Primitive elements in Hopf algebras over the integers

Let $H$ be a Hopf algebra over $mathbb Z$, and assume that $H$ is cocommutative, graded, generated in degree $1$, and connected (its degree-$0$ part is $mathbb Z$).

Are there nice, natural conditions that will enforce that $H$ is a universal enveloping algebra of a Lie algebra over $mathbb Z$?

For example, if $H$ is $mathbb Z$-free, then the Milnor-Moore theorem implies $Hotimesmathbb Q=U(P)$ for $P’$ the space of primitives in $Hotimesmathbb Q$, and presumably $P’=Potimesmathbb Q$ for $P$ the $mathbb Z$-module of primitives in $H$.

I’m sure this works in a much more general setting, but I failed to locate relevant papers or books on Hopf algebras over non-fields.

Note that this question is related to the MO question
Integral Milnor-Moore theorem, though it seems orthogonal.

matrices – Can all finite dimensional non commutative algebras be embedded into matrix rings?

Suppose I have a finite (non-)commutative ring $R/k$ (over a field $k$ of char $0$) with a linear “trace” function $t: R to k$. Can I find square matrices $A_1,dots,A_n$ (of some dimension $r$) so that I have an embedding $f: R to M_r(k)$ compatible with the trace functions on both sides?

One restriction I can see for the trace function on $R$ is that it should be invariant under cyclic permutations : $t(a_1a_2dots a_n) = t(a_2dots a_na_1)$. Is this the only restriction?

ra.rings and algebras – Elements with equal annihilators

Let $R$ be a finite commutative ring with unity. Let $a in R$ and define $C_a = {b in R : ann(a) = ann(b)}$.

I want to know the cardinality of the set $C_a$.

For example,

If $a=0$ then $|C_a| = 1$.

If $a$ is a unit then $|C_a| = |U(R)|$ the set of units of $R$.

If $a$ is a zero divisor then I don’t now the answer.

If $R$ is the ring of integers modulo $n$ and $a$ is a non-zero zero divisor then $C_a = {x in mathbb Z_n : (x,n)=d}$ where $a = md$ for a proper divisor $d$ of $n$.

If $R$ is a reduced ring, then $R cong F_{q_1} times cdots times F_{q_k}$, product of finite fields. Define supp(a) = {1 le i le k : a_i is a unit} where $a = (a_1,dots,a_k) in R$. Then $C_a = {b in R: text{supp} (b) = text{supp} (a)}$. From this cardinality of $C_a$ can be obtained.

Thank you.

at.algebraic topology – Uses for (Framed) E2 algebras twisted by braided monoidal structure

$newcommand{C}{mathcal{C}}$ $newcommand{g}{mathfrak{g}}$
If $C$ is a monoidal category (not necessarily a symmetric monoidal category), it’s possible to define the notion of an algebra object $A$ in $C$, with multiplication operations $$A^{otimes n} (:= Aotimes_C Aotimes_C cdotsotimes_C A)to A.$$

Similarly, if $C$ is a braided monoidal category (resp., a ribbon category), one can define a notion of $E_2$ DG algebra $A$ (resp., framed $E_2$ DG algebra $A$) “twisted” by $C$, consisting of operations $A^{otimes n}to A$ compatible with braiding. (Note: I actually don’t know a reference for this, but it follows from standard “homotopy field theory” arguments involving the Ran space.)

In particular, if $C$ is a braided monoidal (or ribbon) category coming from an associator on a Lie algebra $g$ (with choice of Casimir), there is a whole category of “associator-twisted” $g$-equivariant $E_2$ (resp., framed $E_2$) algebras.

My question is whether algebras of this type have been encountered before. They feel very CFT-ish, and so I’m particularly curious about physics and knot theory applications. In particular, the framed variant should gove some kind of derived 2D TQFT-style invariants.

Any references would be useful. Thanks!

Universality in the class of separable Banach algebras

Let us consider the class of Banach algebras with homomorphisms that are bounded below but not necessarily isometric.

  1. Is there are separable Banach algebra that contains isomorphic images of all separable Banach algebras?

  2. Is there a commutative separable Banach algebra that contains commutative separable Banach algebras?

The trick with bounded the distance between commuting projections (of arbitrary norm) does not work in either case.

Universal property for finite-dimensional Clifford algebras

Here’s my definition of a Clifford algebra:

Definition: Let $B(cdot,cdot)$ be a symmetric bilinear form on a vector space $V$ over $mathbb{K}$ and $Q$ its associated quadratic form. The Clifford algebra associated to the quadratic space $(V,Q)$ is a pair $mathcal{C}l(V,Q)$ where $mathcal{C}l(V,Q)$ is a $mathbb{K}$ associative algebra with identity $1$ and $varphi: V to mathcal{C}l(V,Q)$ is a Clifford map satisfying the following properties:

(a) $varphi(u)varphi(v)+varphi(v)varphi(u) = 2B(u,v)1$ for every $u,v in V$.

(b) The subspace $text{Im}varphi$ generates the algebra $mathcal{C}l(V,Q)$.

(c) For every Clifford map $phi: V to mathcal{A}$ on $(V,Q)$ there exists a homomorphism of algebras $f: mathcal{C}l(V,Q) to mathcal{A}$ such that $phi = fcirc varphi$.

