Questions tagged [noncommutative-algebra]
Non-commutative rings and algebras, non-associative algebras. Can be used in combination with ra.rings-and-algebras
522 questions
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What is a quantum condensed space?
Due to a categorical equivalence involving compact topological spaces and unital commutative $\mathrm{C}^*$-algebras, there is a practise involved in so-called quantum mathematics where a ...
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Algebraic theorems with no known algebraic proofs
What are some good examples of algebraic theorems that have no known algebraic proofs?
A few I know concern classifications of (not necessarily associative) division algebras over $\mathbb{R}$: the ...
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Hilbert symbol of a quaternion algebra given ramified places
I am reading the paper: https://projecteuclid.org/journals/experimental-mathematics/volume-17/issue-3/Derived-Arithmetic-Fuchsian-Groups-of-Genus-Two/em/1227121388.full
in order to find an explicit ...
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Free, easy-to-use program for noncommutative algebra over finite fields
I am looking for a computer program that can handle computations in noncommutative algebra over a finite field of prime order $p$.
My requirements are:
The program should be free, as I do not have ...
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Is this ring isomorphic to a quotient of a group algebra?
Consider the quotient of the free algebra $\mathbb{Q}\langle \alpha, \beta, \gamma, \delta, \varepsilon, \zeta \rangle$ by the two-sided ideal $I$ subject to the relations $$ \alpha\delta=\delta\alpha=...
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Finitely generated module over a skew Laurent polynomial ring $\mathbb{K}[x_1^{\pm 1},x_2^{\pm1}]$
Let $A := \mathbb{K}_{\Lambda}[x_1^{\pm 1}, x_2^{\pm 1}, x_3^{\pm 1}, x_4^{\pm 1}]$, $B := \mathbb{K}_{\Lambda'}[x_1^{\pm 1}, x_2^{\pm 1}, x_3^{\pm 1}]$, and $C := \mathbb{K}_{\Lambda''}[x_1^{\pm 1}, ...
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Right maximal ideals in skew-Laurent rings over division Rings
Let $R$ be a Noetherian domain, and let $D$ denote its division ring. Define $S = D_q[x_1^{\pm 1}, x_2^{\pm 1}]$ as an iterated skew-Laurent polynomial ring with the relation $x_1 x_2 = q x_2 x_1$. Is ...
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Is being graded commutative a necessary condition on $A$ such that $H^*(A)$ is commutative?
If we consider any differential graded algebra $A^\bullet$, then its homology is a graded algebra, since the tensor product interacts well with homology.
A sufficient condidtion for the homology to be ...
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Examples of $\mathbb{N}$-graded algebras whose global dimension is strictly less than the GK dimension
The relationship between the global and GK dimensions of Artin-Schelter regular algebras remains to be mysterious, yet both dimensions are conjectured to be equal.
In a more broad setting, are there ...
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Non-degenerate bilinear pairing of finite dimensional algebras
A finite dimensional algebra (over $\mathbb{C}$, say) is said to be Frobenius if it comes equipped with a nondegenerate bilinear pairing
\begin{align*}
\langle -, - \rangle : A \times A \rightarrow \...
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The state of the art on topological rings - the Jacobson topology
I was recently studying the Jacobson density theorem and I found it quite interesting.
Most textbooks I've seen, including Jacobson's own Basic Algebra, only spend a few lines about the reason why it ...
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Representation functor on modules
Let $k$ be a field and $A$ a unital associative $k$-algebra.
The representation functor associates, to each object in non-commutative geometry, a genuine geometric object on the representation variety ...
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When an algebra isomorphism preserves positive involution
Let $A$ be a $K$-algebra where $K$ is a field with a unique ordering.
We say a $K$-linear involution $*$ is positive if the map $A \to K$ via $a \mapsto tr(a^*a)$ is positive definite with respect to ...
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Recent research on polynomial identities
I work in computational complexity, where I work on the problem of polynomial identity testing over arithmetic circuits. One particular case is when the variables over the polynomial ring don't ...
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Is there anything like a Čech complex for calculating local cohomology over *noncommutative* rings?
$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\colim{colim}$Let $R$ be a ring, and consider a two-sided ideal $I = (r_1, r_2, \dots, r_j)$ in $R$. The corresponding $n$th local cohomology functor ...
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Wedderburn-Malcev principal theorem for graded-finite algebras
Let $k$ be a field and $A$ be a noncommutative $k$-algebra with Jacobson radical $J$. If $A$ is finite-dimensional, the Wedderburn-Malcev says that $A$ has a subalgebra $S$ such that $$A = S \oplus J$$...
