text:
\section{Introduction}
In this article we study a categorical analog of a certain class of Hopf algebras: namely Hopf algebras endowed with an inner product with respect to which the maps of multiplication and comultiplication are adjoint. Examples of such Hopf algebras arise naturally in many contexts. A principal example which we consider throughout the article is the Hopf algebra $\bigoplus_n K(\Rep(S_n))$ with multiplication and comultiplication given by induction and restriction functors and inner product given by the dimensions of the $\Hom$-spaces.  An abstract theory of this kind of  algebras was developed by Zelevinsky in the book \cite{Zelbook} under the name of positive self-adjoint Hopf (PSH) algebras. Certain representation-theoretic information, e.g. decomposition into irreducibles, can be derived from the abstract theory of PSH algebras.

All natural examples of PSH algebras have a categorical origin - in fact, they all appear as Grothendieck groups of categories of representations of sequences of groups or algebras. As such, it is logical to expect that the notion of PSH algebra has a categorical counterpart. In this article we propose a definition of such an object, which we call a symmetric self-adjoint Hopf (SSH) category. We apply this definition to show that any such category is equipped a canonical categorical action.

To construct the categorification of the notion of PSH algebra we observe that the property of self-adjointness allows to essentially only consider the structure of multiplication in the algebra or the category. The Hopf axiom in this approach can be viewed as a property of the multiplication.

On the categorical level this suggests to define a symmetric self-adjoint Hopf (SSH) category structure as a functor $\Hop$ from the category of finite sets $\FinSet$  to $\CatAd$ - the category of graded 2-vector spaces with 1-morphisms that admit adjoints. Such a functor defines the categorical analog of multiplication. $\Hop$ shouldn't preserve composition on the nose, so we have to work in a framework naturally allowing for these kind of functors. For the examples treated in this article it is practicable to define $\Hop$ as a functor of bicategories. To treat the Hopf axiom as a property of multiplication we require that Cartesian squares in $\FinSet$ should go to squares satisfying the Beck-Chevalley condition (\autoref{sec:BC}) in $\CatAd$. This allows us to associate an isomorphism to any image of a Cartesian square under $\Hop$.
In particular, the categorical Hopf axiom is given by the image of a certain Cartesian square of finite sets under $\Hop$. This allows for a concise definition of self-adjoint Hopf category without resorting to the use of complicated coherence relations that appear in the literature on the subject.
The notion of Hopf structure for semisimple abelian categories was first considered by Crane and Frenkel in  \cite{FrenkelCrane} and the coherence diagrams from their definition follow immediately in our framework (see \autoref{frenkelcranehopfcat}).
Moreover, we show that in $\CatAd$ a square satisfies the Beck-Chevalley condition if and only if it is a certain $2$-categorical generalization of a Cartesian square (an \emph{\comma} square). In this sense the Hopf axiom can be thought as a continuity condition on $\Hop$, i.e. the requirement that $\Hop$ preserves \comma squares. The relevant definitions, as well as some properties of SSH categories are contained in \autoref{sec:SSHDef}.

Denote the finite set $\{1,\ldots,n\}$ by $\fset{n}$. We will say that $\Hop$ gives a structure of SSH category on the category $\Hop(\fset{1})$ and that $\Hop(\fset{1})$ is an SSH category. We show in \autoref{ssec:SSHbasicproperties} that the Grothendieck group of an SSH category has a canonical structure of PSH algebra. In this sense the notion of SSH category is a categorification of the notion of PSH algebra. The category $\Hop(\fset{1})$ has a canonically defined symmetric monoidal structure (hence the word \emph{symmetric} in the designation).

Two of the principal examples of PSH algebras considered in the book \cite{Zelbook} are the algebras $\bigoplus_n K(\Rep(S_n))$ and $\bigoplus_n K\left(\Rep(S_n[G])\right)$ where $S_n[G]$ is the wreath product $G^n\rtimes S_n$. The parallel examples of SSH categories that we consider in this work are the category $\PP$ of polynomial functors defined by Friedlander and Suslin in \cite{FrieSusPoly} and the category $\PP_G$ of equivariant polynomial functors defined in \autoref{ssec:defpequivariant}. In characteristic 0 both of these categories are semisimple and the SSH structure on them descends to the PSH structure on $\bigoplus_n K(\Rep(S_n))\cong K(\PP)$ and $\bigoplus_n K\left( \Rep(S_n[G])\right)\cong K(\PP_G)$ respectively. These examples are treated in \autoref{sec:p} and \autoref{sec:pequivariant}.

