nLab twisted intertwiner

Idea

In representation theory, the notion of twisted intertwiners is a generalization of that of intertwiners.

References using the terminology “twisted intertwiner” include Fuchs & Schweigert 2000a (7.2), 2000b (2.2), Felder, Fröhlich, Fuchs & Schweigert 2000 (4.7), but the notion itself is probably older and may (or may not) be known under other names.

Definition

Definition

For $G$ a group consider its linear representations $(\rho, V)$, on vector spaces $V$, with $g \in G \;\vdash\; V \xrightarrow{\rho(g)} V$.

For $(\rho_1,V_1), (\rho_2, V_2) \;\in\; Rep(G)$ a pair of linear representation respectively (all this generalizes straightforwardly to algebras and their modules), a twisted intertwiner

$$(\rho_1, V_1) \xrightarrow{(\eta, \alpha)} (\rho_2, V_2)$$

is

1. a linear map $\eta \colon V_1 \longrightarrow V_2$,

2. an automorphism $\alpha \in Aut(G)$ of the group

such that

(1)$$\underset{g \in G}{\forall} \;\;\; \vdash \;\;\; \eta \circ \rho_1(g) \;=\; \rho_2\big(\alpha(g)\big) \circ \eta \,.$$

Moreover, given a pair of consecutive twisted intertwiners $(\rho_1, V_1) \xrightarrow{(\alpha,\eta)} (\rho_2, V_2) \xrightarrow{(\alpha',\eta')} (\rho_3, V_3)$, their composition is given by the direct product form

(2)$$(\alpha', \eta') \circ (\alpha, \eta) \;=\; \big( \alpha' \circ \alpha ,\, \eta' \circ \eta \big) \,.$$

This reduces to the ordinary notion of intertwiners in the case that the automorphism is the identity map, $\alpha = id_G$.

Properties

Deformations

There naturally are higher order morphisms between twisted intertwiners (these seem not to have been considered in the literature and do not carry an established name — we here refer to them as “deformations”, just to have a word):

Definition

Given a parallel pair of twisted intertwiners $(\alpha, \eta), (\alpha', \eta')\,\colon\, (\rho_1, V_1) \rightrightarrows (\rho_2, V_2)$, then a deformation between these

$$a \;\colon\; (\alpha, \eta) \Rightarrow (\alpha', \eta')$$

is $a \in G$ such that

(3)$$\alpha' \;=\; Ad_a \circ \alpha \,,$$

(where “$Ad$” denotes the adjoint action of a group on itself by inner automorphisms, $Ad_a(g) \coloneqq a g a^{-1}$)

and

(4)$$\eta' \;=\; \rho_2(a) \circ \eta \,.$$

Remark

Conversely, if $(\eta, \alpha) \colon (\rho_1, V_1) \xrightarrow{\;} (\rho_2, V_2)$ is a twisted intertwiner, and $a \in G$, then also

$$\big( \rho_2(a) \circ \eta ,\, Ad_A \circ\alpha\big) \;\colon\; (\rho_1, V_1) \xrightarrow{\;} (\rho_2, V_2)$$

is a twisted intertwiner:

$$\begin{array}{rcl} \eta' \circ \rho_1(g) &\equiv& \big(\rho_2(a) \circ \eta\big) \circ \rho_1(g) \\ &=& \rho_2(a) \circ \eta \circ \rho_1(g) \\ &=& \rho_2(a) \circ \rho_2\big(\alpha(g)\big) \circ \eta \\ &=& \rho_2(a) \circ \rho_2\big(\alpha(g)\big) \circ \rho_2(a)^{-1} \circ \rho_2(a) \circ \eta \\ &=& \rho_2\big(a \cdot \alpha(g) \cdot a^{-1} \big) \circ \eta' \\ &\equiv& \rho_2\big( \alpha'(g) \big) \circ \eta' \,. \end{array}$$

In other words, the group $G$ acts on the hom-sets of twisted intertwiners by deformations.

Lemma

For linear representations $(\rho_1, V_1)$, $(\rho_2, V_2)$, deformation (Def. ) of twisted intertwiners $(\rho_1, V_1) \xrightarrow{\;} (\rho_2, V_2)$ is an equivalence relation, and composition of twisted intertwiners (2) passes to its equivalence classes.