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Which CAS can do basic non-commutative differential algebra?
This is a repost of my question at MSE from 7 months ago, to which I haven't been able to find an answer yet.
I am looking for a CAS (possibly incl. additional packages/libraries) that can compute ...
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Star-algebra isomorphism
I have asked this question: When an algebra isomorphism preserves positive involution, but now I want to modify it.
Let $A$ and $B$ be $K$-algebras where $K$ is a field with a unique ordering.
We say ...
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Non-commutative Gorenstein Koszul algebras
I was wondering if there exists a characterization of finite-dimensional Gorenstein Koszul algebras in terms of their Koszul dual?
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Noncommutative transcendence degree of representation algebras
Let $G$ be a reductive group, for example $\text{GL}_n(\mathbb{C})$. Let $V$ denote its defining representation, and let $R$ denote the tensor algebra on the irreducible representations of $G$. It may ...
- 2,907
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Determining the multiplication via addition and some unary operation
It is known that the addition operation in a skew-field $F$ (more generally, in a quasifield) is uniquely determined by the multiplication operation and the unary involutive operation $1_{-}:F\to F$, ...
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Trace map on Ext group
Let $R$ be a (possibly non-commutative) unital ring and $M$ be a perfect left $R$-module. Then, we have the trace map
$$
\operatorname{Tr}\colon \mathrm{Hom}_R(M,M)\to R/[R,R]\,.
$$
According to the ...
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Matrix ring isomorphisms of different sizes
Do there exist (unital, associative, noncommutative) rings $R$ and $S$, where $\mathbb{M}_2(R)\cong \mathbb{M}_3(S)$, but these matrix rings are not isomorphic to $\mathbb{M}_6(T)$ for any ring $T$?
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Is there a notion of point in noncommutative geometry?
It is not clear to me whether there is a general notion of point in NCG. I have heard (more through physics) that the notion of a point becomes meaningless or ill-defined in noncommutative spaces, but ...
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Simple-direct-injective modules
A right $R$-module $M$ is called a simple-direct-injective module if it satisfies any of the following equivalent conditions:
For any simple submodules $A,B$ of $M$ with $A \cong B \subseteq^{\oplus} ...
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Faithful flatness and non-commutative algebras
$\DeclareMathOperator\Spec{Spec}$When dealing with commutative algebras, a usefull criterion for faithful flatness is the following:
Let $f:A\rightarrow B$ be a morphism of commutative algebras. Then $...
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Does Wedderburn's Little Theorem hold constructively?
Wedderburn's Little Theorem states that every finite division ring is commutative. Perhaps even more surprising, this implies that every finite reduced ring is commutative.
The proofs that I am aware ...
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Algebra with three anti-commutator relations
Let $u,v,w \in \mathbb{F}_p^{\times}$. Consider the $\mathbb{F}_p$-algebra $V$ generated by $ a,b,c$ and the relations
$$u a^2 = bc + cb$$
$$v b^2 = ac + ca$$
$$w c^2 = ab + ba$$
Is $V$ generated by ...
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Formal smoothness of path algebras and connections
Let $k$ be a field of characteristic zero and $A = kQ$ the path algebra associated with a quiver $Q$. The algebra $A$ is said to be formally smooth over $k$ if
$$
\Omega^1_kA = \operatorname{Ker}(\...
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Particular example of a quadratic extension of a nonunital ring
I want to construct a concrete non-unital ring $R$ with the following properties:
$R$ is a noncommutative non-unital ring with a right unite $r$ i.e $t.r=t$ for any $t\in R$.
$S\subset R$ is a ...
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A problem in commutative algebra whose solution requires algebraic geometry (resp., noncommutative algebra)?
One can argue that commutative algebra is affine algebraic geometry. However, a great deal of commutative algebra generalizes to non-commutative algebra, and in that setting there is little geometry, ...
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Minimal ideals and subalgebras of semisimple algebras
I'm considering an algebra to be a ring which is also a vector space over some field $F$, and the algebra $A$ is said to be semisimple if it is semisimple as a ring, i.e., $A$ can be written as a ...
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A note on orders in quaternion algebras
Definition. An algebra $B$ over a field $F$ is a quaternion algebra if there exists $i,j\in B$ such that $1,i,j,ij$ is a basis for $B$ as a vector space over $F$, where $i^2=a,j^2=b;a,b\in F^\times$.
...
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Semi-simple algebras over operads
I believe people thought about this questions, however I couldn't find any reference. I appreciate if someone could direct me to some detailed discussions about it.
