The noncommutative-algebra tag has no usage guidance.

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### Determinants of octonionic hermitian matrices

For quaternionic hermitian matrices (i.e. quaternionic square matrices $(a_{ij})$ satisfying
$a_{ji}=\bar a_{ij}$) there is a nice notion of (Moore) determinant which can be defined as follows.
...

**13**

votes

**2**answers

2k views

### Left and right eigenvalues

A quaternionic matrix $A$ gives rise to a
function $\mathbb{H}^n \to \mathbb{H}^n$
given by $x \mapsto A \cdot x$. This is real linear,
but not complex- or quaternionic-linear
(in general) if we ...

**7**

votes

**2**answers

415 views

### Quantum Grassmannians?

In noncommutative algebraic geometry a commonly studied family of objects are quantum projective spaces. Theses are certain deformations of the homogeneous coordinate ring of $\mathbb{CP}^n$. For ...

**16**

votes

**1**answer

418 views

### Koszul complex for non-Koszul algebras

Let $A$ be a graded, connected, locally finite, quadratic algebra over a field $k$; that is, $A$ may be presented as $T(V)/I$, where $V = A_1$ is a finite dimensional $k$ vector space, and the ideal ...

**4**

votes

**1**answer

58 views

### reduced norm from degree 3 division algebra

Let $D$ be a degree $3$ division algebra over a field $k$ of char not 2 and 3.
Any such division algebra is cyclic. I am interested in knowing the cases when the reduced norm map $Nrd : D^* ...

**5**

votes

**1**answer

97 views

### Why is $\operatorname{nr}_{F[G]}:K_1(F[G])\to Z(F[G])^\times$ a bijection?

Let $A$ be a finite dimensional semisimple $F$-algebra and $K_1(A)$ the Whitehead group of $A$.
By splitting $A$ into its Wedderburn components, the reduced norm map $\operatorname{nr}_A:K_1(A)\to ...

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vote

**0**answers

55 views

### GK dimension of generalized Weyl algebras

I believe that the GK dimension for a generalized Weyl algebra $D(\sigma,a)$ is just the GKdim$(D) + 1$.
Does anyone have a reference for this?
I can find partial results, and I am sure this is ...

**3**

votes

**0**answers

100 views

### When a ring is a polynomial ring?

In the paper (2.11) the authors show that if $k^*$ is a separable algebraic extension of $k$ and $x_1,x_2, \ldots, x_n$ are indeterminates over $k^*$ and a normal one dimensional ring $A$ with $k ...

**4**

votes

**0**answers

63 views

### An order in $\mathbb Q[G]$ which is a maximal $\mathbb Z_p$-order in $\mathbb Q_p[G]$ for finitely many primes $p$

Let $G$ be a finite group and $S$ a finite set of prime numbers. I know that every separable $\mathbb Q$-algebra $A$ contains a maximal $\mathbb Z$-order but I wonder if the following is true.
Is ...

**0**

votes

**1**answer

93 views

### Example of noncommutative central reduced rings which is not reduced

A ring $R$ is called central reduced if every nilpotent element is central. Ungor et al. math.RA 14 Dec 2013 has given an example of a commutative ring which is central reduced but not reduced. Can we ...

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votes

**0**answers

77 views

### Center Picard group non-commutative algebra

I am wondering if there is a way to describe the center of the Picard group of a non-commutative algebra.
Namely, let $A$ be a finitely generated algebra over a field $k$. Denote by ...

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vote

**0**answers

57 views

### Completion of an algebra

Based on arXiv:math/9802041v1, there is a definition for $NC$-filtration and $NC$-completion of an associated algebra over the complex numbers:
Let $R$ be an associative algebra and $R^{\rm Lie} = ...

**2**

votes

**2**answers

330 views

### Are morphisms of finite length modules determined by the behaviour of the simple modules?

Assume we have a noncommutative ring $R$ with exactly 2 non-isomorphic simple left modules $S_1$ and $S_2$ (up to isomorphism) and an $R$-bimodule $M$, which switches the simples, i.e. $M\otimes_R ...

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votes

**3**answers

1k views

### How to work with co-multiplication?

Let $C$ be a coalgebra and $\Delta: C \to C\otimes C$ a co-multiplication map. Then, due the co-associative property we can consider $\Delta^m$. But how is defined $\Delta^{m}: C \to C^{\otimes m}$???
...

**6**

votes

**1**answer

131 views

### Trivial algebras given by generators and relations

Let $V$ be a finite dimensional vector space over a field $K$ of characteristic zero. Assume that we are given a set of (not necessarily homogeneous) elements $f_1,\ldots f_n$ in the tensor algebra ...

**12**

votes

**1**answer

300 views

### Non-commutative Galois theory

Recall that an finite-dimensional algebra $A$ over a field $k$ is central simple iff there is an iso
$A \otimes_k A^{op} \cong M_n(k)$
where $A^{op}$ is the opposite ring and $M_n(k)$ is the matrix ...

