# Deformations of smash products

$\newcommand\k\kappa$ $\newcommand\gl{\mathfrak{gl}}$

# Literature

## Known examples
- [EGG] https://arxiv.org/pdf/math/0501192.pdf : "infinitesimal Hecke algebras" (_"infinitesimal Cherednik algebras" for type_)
- $(Sp_{2n}, \mathbb{C}^{2n})$: one new deformation in every second degree
- $(GL_n, \mathbb{C}^n\oplus (\mathbb{C}^n)^*)$ _(this one is irrelevant for us at the moment, as the module is not simple)_
- [Ts] https://www.math.purdue.edu/~otsymbal/Papers/Journal_Paper_5.pdf :
- $(O_n,\mathbb{C}^n)$ (type B and D): one new deformation in every second degree
- degree $1$ deformations are classificed as _symmetric pairs_ (symmetric spaces) https://en.wikipedia.org/wiki/Symmetric_space#Classification_result

# Theory

## Motivation

- $TV=k\langle v_1,\dots,v_n\rangle$; a free $k$-algebra
- $H$ is always quadratic algebra; $H=k\langle h_1,\dots,h_m: \text{relations of the form }h_i h_j = h_k \text{ (group algebra) or } h_i h_j = h_j h_i + \sum_k a_k h_k \text{ (Lie algebra)}\rangle$

__Example.__ $M:=Mat^{2\times 2}(\mathbb{C})$. Then for $A,B\in M$, we can compute the _commutator_
$$
[A,B] = AB-BA \in M
$$

For instance, $A=E_{i,j}\in M$ and $B=E_{k,l}\in M$.
Then $E_{i,j}E_{k,l}=\delta_{j,k} E_{i,l}$, so
$$
[E_{i,j}, E_{k,l}] = \delta_{j,k} E_{i,l} - \delta_{i,l} E_{k,j}
$$

e.g.:
$$
E_{1,2} E_{1,2} = 0
$$

$$
[E_{i,j},E_{i,j}] = 0
$$

$$
[E_{1,2},E_{1,1}] = - E_{1,2}
$$

$$
[E_{1,2},E_{2,1}] = E_{1,1} - E_{2,2}
$$

In other words:
$$
E_{1,2} E_{2,1} = E_{2,1} E_{1,2} + (E_{1,1}-E_{2,2})
$$ which is a relation of the form $h_i h_j = h_j h_i + \sum_k a_k h_k$

(In our case, $\{E_{i,j}\}=\{E_{1,1},E_{1,2},E_{2,1},E_{2,2}\}$.)

$(M,[\cdot,\cdot])$ is a __Lie algebra__.
This one is also called $\mathfrak{gl}_2$ ("general linear").
If we consider $n\times n$-matrices, we call it $\mathfrak{gl}_n$.

> Other intersesting cases include $\mathfrak{sl}_n$ (special linear; $\mathfrak{gl}_n$ with condition that $tr A = 0$), or $\mathfrak{so}_n$ (special orthogonal; condition $A^T=-A$).

We set $H=k\langle E_{i,j} \text{ for }1\leq i,j\leq n: E_{i,j} E_{k,l} = E_{k,l} E_{i,j} - [E_{i,j},E_{k,l}] \text{ for } (i,j)>(k,l) \rangle$.

(For $\gl_2$, there are $6$ relations...)

$H$ is called __universal enveloping algebra__ of $M=\gl_2$.