Suppose $V$ is finite dimensional, with dimension $n$. Then $mathcal{C}l(V,Q)$ has dimension $2^{n}$ and is generated by $1$, $varphi(x_{i})$, $varphi(x_{i})varphi(x_{j})$, ($i<j$),…, $varphi(x_{1})cdots varphi(x_{n})$, where ${x_{1},…,x_{n}}$ is a basis for $V$. Moreover, one can prove that the Clifford map $varphi$ is injective. Thus, we may identify $V$ with the image of $varphi$, so that $V$ can be treated as a subspace of $mathcal{C}l(V,Q)$. Thus, $varphi(x_{i})$ becomes simply $x_{i}$.

In the literature, we often find the definition of a Clifford algebra with dimension $2^{n}$ as an algebra generated by (the generators) $1$, $x_{i}$, $x_{i}x_{j}$ ($i<j$),…,$x_{1}cdots x_{n}$, and these elements satisfy:
$$x_{i}x_{j} + x_{j}x_{i} = 2delta_{ij}$$
Let’s call this definition our second version of a Clifford algebra.

As I stressed before, this second case is a simplified version of the above defition, in which $V$ has dimension $n$. The generators $1$, $x_{i}$, $x_{i}x_{j}$ ($i< j$),…, $x_{1}cdots x_{n}$ are identifications of $1$, $varphi(x_{i})$, $varphi(x_{i})varphi(x_{j})$, ($i<j$),…, $varphi(x_{1})cdots varphi(x_{n})$ and $B$ is a bilinear form in which the basis of $V$ is orthonormal.

In summary, the second version follows from the first. But taking the second version as the definition of a Clifford algebra seems a little odd to me because it does not mention property (c) of the above definition, which plays an important role in the abstract theory of Clifford algebra. So my question is: does property (c) holds trivially in the case of finite dimensional vector spaces $V$, so it does not have to be demanded in the definition of such a Clifford algebra, as the second version seems to imply? If not, why is this property commonly omitted in so many references?

oa.operator algebras – Trying to understand Haagerup tensor product $B(H)otimes_{rm h}B(K)$

I’m self reading Haagerup tensor product of operator spaces. Understanding it properly, I’m trying some examples. Let $H$ And $K$ be Hilbert space. Let $B(H)$ and $K(H)$ denotes the spaces of bounded and compact operators on $H$?

Can someone explain me what is $B(H)otimes_{rm h}B(K)$ and $B(H)otimes_{rm h}K(H)$? Are these spaces completely isometric to some well known operator space?

Is there any reference/lecture notes where I can find these kind of stuff?

P.S: The same question was first asked on MSE here.

nt.number theory – Explicit construction of division algebras of degree 3 over Q

In his book Introduction to arithmetic groups, Dave Witte Morris implicitly gives a construction of central division algebras of degree 3 over $mathbb{Q}$ in Proposition 6.7.4. More precisely, let $L/mathbb{Q}$ be a cubic Galois extension and $sigma$ a generator of its Galois group.If $p in mathbb{Z}^+$ and $p neq tsigma(t)sigma^2(t)$ for all $t in L$, then
$$ D=left{ begin{pmatrix}
x & y & z\
psigma(z) & sigma(x) & sigma(y)\
psigma^2(y) & psigma^2(z) & sigma^2(x)
end{pmatrix} :(x,y,z)in L^3 right}

is a division algebra.

On page 145, just before Proposition 6.8.8, Morris claims that it is knows that every division algebra of degree 3 arises in this manner. This should follow from the fact that every central division algebra of degree 3 is cyclic. I could not find this explicit construction in my references (e.g. Pierce – Associative Algebras, though maybe I missed something) and I would like to know if there is a reference or a quick way to see that this exhausts all central division algebras of degree 3 over $mathbb{Q}$.

ra.rings and algebras – Representation theory terminology question

For a paper I’m writing, I need a term for a representation-theoretic concept that I’m sure someone has thought of before, so I thought I’d ask here rather than just make something up.

Let $G$ be a group and $R$ be a commutative ring. Consider an $R(G)$-module $V$. For any ideal $I$ of $R$, we have the submodule
$$I V = {text{$c cdot v$ $|$ $c in I$ and $v in V$}}.$$
What is the term for $R(G)$-modules $V$ such that all submodules are of this form? The ones I’m interested in have the additional property that if you ignore the $G$-action, then they are free $R$-modules (though not finitely generated!), but I doubt this matters for this question.

For an easy example, if $R = mathbb{Z}$ and $G = text{GL}(n,mathbb{mathbb{Z}})$, then $mathbb{Z}^n$ has this property.

If $R$ is a field, then this reduces the the usual notion of an irreducible representation, so I think of this as a version of irreducibility. But looking through my ring-theory books, I can’t find it anywhere.

brauer groups – A local-to global principle for splitting of Azumaya algebras

Let $S$ be a finitely generated domain with the field of fractions $F.$ Let X be a smooth,
geometrically connected affine variety over $S.$ Let $A$ be an Azumaya algebra over $X.$
Assume that for all large enough primes $p,$ $A_p$ splits over $X_p$-the reduction modulo $p$ of $X.$ Does this assumption imply that $A_{overline{F}}$ splits over $X_{overline{F}}?$ My naive guess is that the answer should be “yes”. Any suggestions or references would be
greatly appreciated.