**14**

votes

**3**answers

3k views

### Binomial Expansion for non-commutative setting

What could be a reference about binomial expansions for non-commutative elements?
Specifically, where can I find a closed formula for the expansion of $(A+B)^n$ where $[A,B]=C$ and $[C,A]=[C,B]=0$?
...

**7**

votes

**3**answers

612 views

### Motivation for the Preprojective Algebra

Let $Q=(Q_0,Q_1)$ be a quiver and $k$ a field. We construct a new quiver $\bar{Q}$ in the following way. Let the vertices of $\bar{Q}$ be the same as the vertices of $Q$, and let the arrows of ...

**1**

vote

**1**answer

50 views

### Under what assumptions can endomorphisms of $M/IM$ be realized as a subquotient of endomorphisms of $M$?

Suppose we have an algebra $A$ (unital, associative), with an ideal $I \leq A$ and a finitely generated module $M$ over $A$.
It is possible to obtain both $\mathrm{End}_A(M)$ and ...

**4**

votes

**1**answer

80 views

### Filtered Colimit of associative $k$-algebras that are domains

Let $C$ be a filtered subcategory of the category of commutative algebras over a fixed field $k$ whose objects are all integral domains.
Then the colimit of the obvious diagram is an integral domain.
...

**2**

votes

**0**answers

96 views

### Examples of noncommutative Bezout domains

I would like to see some (or many!) examples of noncommutative Bezout domains (one-sided principal ideals sum to one-sided principal ideals). I've read somewhere that it's not easy to find an example ...

**5**

votes

**5**answers

565 views

### Elementary linear algebra over a (possibly skew) field $K$

I have a number of questions which seem linked to me, about basic (?) linear algebra:
Given a field (possibly skew) $K$, and an superfield $L$, one can do linear matrix algebra with coefficients in ...

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votes

**0**answers

96 views

### Is there an introduction to non-commutative geometry for non-physics and mathematics students?

I am looking for a simple explanation as how spectral triples give rise to the definition of distance using Dirac operators?

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votes

**0**answers

167 views

### Factorisation of twisted polynomials

Let $K=\mathbb{C}((t))$ and let $K_m=\mathbb{C}((t^{1/m}))$. let $K\{x\}$ denote the ring of twisted polynomials. The addition in this ring is defined as usual, but the multiplication is adjusted by ...

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votes

**0**answers

105 views

### Quotient modules of polynomial rings by maximal one-sided ideal

Let $R[X]$ be a ring of polynomials over an associative unital ring $R$ which is not necessarily commutative. Let $M$ be a maximal left ideal in $R[X]$. It is easy to see that if the intersection of ...

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48 views

### Injective Dimension of a quotient of the quantum plane

I am wondering what the injective dimension of the following ring is:
$$\frac{A}{(ax, bx)}$$
where $A$ is the so-called quantum plane $k\langle x, y \rangle/(xy + yx)$ with $k$ a field, and $a, b$ are ...

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votes

**2**answers

653 views

### How to make the Capelli's identity less mysterious?

The formulation of the Capelli's identity is very elementary; it has important applications in invariant theory and representation theory, see http://en.wikipedia.org/wiki/Capelli%27s_identity
To ...

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votes

**1**answer

95 views

### Quaternion algebra in characteristic $p$

Given a prime number $p$, can you give me concrete examples of fields $\mathbf F$ of characteristic $p$ and quaternion algebras $\mathbb H(\mathbf F)$ over $\mathbf F$ such that $\mathbb H(\mathbf F)$ ...

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87 views

### Differentially closed fields

Let F be a field. Recall that an additive map $d: F\rightarrow F$ is said to be a derivation if $d(ab)=ad(b)+d(a)b$.
Now let $F$ be a ring and let $d$ be a derivation of $F$. Examples I have in mind ...

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votes

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### What makes a theorem *a* “nullstellensatz.”

I know what the (Hilbert) Nullstellensatz says. A MathSciNet search on "nullstellensatz" turns up nearly 200 papers, with only a minority offering either new proofs or new applications of the classic ...

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votes

**0**answers

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### Does an hereditary scalar extension indicate the original algebra is hereditary?

Let $A$ be a finite-dimensional algebra over a field $\mathbb{F}$ of characteristic zero and let $\mathbb{K}/\mathbb{F}$ be a finite galois extension.
Assume we know that ...

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votes

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### Maximal ideal that annihilates entire ring

Does there exist a ring $R$ with a nonzero maximal ideal $M$ such that $R^2=R$ and $MR = RM = 0$?
Here $R$ is associative but does not have an identity (obviously). It seems a simple enough question ...

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votes

**8**answers

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### Simplest examples of rings that are not isomorphic to their opposites

What are the simplest examples of
rings that are not isomorphic to their
opposite rings? Is there a science to constructing them?
The only simple example known to me:
In Jacobson's Basic Algebra ...

