ct.category theory – Algebras for general transfors

Algebras for endofunctors bridge the gap between functors acting on a category and structures defined in it. An algebra for an endofunctor $F$ is instantiated by some morphism $Fa to a$, and more crucially a morphism of algebras is a map $ato b$ between their codomains making the evident diagram commute. This construction is intrinsically homotopical, so it makes good sense to loosen the restriction from just endofunctors.

Do you know of any writing about algebras for more general transfors? Seems that many 2-categorical results about algebras over monads, lax morphisms, et cetera, would enjoy a more full account of transfors and their algebras.

ra.rings and algebras – Tensor product of positive linear maps is positive

Let $pi_1: A_1 to B_1$ and $pi_2: A_2 to B_2$ be positive linear maps between complex $*$-algebras. Is the mapping
$$pi_1 otimes pi_2: A_1 otimes A_2 to B_1 otimes B_2$$
again positive?

I.e., if $sum_{i=1}^n x_i otimes y_i in A_1 otimes A_2$, do we have
$$sum_{i,j=1}^n pi_1(x_i^*x_j)otimes pi_2(y_i^*y_j) ge 0$$

I know the proof for the case that $B_1 = B_2 = mathbb{C}$. In that case, we can work with diagonalisation of matrices.

hopf algebras – Do chains send homotopy inverse limits of spaces to homotopy inverse limits of $E_infty$-coalgebras?

Let $X_bullet := … X_2 to X_1$ be a tower of connected and simple spaces
with the following properties:

The induced tower $H_ast(X_bullet; mathbb{F}_p)$ of graded $mathbb{F}_p$-vector spaces
is Mittag-Leffler and lifts to a tower of graded abelian hopf algebras over $mathbb{F}_p$.
The induced tower $pi_1(X_bullet)$ is Mittag-Leffler.

By a theorem of Goerss the canonical morphism $$ H_ast(holim X_bullet; mathbb{F}_p)to lim H_ast(X_bullet; mathbb{F}_p)$$ is an isomorphism, where the limit on the right hand side is taken in the category of graded abelian hopf algebras over $ mathbb{F}_p$.

The limit in the category of graded abelian hopf algebras over $ mathbb{F}_p$ forgets to the limit in the category of graded cocommutative coalgebras over $mathbb{F}_p$ but does generally not forget to the limit in graded $mathbb{F}_p$-vector spaces.

Is there a similar result on the chain level?

Is it true under the assumptions on $X_bullet$ that the canonical morphism $$ C_ast(holim X_bullet; mathbb{F}_p)to holim C_ast(X_bullet; mathbb{F}_p)$$ is a quasi-isomorphism, where
$C_ast(-; mathbb{F}_p)$ are chains with $mathbb{F}_p$-coefficients, and the homotopy limit on the right hand side is taken in the $infty$-category of $E_infty$-coalgebras over $mathbb{F}_p$?

This would follow of course from Goerss theorem if homology $H_ast$ would send
the homotopy limit in $E_infty$-coalgebras (over $mathbb{F}_p$) of the tower $ C_ast(X_bullet; mathbb{F}_p)$ to the limit in graded cocommutative coalgebras over $mathbb{F}_p$.

Does one know such a result?

Can one say more if one additionally assumes that the tower $X_bullet := … X_2 to X_1$
of spaces refines to a tower of grouplike $E_infty$-spaces?

Under this assumption the induced tower $C_ast(X_bullet; mathbb{F}_p)$
is a tower of $E_infty$-hopf algebras over $mathbb{F}_p$, i.e. abelian group objects (in the derived sense) in the $infty$-category of $E_infty$-coalgebras over $mathbb{F}_p.$

Therefore by Goerss theorem the canonical morphism $$ C_ast(holim X_bullet; mathbb{F}_p)to holim C_ast(X_bullet; mathbb{F}_p)$$ would be a quasi-isomorphism if homology $H_ast$ would send
the homotopy limit in $E_infty$-hopf algebras of the tower $ C_ast(X_bullet; mathbb{F}_p)$ to the limit in graded abelian hopf algebras over $mathbb{F}_p$.

Does one know such a result?

ra.rings and algebras – rings isomorphism

ra.rings and algebras – rings isomorphism – MathOverflow

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!

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