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Results tagged with ra.rings-and-algebras
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user 12858
Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
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Invertibility of all left multiplication maps in non-unital rings
Suppose that $R$ is a ring, not necessarily commutative nor associative. Assume that for every non-zero $a \in R$, the left multiplication map
$$ \lambda_a \colon R \to R \colon x \mapsto ax $$
is inv …
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2
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Tits-Kantor-Koecher construction for Jordan algebra of symmetric bilinear form
This is contained in Jacobson's Blue Book (Structure and Representations of Jordan Algebras, AMS Colloquium Publications, 1968), as Exercise 1 on p. 342, for arbitrary fields, and with no assumptions …
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2
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polarization/linearization as in jordan forms
The principle of polarizing is given by what you wrote yourself in the third paragraph. If $p$ is a homogeneous polynomial of degree $n$, then:
The clearest formulation is to take $p(x+ \lambda y) …
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4
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Group representation with algebra structure
A classification is too much to hope for, but the representation theory tells you whether such an algebra structure can exist: if $V$ is your $G$-representation, then an algebra product corresponds to …
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3
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2
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919
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Skew fields inside quaternion division algebras
Suppose that $Q$ is a quaternion division algebra with center $k$, where $k$ is an arbitrary commutative field (let's say with $\operatorname{char}(k) \neq 2$ if necessary). Assume that $D$ is an arbi …
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7
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An algebra map between Hopf algebras that does not commute with the counit
Such a map can certainly exist. For instance, take the $k$-algebra $G = k \times k$, with
$$ \begin{aligned}
&\Delta(1,0) = (1,0) \otimes (1,0) + (0,1) \otimes (0,1), \\
&\Delta(0,1) = (1,0) \otimes ( …
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12
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Simplest examples of rings that are not isomorphic to their opposites
Here is an explicit example of a central simple algebra over $\mathbb{Q}$ not isomorphic to its opposite (which is merely a detailed example of what Pete explained).
First take a cubic cyclic Galois …
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18
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4
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For which rings $R$ is $\mathrm{SL}_n(R)$ generated by transvections?
Let $R$ be a commutative ring $R$ with $1$, and $n \geq 2$ an integer.
Under which conditions is the group $\operatorname{SL}_n(R)$ generated by transvections?
(A transvection is a matrix with $1$ e …
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12
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Accepted
For which rings $R$ is $\mathrm{SL}_n(R)$ generated by transvections?
I'm answering my own question based on the excellent reference given by Max and the additional comments of Jim Humphreys. There is nothing new in my answer, but I think it's useful to close the questi …
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6
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Accepted
Chirality of octonion algebras
Perhaps this is more a question about the Fano plane than about the (real) octonions. Notice that the automorphism group of the Fano plane is the simple group $\operatorname{GL}(3, \mathbb{F}_2) \cong …
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12
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Divisibility and factorization in rings that are not integral domains
In my course notes for an undergraduate course "Algebra I", I wrote at the point when I'm introducing the notion of divisibility in rings (in a section on unique factorization):
We want to study f …
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12
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Applications of Jordan algebras
They turn up quite often in the study of (exceptional) linear algebraic groups. The most famous instance of this is the fact that algebraic groups of type $F_4$ are precisely the automorphism groups o …
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3
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1
answer
303
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ABA-product of matrices and length of chains of principal inner ideals
Let $k$ be a field, $p,q$ positive integers, and let $R$ be the space of $(p \times q)$-matrices over $k$, and $S$ be the space of $(q \times p)$-matrices over $k$. For every matrix $A \in R$, we defi …
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Cayley-Dickson form of a quaternion
I believe there is a good reason why mathematicians don't use the terminology "simplex-part" and "perplex-part": they are not canonical! Indeed, algebraically there is no way to distinguish the elemen …
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16
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vector to diagonal matrix
I'm not sure whether it answers your question, but here is a "matrix procedure" to transform the column vector $v$ into a diagonal matrix $D$:
Let $E_i$ be the $n \times n$ matrix with a $1$ on posit …
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