In \autoref{prop:heisdouble} we show (following Zelevinsky) that any positive self-adjoint Hopf algebra $A$ admits a canonically defined action of the Heisenberg double $(A,A)$. This action is given by multiplication by the elements of $A$ and their adjoints which satisfy certain relations. The main result of this work is the proof in \autoref{sec:heisenbergaction} that every SSH category admits a categorical analog of such an action. To elaborate, a certain reformulation of the relations between multiplication and comultiplication giving the Heisenberg double action in $\ref{heisenbergrelation}$ allows us to interpret the categorical action as a statement about the existence of a certain 2-isomorphism in Theorem \ref{th:deltam}. This 2-isomorphism arises as a consequence of certain squares - which are part of the SSH structure - being Beck-Chevalley squares.

A remarkable feature of this isomorphism is that it arises on the category level from a canonical square with a non-invertible 2-morphism satisfying the Beck-Chevalley condition. Such a square is a purely categorical construct which is not detectable on the vector space level, where it descends to a non-commutative diagram. In other words the Heisenberg double action on the vector space level can only be formulated in terms of relations between multiplication and co-multiplication, whereas on the category level multiplication is enough and hence we are not required to choose an adjoint representing the co-multiplication.

To describe this approach in a more conceptual way we introduce a new category $\Heis{\Hop}$ that we call a \emph{categorical Heisenberg double} associated to the functor $\Hop$ equipped with a canonical projection to $\CatAd$. We use certain conjectural properties of this category stated in \S\ref{heuristics} to compute this square as an $\comma$ square in $\CatAd$. We conjecture that the category $\Heis{\Hop}$ is a categorification of the algebraic notion of Heisenberg double in the sense that the $2$-morphism structure of the action is given by the squares and relations between them in $\Heis(\Hop)$. It is our hope that $\Heis{\Hop}$ will be useful for constructing new canonical categorifications given the generalization of the notion of Hopf category to various frameworks such as dg- or stable $\infty$-categories.

The simplest and the most familiar example of the Heisenberg double construction is the Heisenberg algebra of infinite rank. It is the Heisenberg double associated to the PSH algebra $\Lambda$ of symmetric functions. Note that $\Lambda \cong \bigoplus_n K(\Rep(S_n))\cong K(\PP)$. The Heisenberg algebra is traditionally defined in the literature in terms of an infinite number of generators and relations; there are several common presentations, some of which, as well as their relation to the generator-independent definition used in this article we outline in \autoref{ssec:generators}.
Khovanov proposed, in \cite{Khov}, an approach to categorification of the Fock space action of this algebra in a diagrammatic language of generators and relations for endofunctors of a category and $2$-morphisms between them. In \cite{hongyacobiPoly} Hong and Yacobi describe this action on the category $\PP$ of polynomial functors. Since $\PP$ is our basic example of an SSH category, it is natural to compare our construction of categorical Heisenberg double action to the former. In \autoref{ssec:FockP} we explicitly construct the morphism lifting the Heisenberg relation in this case.

In fact, the original motivation for this article was our desire to understand which parts of Khovanov's construction (and it's realisation on $\PP$ by \cite{hongyacobiPoly}) can be derived from the categorical version of a PSH structure via categorical constructions. The 2-isomorphisms we construct in Theorem \ref{th:deltam} are part of Khovanov's 2-morphism structure - these 2-morphisms are exactly what is needed to construct an embedding of the Heisenberg algebra into the Grothendieck group of Khovanov's Heisenberg category. Moreover, it is possible to recover all of the graphical calculus constructed in \cite{Khov} from an SSH structure on a category. Given a category with a SSH structure and an analog of a primitive element in the algebra one can construct the Heisenberg action in the sense of Khovanov by choosing adjunction data satisfying certain compatibility conditions. We elaborate on this in a forthcoming article.

In the article \cite{savageyacobi} by Savage and Yacobi the authors propose to consider the isomorphisms \eqref{list:categoricalFock} (which they call "categorified Mackey relations") for categorifying the Fock space representation of the Heisenberg doubles given by the $K$-groups of categories of modules over towers of algebras and construct these for a certain class of examples. It would be interesting to attempt to formulate their results in the generalization of the SSH categories framework described here to the non-semisimple category case, and this might provide additional insight into the structure inherent in these examples (see also \autoref{rem:leftright}).

The Heisenberg algebra action and its categorification also appears naturally in several places (\cite{Grojnowski}, \cite{Nakajima} and the categorical version in \cite{CautisLicataHilbert}, also \cite{CautisLicataQuantum} to cite a few). It would be interesting to compare our construction to the existing results.