Let us also consider $V=Mat^{2\times 1}(\mathbb{C})$. We have an __action__ of $M=\gl_2$ on $V$: for $A\in M$ and $v\in V$, we can compute
$$ A\cdot v = Av \in V
$$

Set $v_1=(1,0),v_2=(0,1)$ to be the standard basis of $V$. Then
$$
E_{i,j} \cdot v_k = \delta_{j,k} v_i
$$

$TV\rtimes H$, the __smash product__ of the tensor algebra of $V$ with $H$ is then the algebra
$$
k\langle E_{i,j}, v_i : \text{relations from $H$ plus } E_{i,j} v_k = v_k E_{i,j} + \delta_{j,k} v_i \rangle
$$

$A_0 = SV\rtimes H$, the __smash product__ of the symmetric algebra of $V$ with $H$ is then the algebra
$$
k\langle E_{i,j}, v_i : \text{relations from $TV\rtimes H$ plus } v_i v_j = v_j v_i \rangle
$$

There are two kinds of generators:
- those from the Hopf algebra $H$
- those from the $H$-module $V$

There are three kinds of relations:
- between generators of $H$
- between generators of $H$ and $V$
- between generators of $V$

These relations imply that $S(V)\rtimes H$ has a basis given by ordered monomials in the generators of $H$ and $V$, the __PBW basis__. The basis representation with respect to this basis is called __normal form__.

We also want to consider:
$A_\k = TV\rtimes H$, a __deformation__ of $A_0$
$$
k\langle E_{i,j}, v_i : \text{relations from $TV\rtimes H$ plus } v_i v_j = v_j v_i + \kappa(v_i,v_j) \rangle
$$

Such $\k$ will have to satisfy $\k(v_i,v_j)=-\k(v_j,v_i)$. The elements $\k(v_i,v_i)$ may be chosen in $H$, that is, any polynomial in the $\{E_{i,j}\}_{i,j}$.

As a special case, we get $A_{\k=0}=A_0$.

$\k$ is called __PBW deformation__ if $A_\k$ still has a PBW basis, i.e., if the ordered monomials still give linearly independent elements.

:::info
__Main questions__
- Which $\k$ are possible? Which PBW deformations are there?
- Given $\k$, what is the center of $A_\k$?
:::

## Abstract set-up

$\renewcommand\o{\otimes}$

$$ A = (TV\rtimes H)/I $$

$TV := T(V) :=$ tensor algebra; for a vector space $V$ (for us usually finite-dimensional) this is $k\oplus V\oplus V\otimes V \oplus \cdots$, or $k\langle v_1,\dots,v_n\rangle$ (non-commutative) for a basis $v_1,\dots,v_n$

## Tensor algebra

$H$, a Hopf algebra; for us, there are two cases:
- $H=kG$, a group algebra,
- or $H=U(L)$, a universal enveloping algebra

$V$ is $H$-module; there is an _action_ $H\otimes V\to V, (h,v)\mapsto h\cdot v=hv$, s. th. $1\cdot v=v$, $(ab)\cdot v=a\cdot (b\cdot v)$.

If $H$ is as above, then such an action induces one on $TV$.
- $G\ni g\cdot (v\otimes w)=gv\otimes gw$
- $G\ni g\cdot 1=1$
- $L\ni x\cdot (v\otimes w)=xv\o w + v\o xw$
- $L\ni x\cdot 1=0$

This implies
- $g(v_1\o\dots\o v_n) = gv_1\o\dots\o gv_n$
- $x(v_1\o\dots\o v_n) = xv_1\o\dots\o v_n+\dots+v_1\o\dots\o xv_n$

:::success
$TV$ is an $H$-module algebra
- $g(vw)=(gv)(gw)$
- $x(vw)=(xv)w+v(xw)$
:::

## Semidirect products

$H$ a Hopf algebra.
$B$ an $H$-module algebra (for us, $B=TV$).

__Def.__ $B\rtimes H := B\o H$ as a vector space, with multiplication
$$
(b\o h)(b'\o h')=b(h_1\cdot b')\o h_2 h'
$$ where $h_1\o h_2=g\o g$ for $g\in G$ and $h_1\o h_2=x\o 1+1\o x$ (in $H\o H$) for $x\in L$ (in Sweedler's notation).