Proof

That we have an equvalence relation is clear. To see that it is respected by composition, consider

$$(\eta, \alpha), (\eta', \alpha') \,\colon\, (\rho_1, V_1) \rightrightarrows (\rho_2, V_2)$$

with

$$a \,\colon\, (\eta, \alpha) \Rightarrow (\eta', \alpha')$$

and

1. another $(\widetilde{\eta}, \widetilde{\alpha}) \,\colon\, (\rho_2, V_2) \xrightarrow{\;} (\rho_3, V_3)$, then

   $$\widetilde{\alpha}(a) \;\colon\; (\widetilde{\eta}, \widetilde{\alpha}) \circ (\eta, \alpha) \Rightarrow (\widetilde{\eta}, \widetilde{\alpha}) \circ (\eta', \alpha')$$

   because:

   $$\begin{array}{rcl} \widetilde{\eta} \circ \eta' &=& \widetilde{\eta} \circ \rho_2(a) \circ \eta \\ &=& \rho_3\big( \widetilde{\alpha}(a) \big) \circ \widetilde{ \eta } \circ \eta \end{array}$$

   and

   $$\begin{array}{rcl} \widetilde{\alpha} \circ \alpha' &=& \widetilde{\alpha} \circ Ad_a \circ \alpha \\ &=& Ad_{\widetilde{\alpha}(a)} \circ \widetilde \alpha \circ \alpha \,, \end{array}$$

2. another $(\widetilde{\eta}, \widetilde{\alpha}) \,\colon\, (\rho_0, V_0) \xrightarrow{\;} (\rho_1, V_1)$, then we have

   $$a \;\colon\; (\eta, \alpha) \circ (\widetilde{\eta}, \widetilde{\alpha}) \Rightarrow (\eta', \alpha') \circ (\widetilde{\eta}, \widetilde{\alpha})$$

   immediately:

   $$\begin{array}{rcl} \eta' \circ \widetilde{\eta} &=& \rho_2(a) \circ \eta \circ \widetilde{\eta} \end{array}$$

   and

   $$\begin{array}{rcl} \alpha' \circ \widetilde{\alpha} &=& Ad_a \circ \alpha \circ \widetilde{\alpha} \,. \end{array}$$

Remark

This means that

• there is a 2-category of representations, twisted intertwiners and deformations

• with a homotopy category of representations and deformation classes of twisted intertwiners.

This becomes manifest by recasting the original component definition (1) of twisted intertwiners as characterizing certain slice morphisms in Cat, to be discussed below.

Category theoretic description

We discuss how the properties of twisted intertwiners above becomes manifest when understanding them as a natural construction in category theory, via slices of the 2-category Cat of categories, functors and natural transformations:

Writing $\mathbf{B}G$ for the delooping groupoid of $G$ and $Vec$ for the category of vector spaces (over a given ground field), the ordinary category Rep of $G$-representations and ordinary intertwiners $\eta \colon \rho_1 \Rightarrow \rho_2$ between these is equivalently the functor category

$$Rep(G) \;\;\simeq\;\; Func\big( \mathbf{B}G ,\, Vec \big) \,.$$

In contrast, the category of $G$-representations with twisted intertwiners between them is the iso-comma (2,1)-category between $\mathbf{B}G \,\colon\, \mathbf{B}Aut(\mathbf{B}G) \longrightarrow Grpd$ (on the left the delooping of the automorphism 2-group of $G$) and $Vec \,\colon\, \ast \to Grpd$, whose 1-morphisms are diagrams in Grpd of this form:

hence natural transformations

whose commuting naturality square is equivalent to the component equation (1),

and whose composition by whiskering

is the natural transformation

which reflects the composition law (2),

while the 2-morphisms are the evident paper-cup pasting diagrams

where

gives the relation (3),

from $(\alpha, \eta)$ to the whiskered (now on the other side) composite transformation

exhibiting the relation (4).

References

The component-definition (1) is considered (broadly in a context of 2d CFT) in:

• Jürgen Fuchs, Christoph Schweigert: Symmetry breaking boundaries II. More structures; examples, Nucl. Phys. B 568 (2000) 543-593 [arXiv:hep-th/9908025, doi:10.1016/S0550-3213(99)00669-0]

• Giovanni Felder, Jürg Fröhlich, Jürgen Fuchs, Christoph Schweigert: The geometry of WZW branes, J. Geom. Phys. 34 (2000) 162-190 [doi:10.1016/S0393-0440(99)00061-3, arXiv:hep-th/9909030]

• Jürgen Fuchs, Christoph Schweigert: Lie algebra automorphisms in conformal field theory, in Conference on Infinite Dimensional Lie Theory and Conformal Field Theory (May 2000) [arXiv:math/0011160]