\subsection*{Generalization and future applications}
\label{rem:stable}
The setting of 2-vector spaces considered in this article is rather restrictive and doesn't include many important examples which should naturally exhibit an SSH structure, such as the category of polynomial functors over a field of positive characteristic. The Grothendieck group of this category won't be PSH, but will be a self-adjoint Hopf algebra. It seems that the correct generalization for this setting is that of stable $\infty$-categories. The main reason this seems reasonable as a generalization is that the equivalence between \comma and Beck-Chevalley squares should carry over easily to that setting. The problem with transferring everything directly to the stable $\infty$ setting is the lack of a framework for working with the $(\infty,2)$ category of all stable $\infty$ categories and computing things like \comma squares, or their appropriate generalization. The setting of stable $\infty$-categories would allow us to work with categories of complexes of objects, thus for example providing an approach to categorification of Boson-Fermion correspondence by using the connection to categorified Heisenberg action described below.

Zelevinsky's main theorem about PSH algebras is that any such algebra is isomorphic to the tensor product of many copies of the PSH algebra $\Lambda$ of symmetric functions (see \cite{Zelbook} \S2 for the precise formulation). His proof is somewhat combinatorial in nature, and gives the morphism to the tensor product only up to a non-canonical choice. In \autoref{sec:pequivariant} we construct a canonical equivalence of SSH categories $\PP_G$ and $\PP^{\otimes\Irr G}$ which descends to the isomorphism from Zelevinsky's theorem. It would be interesting to understand what analog does Zelevinsky's decomposition theorem have in general on the categorical level. As before one can hope to get especially interesting and new results outside of the semisimple setting. For example, in the case of the category of polynomial functors over a field of positive characteristic, the decomposition theorem doesn't hold on the vector space level, because the K-group lacks the positivity property. This case is interesting because of its connection to the theory of modular representations of symmetric groups.

As we noted earlier, an SSH structure naturally defines a symmetric monoidal structure on the category. The reason for it is our choice of $\FinSet$ as the indexing category. Modifying the definition to get braided monoidal structure on the underlying category would produce a framework for dealing with interesting examples such as categories of representations of quantum groups, or of $\GL(n,\FF_q)$, and categorical objects connected to Hall algebras.

\subsection*{Acknowledgments} The idea of using the Beck-Chevalley condition was suggested to us by the lectures of Nick Rozenblyum in the Caesarea 2013 workshop on Grothendieck operations and $\infty$-categories. We thank him for several helpful discussions which helped shape our approach in this paper. We thank Oded Yacobi and Jiuzu Hong for introducing us to the subject of categorification and the notion of polynomial functors, Emily Riehl and Vladimir Hinich for helpful comments and Ross Street for suggesting ways to make our article more rigorous by using bicategories. We thank our colleagues Sasha Yom Din and Evgeny Musicantov for their mathematical and moral support.

Finally, we thank our advisor Joseph Bernstein who influenced our approach to mathematics through many discussions on this and other topics.

\section{Notations}
\begin{itemize}
\item $\kk$ - a field of characteristic 0
\item $\Vect$ - the category of finite dimensional vector spaces over $\kk$

\item $\fset{n}$ - the set $\{1,2,\ldots,n\}$
\item $\pset{n}$ - the set  $\{1,2,\ldots,n, *\}$ in the category of pointed finite sets $\FinSet_*$
\item $\Sh(S)$ - the category of sheaves of finite dimensional vector spaces on $S$
\item identity morphisms will be drawn as $\longequal$ in diagrams
\item we will refer to a square of the form
\smallttik{
A \ar{r} \ar{d} \& B \ar{d} \ar[Rightarrow,shorten <=0.7em,shorten >=0.7em]{dl}[above,sloped]{\alpha}\\
C \ar{r} \& D
}
as $\alpha$

\end{itemize}


TITLE: Factorization of fields to finitely generated ideals
QUESTION [1 upvotes]: Let $K$ be a field and $I=\langle 2, x^2, x^3\rangle$ an ideal of $K[x]$. How does $K[x]/I$ look like? I think I have a quite poor grasp of how factorization works in rings and I would very much appreciate an explanation.

REPLY [2 votes]: First, note that if $x^2 \in I$, then $x^3 = x \cdot x^2\in I$ by the definition of an ideal. So we can remove $x^3$ from the generators: $I = \langle 2, x^2 \rangle$.
Next, consider two cases:

$2$ is a unit (invertible) in $K$. Then, since $2 \in I$, we have $2 \cdot {1 \over 2} = 1\in I$. An ideal containing $1$ coincides with the whole ring, since $\forall x \in K[x]: x = 1 \cdot x \in I$. In this case, $I = K[x]$ and $K[x]/I = 0$ (the zero ring).
$2$ is not a unit in $K$, which can happen only if $2 = 0$ in $K$ (that is, the characteristic of the field is $2$). In this case we can remove $2$ from the set of generators, since any ideal contains $0$ by definition: $I = \langle x^2 \rangle$. Now, any element of $K[x]/I$ can be represented by $a + bx$, where $a,b \in K$ and the multiplication rule is derived from $x^2 \in I$, which translates to $x^2 = 0$ in the quotient ring: $(a+bx)(c+dx)=ac+(ad+bc)x+bdx^2 = ac+(ad+bc)x$. This formula, with the obvious addition rules, fully describes the ring $K[x]/I$.