So the above is
$$
(b\o g)(b'\o h')=b(g\cdot b')\o g h'
$$ and $$
(b\o x)(b'\o h')=b(x\cdot b')\o h'+bb'\o x h'
$$

__Notation.__ We identify $B=B\otimes 1\subset B\rtimes H$ and $H=1\o H\subset B\rtimes H$.

__Expl.__ $G, K$ finite groups, $G$ acts on $K$ by automorphisms.
$$
k(K\rtimes G)\cong kK\rtimes kG
$$ where $K\rtimes G$ is the following group structure on the $K\times G$:
$$ (k,g)(k',g') = (k(g\cdot k'), gg')
$$

__Prop.__ $K, G\le F$, $K\triangleleft F$, $KG=F$, $K\cap G=\{1\}$ $\Rightarrow$ $F=K\rtimes G$

__Expl.__ $L$ a Lie algebra with a module $V$ via $\cdot$.
$B=TV$, $H=U(L)$, $B\rtimes H$ is the algebra generated by $v_1,\dots,v_n$ (a basis of $V$), $x_1,\dots,x_m$ (a basis of $L$), with the relations
$$ x_ix_j-x_jx_i=[x_i,x_j]=[x_i,x_j]_L
$$
$$ x_i v_k = (1 \otimes x_i)(v_k \otimes 1) = ((x_i)_1\cdot v_k) (x_i)_2 = (x_i\cdot v_k) 1 +(1\cdot v_k) x_i = (x_i\cdot v_k) + v_k x_i
\\
\Leftrightarrow [x_i,v_k]=x_i\cdot v_k
$$
(In fact, these are defining relations.)

__Expl.__ $G$ a finite group with a module $V$ via $\cdot$.
$B=TV$, $H=kG$, $B\rtimes H$ is the algebra generated by $v_1,\dots,v_n$ (a basis of $V$), $g_1,\dots,g_m$ (a basis of $G$), with the relations
$$ g_ig_j = g_{i \circ j}
$$
$$ g_i v_k = (1 \otimes g_i)(v_k \otimes 1) = ((g_i)_1\cdot v_k) (g_i)_2 = (g_i \cdot v_k) g_i
$$
(In fact, these are defining relations.)

## The ideal $I$

$\kappa:V\o V\to H$,
$$
I = I_\kappa = ( [v_i,v_j]-\kappa(v_i,v_j) )
$$

So
$$
A = A_{H,V,\kappa} = (TV\rtimes H)/I_\kappa .
$$

Some observations:
- $\kappa$ has to be skew-symmetric; $\kappa: V\wedge V\to H$
- $\kappa$ is determined by the image $(v_i\o v_j)_{i<j}$

__Question 1.__ We want to be able to compute in the algebra $A_{H,V,\kappa}$.

__Expl.__ $\kappa=0$: $A_{H,V,0}=SV\rtimes H$.

We don't want "too many cancellations"; we want $A$ to be a PBW deformation of $A_{H,V,0}$.

$A_{H,V,\kappa}$ is a filtered algebra, where we assign elements from $V$ degree $1$ and elements from $H$ (or $L$) degree $0$.

So $F_i(A)=\overline{F_i(TV)\rtimes H} = \overline{(T^{\leq i}V)\rtimes H}=\bigoplus_{0\leq j\leq i} \overline{V^{\o j} \rtimes H}$.

$\newcommand\gr{\operatorname{gr}}$
__Def.__ $A=\bigcup_i F_i(A)$ a filtered algebra. Then $\gr A:=\bigoplus_i F_i(A)/F_{i-1}(A)$ is the _associated graded algebra_ (with a special product).

__Thm.__ $\gr U(L)\cong S(L)$.

__Def.__ $A$ is a _PBW deformation_ of $A_0=SV\rtimes H$ if $\gr A\cong A_0$ (as $\mathbb{N}$-graded algebras).

__Thm.__ $A$ is a PBW deformation of $A_0$ $\Leftrightarrow$ a certain Jacobi identity holds.

__Question 2.__ Given $(H,V)$ (or $(L,V)$), which $\kappa$ yield PBW deformations of $A_0$? Describe the (projective variety / moduli space) of all deformations.