**3**

votes

**1**answer

56 views

### Quaternion orders such that every proper ideal is invertible

Let $B$ be a quaternion algebra over $\mathbb{Q}$ and let $\mathcal{O} \subset B$ be an order.
A lattice in $B$ is (left) proper over $\mathcal{O}$ if its left order is equal to $\mathcal{O}$. We ...

**6**

votes

**2**answers

265 views

### non commutative polynomial which is zero for all matrix evaluation

I want to work on $K$ an algebraic closed (commutative) field of characteristic zero (even if it seems to be more general).
We can define the free K-algebra of polynomials in non commutative ...

**4**

votes

**2**answers

249 views

### unitary reduction of $q$-normal matrices

The unitary reduction of normal matrices is a well-known fact: if $A\in M_n(\mathbb C)$ commutes with its Hermitian adjoint $A^*$, then there exists a unitary $U\in\mathbb U_n$ and a diagonal matrix ...

**0**

votes

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180 views

### Basic question about power series and complete group algebras

This is a pretty basic question, but I suspect it might be too exotic for math.stackexchange.
Let $\mathbb{Z}_p$ be the $p$-adic integers. For free pro-$p$ group $F_r$ of rank $r$, we can consider ...

**7**

votes

**1**answer

109 views

### Center of $k$-algebra with two generators and sole defining relation $yx - xy = 1$ when $\text{char}\,k > 0$

Let $A(k)$ be a $k$-algebra with two generators, $x$, $y$, and one defining relation: $yx - xy = 1$. What is the center of the algebra $A(k)$ in the case $\text{char}\,k > 0$?

**1**

vote

**1**answer

107 views

### dg-resolution of the polynomial algebra

I am intersted in constructing a cofibrant resolution of the commutative polynomial algebra in some number of variables in the category of dg-algebras(not necceserily commutative).
The resolutions ...

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votes

**2**answers

153 views

### Rings all of whose torsion modules are cyclic

Let us call a (possibly non-commutative) ring $R$ "very good" if every finitely generated torsion left $R$-module is cyclic. Here is an example of such a ring:
Let $k=\mathbb{C}((t))$ and let ...

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vote

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116 views

### Are these quaternion algebras definite or indefinite?

By investigating a different problem I have ended up looking at Quaternion algebras and have a lot to learn about them. Before I do, however, I want to see if my idea has any hope of being useful. So ...

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votes

**1**answer

245 views

### Graded Hopf algebras and H-spaces

Let $k$ denote an algebraically closed field of characteristic $0$. Suppose $K=\bigoplus_{i\geq 0}K(i)$ is a Hopf $k$-algebra which admits a connected Hopf-grading (that is, a grading which is both ...

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votes

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330 views

### Projectives in the category of discrete G-modules

If $G$ is a profinite group, then the category $Mod(G)$ of discrete $G$-modules has sufficiently many injectives (Neukirch, Schmidt, Wingberg: Cohomology of Number Fields, 2.6.5).
Since the cited ...

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votes

**0**answers

182 views

### Discriminants of Clifford algebras

I have a Clifford algebra defined over a field of characteristic not equal to $2$. Is there a formula for its discriminant in terms of the corresponding symmetric bilinear form (or in terms of its ...

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votes

**6**answers

3k views

### Definition of an algebra over a noncommutative ring

I've tried in vain to find a definition of an algebra over a noncommutative ring. Does this algebraic structure not exist? In particular, does the following definition from ...

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vote

**1**answer

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### Graded category O for for rational Cherednik algebras, but at t=0

The paper [1] introduced the category $\mathcal{O}$ for rational Cherednik algebras $H_{t,c}(W)$. This construction is tailored for the $t=1$ case (equivalently, the $t\neq 0$ case). The general setup ...

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245 views

### Are all separable algebras Frobenius algebras?

Let $\mathcal C$ be a [added later: semi-simple] tensor category, and let $A=(A,m:A\otimes A\to A,i:1\to A)$ be an algebra object in $\mathcal C$.
The algebra is...
Separable if there is an ...

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**0**answers

82 views

### Centers of Noetherian Algebras and K-theory

I'll start off a little vauge: Let $E$ be a noncommutative ring which is finitely generated over its noetherian center $Z$. Denote by $\textbf{mod}\hspace{.1 cm} E$ the category of finitely ...

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### Hochschild Cohomology of the Quantum Torus

I would like some advice on how to compute directly, or by a higher powered method the Hochschild Cohomology groups of the quantum torus using the stated complex I have found. I think there are ...

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votes

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223 views

### Must a finitely generated projective module over a group ring with vanishing coinvariants be trivial?

Let $G$ be a (possibly infinite) group. Let $\mathbb{Z}[G]$ be its integral group ring and let $P$ be a finitely generated projective module over $\mathbb{Z}[G]$. Suppose that the coinvariants of $P$ ...