TITLE: Scaling property of  Fourier series and Fourier Transform
QUESTION [4 upvotes]: This question about the intuition behind the scaling property of the Fourier transform made me wonder about the corresponding notion for a Fourier series.
The Fourier transform of $f(ax)$ is $\frac{1}{|a|} \mathcal{F(\frac{u}{a})}$. If $a>1$ then the graph of $f(ax)$ is $f$ compressed and so its Fourier transform has frequencies that are higher. However they are scaled down in magnitude.
On the other hand take $f(x)= \cos x$. Its fourier series is trivially itself, with the coefficient of $\cos x$ being $1$. Scaling it to $f(ax) = \cos ax$ where $a>1$ is an integer still has a trivial fourier series, and the coefficient of $\cos ax$ is $1$. This is unlike the Fourier transform where the "coefficients" of each frequency get scaled as the formula shows: $\frac{1}{|a|} \mathcal{F(\frac{u}{a})}$.
Is there a conceptual way to explain this discrepancy?

REPLY [6 votes]: Let $a\neq 0$.  For a function $f$ define $m_af$ by $(m_af)(x) = f(ax)$.  Then the scale factor that appears in a Fourier transform of $m_af$ is directly related to the norm of $m_a$ since $$||m_af||_2 = ||m_a||\cdot ||f||_2$$ and a Fourier transform on $\mathbb{R}$ or $S^1$ is (up to a normalization factor) unitary.  Now on $\mathbb{R}$ we have $||m_a|| = |a|^{-\frac{1}{2}}$ while on $S^1$ (and $a \in \mathbb{N_{>0}}$) $||m_a||=1$.
As an amusing aside, if you take the Fourier transform of $\cos$ as a tempered distribution then $$\mathcal{F}(m_a\cos) = \tfrac{1}{2|a|}(m_{a^{-1}}\delta_1 + m_{a^{-1}}\delta_{-1})= \tfrac{1}{2|a|}(|a|\delta_a + |a|\delta_{-a}) = \tfrac{1}{2}(\delta_a + \delta_{-a})$$ by the transformation property of the Dirac distribution as noted in a comment.  So in this case there is a factor $|a|^{-1}$ as usual although it is not directly visible.


TITLE: Ring of integers of cubic number field
QUESTION [13 upvotes]: I want to show that the ring of integers of the cubic number field $K = \mathbb Q(\alpha)$, where $\alpha$ is a root of $f = X^3 - X - 2$, is equal to $\mathbb Z[\alpha]$.
$(1, \alpha, \alpha^2)$ forms a $\mathbb Q$-basis of $K$ consisting of integers. I know $\mathbb Z[\alpha] \subseteq \mathcal O_K$. To show equality, I'm guessing the following result is useful:

Let $K$ be a number field of degree $n$, $(\alpha_1, \dots, \alpha_n)$ a $\mathbb Q$-basis of $K$ consisting of integers. Suppose there exist $m \in \mathbb N, m > 2$ and $k_1, \dots, k_n \in \mathbb Z$ with $gcd(m, k_1, \dots, k_n) = 1$ such that $$\frac{k_1 \alpha_1 + \dots + k_n \alpha_n}{m} \in \mathcal O_K$$ then $m^2 \,\big\vert\, disc(\alpha_1, \dots, \alpha_n)$

I have $disc(\alpha_1, \dots, \alpha_n) = -104$, so $m = 2$, hence I want to show that I cannot have $$\frac{a + b \alpha + c \alpha^2}{2} \in \mathcal O_K$$ for integers $a, b, c \in \mathbb Z$ of which at least one is odd. But I'm getting stuck here. How do I show these aren't integral elements? Can anyone give me a hint on how to proceed?

REPLY [0 votes]: Other answers are too advanced. Checking definitions is enough for this question.
We have $\mathrm{tr}\,\frac12(a+b\alpha+c\alpha)=\frac32a+c$, so $a$ is even, leaving $\frac12(b\alpha+c\alpha^2)$ to be considered. We have $\mathrm{N}(\frac12\alpha(b+c\alpha))=-\frac14(b^3-bc^2+2c^3)$, which is an integer iff $b,c$ are both even.