## Embedding of orthogonal groups
$\newcommand\C{\mathbb{C}}$ $\newcommand\so{\mathfrak{so}}$ $\newcommand\sl{\mathfrak{sl}}$

$$
[\begin{pmatrix} A & 0\\ 0 & 0 \end{pmatrix}
, \begin{pmatrix} 0 & v\\ -v^t & 0 \end{pmatrix}]
= \begin{pmatrix} 0 & Av\\ 0 & 0 \end{pmatrix}
- \begin{pmatrix} 0 & 0\\ -v^t A & 0 \end{pmatrix}
= \begin{pmatrix} 0 & Av\\ -(Av)^t & 0 \end{pmatrix}
$$

This shows that $\C^n\to\so_{n+1}$ is $\so_n$-linear.

$\so_{n+1}\cong\so_n\oplus\C^n$, where $\so_n$ is Lie subalgebra, and $\C^n$ is the "standard" module of $\so_n$.

Note that $[v,w]\in\so_n$; this gives a second natural choice for $\k$:
$$
\k(v,w) := [v,w]_{\so_{n+1}} \in \so_n
$$ Concretely, $[e_i,e_j]=E_{ij}-E_{ji}$.

In this situation, $A_\k=(TV\rtimes U(\so_n))/I_\k\cong U(\so_{n+1})$.
(So by the classical PBW theorem, $A_\k$ is a PBW deformation of $A_0=SV\rtimes U(\so_n)$ in this case; as a filtered vector space, $A_\k\simeq S(\so_{n+1})\simeq S(V)\rtimes U(\so_n)$!)

Similarly: $\sl_{n+1}=\gl_n\oplus\C^n\oplus(\C^n)^*$.

More generally: "symmetric spaces".

## Basis of $(I \o V) \cap (V \o I)$
for Theorem 0.4 of [paper](https://arxiv.org/pdf/1308.6011.pdf)

$I = (v_i \o v_j - v_j \o v_i \mid 1 \leq i,j \leq n) \trianglelefteq V \o V$

A basis of $(I\o V)\cap (V\o I)$ is given by $v_i v_j v_k + v_j v_k v_i + v_k v_i v_j - v_k v_j v_i - v_j v_i v_k - v_i v_k v_j$ for $1 \leq i < j < k \leq n$.

$\newcommand\s\sigma$
This is isomorphic to $\Lambda^3 V$: $v_i\wedge v_j\wedge v_k\mapsto \sum_{\s\in S_3} (-1)^\s v_{\s(i)}\o v_{\s(j)}\o v_{\s(k)}$

## $\kappa$ is $H$-invariant
for Theorem 0.4 of [paper](https://arxiv.org/pdf/1308.6011.pdf)
__Def.__ $h \cdot \kappa(r) = \kappa(h \cdot r)$ for all $r \in I$, $h \in H$

In our case with $I = (v_i \o v_j - v_j \o v_i \mid 1 \leq i,j \leq n) \trianglelefteq V \o V$
this is equivalent to

$h\in L\subset U(L)$
$h \cdot \kappa(v_i,v_j)$
$= h \cdot \kappa(v_i\o v_j-v_j\o v_i)$
$\overset{!}{=} \kappa(h \cdot (v_i\o v_j-v_j\o v_i))$
$= \kappa(h \cdot (v_i \o v_j)) - \kappa(h \cdot (v_j \o v_i))$
$= \kappa((h \cdot v_i) \o v_j + v_i \o (h \cdot v_j)) - \kappa((h \cdot v_j) \o v_i + v_j \o (h \cdot v_i))$
$= \kappa((h \cdot v_i) \o v_j - v_j \o (h \cdot v_i)) + \kappa(v_i \o (h \cdot v_j) - (h \cdot v_j) \o v_i)$
$= \kappa(h \cdot v_i,v_j) + \kappa(v_i,h \cdot v_j)$