The categories of associative ...
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Wedderburn theorem for finite-dimensional algebras over the complex numbers
I'm trying to understand how to apply the Wedderburn theorem in the context of unitary algebras over $\mathbb{C}$ that are finite-dimensional and semisimple. Let $\mathcal{A}$ be a $\mathbb{C}$-...
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Multiplicative bases, path algebras, and Ext algebras
I am interested in understanding when a multiplicative basis exists for finite dimensional algebras over an algebraically closed field, and, in particular, Ext-algebras that are finite dimensional.
It ...
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Composition of Frobenius $n$-homomorphisms, characteristic-free?
This question is, as so often, a crossbreed of curiosity and laziness. The
former has led me to an interesting, but somewhat unsatisfactory paper by
Khudaverdian and Voronov
(arXiv:2002.02395v2) and, ...
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Is anything known about the center of the Fomin-Kirillov algebra?
Let $\mathcal{B}_{\mathbb{S}_m}$ be the quotient of the Fomin-Kirillov algebra so that its pairing becomes certainly nondegenerate. This algebra is conjecturally isomorphic to the Fomin-Kirillov ...
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On a lemma of projective dimension
Let $R$ be a finite-dimensional algebra, and $A=R\oplus A_1\oplus A_2\oplus \dotsb$ be an $\mathbb{N}$-graded algebra which is locally finite (i.e. all $A_i$'s are of finite dimension). Let $\text{...
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On the conditions for Artin-Schelter Gorenstein algebras
Let $ k $ be a field and $ A $ a connected graded $ k $-algebra ($ A $ is associative, but not assumed to be commutative).
The algebra $ A $ is called Artin-Schelter Gorenstein* of dimension $ d $ if ...
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Is there a $3$-commutative algebra?
Let $k$ be a field of characteristic $0$. If $m\ge2$, I denote $P_m$ the standard polynomial in $m$ non-commutating indeterminates:
$$P_m(X_1,\dotsc,X_m)=\sum_{\sigma\in\mathfrak S_m}\epsilon(\sigma)...
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4
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Kaplansky inverse element theorem on group C-star algebra
In a class talking about $C^*$ algebra and (higher) index theory, I heard a theorem
(related to Kaplansky, proved?), that is
Suppose $\Gamma$ is a group (admitting Haar measure if necessary) while $\...
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On the structure of an algebra as a bimodule
$\DeclareMathOperator\End{End}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Ker{Ker}\newcommand{\bi}{\mathrm{bi}}\newcommand{\op}{\mathrm{op}}$Let $K$ be a field (say of characteristic zero), and $...
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Gluing data for modules over a ring with idempotents
Let $A$ be a ring. If $e$ is an idempotent, then there is an abelian recollement involving the categories $A\text{-}\mathrm{Mod}$ and $eAe\text{-}\mathrm{Mod}$. This is Example 2.7 in Homological ...
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Arithmetic derivatives and non-commutative generalizations
In the theory of arithmetic derivatives, in the simplest case an arithmetic derivative on $\mathbb{N}$ is defined via the rule $(a \times b)'= a \times b' + a' \times b$, mirroring the product rule ...
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On the deformation theory of associative algebras
Let us start by recalling the notion of a formal deformation:
Let $K$ be a field of characteristic zero and $A$ be an associative $K$-algebra. Consider a commutative augmented $K$-algebra $R$, with ...
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Are the trace relations among matrices generated by cyclic permutations?
Let $X_1,\dots,X_n$ be non commutative variables such that $\operatorname{tr} f(X_1,\dots,X_n) = 0$ whenever the $X_i$ are specialized to square matrices in $M_r(k)$ for any $r \geq 1$. Does this ...
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431
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Smallest faithful representation of an upper-triangular matrix quotient
This is a curiosity question that came out of teaching abstract algebra.
Let $F$ be a field, and $n>1$ an integer.
Let $F^{n \leq n}$ be the $F$-algebra of all upper-triangular $n\times n$-matrices ...
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Confusion around a (necklace) cobracket in Ginzburg's article Calabi-Yau Algebras
Something has been puzzling me for quite a while in Ginzburg's article Calabi-Yau Algebras.
At some point he considers the free graded algebra $\mathbb{C}\langle x_1, \dots, x_n, \theta_1, \dots \...
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Subtraction-free identities that hold for rings but not for semirings?
Here is a concrete, if seemingly unmotivated, aspect of the question I am interested in:
Question 1. Let $a$ and $b$ be two elements of a (noncommutative) semiring $R$ such that $1+a^3$ and $1+b^3$ ...
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