if, for every natural number $n$, there is no right $R$-module monomorphism $R^{n+1}\to R^n$. ...
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Kaplansky's theorem and Axiom of choice
Kaplansky in his paper titled by Projective Modules gave an important and essential theorem as follow:
Theorem: Let $R$ be a ring, $M$ an $R$-module which is a direct sum of (any number of) countably ...
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Finite dimensional real division algebras
A celebrated theorem of Milnor and Kervaire asserts that any finite dimensional (not necessarily associative, unital) division algebra over the real numbers has dimension 1,2,4 or 8. This result is ...
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Analog of Haar element in an algebra
In a Hopf algebra $H $ (over some field $ k $), there is the notion of a Haar element $ h \in H$. This is an element of the algebra which has the property that if $ V $ is a representation of $ H $, ...
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