Isotypic faithful 2-representations of ${\mathcal {J}}$-simple fiat 2-categories

Abstract

We introduce the class of isotypic 2-representations for finitary 2-categories and the notion of inflation of 2-representations. Under some natural assumptions we show that isotypic 2-representations are equivalent to inflations of cell 2-representations.

Appendix: Weakly fiat potpourri

In this section, we collect appropriate reformulations of statements from [17–19] (originally for fiat 2-categories) in the more general setup of weakly fiat 2-categories. Let ${\mathscr {C}}$ be a weakly fiat 2-category as defined in Sect. 2.5 with a weak equivalence $*$.

We write ${\mathbf{P}}$ for the direct sum of all principal representations of ${\mathscr {C}}$. Isomorphism classes of indecomposable projective and simple objects in $\overline{{\mathbf{P}}}$ are indexed by elements in ${\mathcal {S}}({\mathscr {C}})$ and denoted by $\hat{P}_{{\mathrm {F}}}$ and $\hat{L}_{{\mathrm {F}}}$.

Proposition 26

Let ${\mathrm {F}},{\mathrm {G}},{\mathrm {H}}\in {\mathcal {S}}({\mathscr {C}})$.

• (i) The inequality ${\mathrm {F}}\,\hat{L}_{{\mathrm {G}}}\ne 0$ is equivalent to ${}^*{\mathrm {F}}\le _R {\mathrm {G}}$ and also to ${\mathrm {F}}\le _L {\mathrm {G}}^*$.

• (ii) The inequality $[{\mathrm {F}}\,\hat{L}_{{\mathrm {G}}}:\hat{L}_{{\mathrm {H}}}]\ne 0$ implies ${\mathrm {H}}\le _L {\mathrm {G}}$.

• (iii) If ${\mathrm {H}}\le _L {\mathrm {G}}$, then there is ${\mathrm {K}}\in {\mathcal {S}}({\mathscr {C}})$ such that $[{\mathrm {K}}\,\hat{L}_{{\mathrm {G}}}:\hat{L}_{{\mathrm {H}}}]\ne 0$.

• (iv) If $\hat{L}_{{\mathrm {F}}}$ occurs in the top or socle of ${\mathrm {H}}\,\hat{L}_{{\mathrm {G}}}$, then ${\mathrm {F}}$ is in the same left cell as ${\mathrm {G}}$.

• (v) If ${\mathrm {F}}\in {\mathscr {C}}({\texttt {i}},{\texttt {j}})$, then there is a unique (up to scalar) non-zero homomorphism $\hat{P}_{\mathbbm {1}_{{\texttt {i}}}}\rightarrow {\mathrm {F}}^*\,\hat{L}_{{\mathrm {F}}}$, in particular, ${\mathrm {F}}^*\,\hat{L}_{{\mathrm {F}}}\ne 0$.

Proof

Mutatis mutandis [17, Lemmata 12–15]. $\square$

Proposition 27

Let ${\mathcal {L}}$ be a left cell of ${\mathscr {C}}$ and ${\texttt {i}}={\texttt {i}}_{{\mathcal {L}}}$.

• (i) There is a unique submodule K of $\hat{P}_{\mathbbm {1}_{{\texttt {i}}}}$ such that

  • (a) every simple subquotient of $\hat{P}_{\mathbbm {1}_{{\texttt {i}}}}/K$ is annihilated by any ${\mathrm {F}}\in {\mathcal {L}}$;

  • (b) the module K has simple top $\hat{L}_{{\mathrm {G}}_{{\mathcal {L}}}}$ for some ${\mathrm {G}}_{{\mathcal {L}}}\in {\mathcal {S}}({\mathcal {C}})$ and ${\mathrm {F}}\, \hat{L}_{{\mathrm {G}}_{{\mathcal {L}}}}\ne 0$ for any ${\mathrm {F}}\in {\mathcal {L}}$.

• (ii) For any ${\mathrm {F}}\in {\mathcal {L}}$ the module ${\mathrm {F}}\, \hat{L}_{{\mathrm {G}}_{{\mathcal {L}}}}$ has simple top $\hat{L}_{{\mathrm {F}}}$.

• (iii) We have ${\mathrm {G}}_{{\mathcal {L}}}\in {\mathcal {L}}$.

• (iv) For any ${\mathrm {F}}\in {\mathcal {L}}$ we have ${}^*{\mathrm {F}}\le _R {\mathrm {G}}_{{\mathcal {L}}}$ and ${\mathrm {F}}\le _L {\mathrm {G}}_{{\mathcal {L}}}^*$.

• (v) We have ${\mathrm {G}}^*_{{\mathcal {L}}}\in {\mathcal {L}}$.

Proof

Mutatis mutandis [17, Proposition 17]. $\square$

The element ${\mathrm {G}}_{{\mathcal {L}}}\in {\mathcal {L}}$ is called the Duflo involution in ${\mathcal {L}}$.

Proposition 28

Let ${\mathcal {J}}$ be a strongly regular two-sided cell in ${\mathscr {C}}$ and ${\mathcal {L}}$ a left cell in ${\mathcal {J}}$. Then for ${\mathrm {G}}\in {\mathcal {L}}$ the following assertions are equivalent.

• (a) ${\mathrm {G}}={\mathrm {G}}_{{\mathcal {L}}}$.

• (b) ${\mathrm {G}}^*\in {\mathcal {L}}$.

• (c) $\{{\mathrm {G}}\}={\mathcal {L}}\cap {}^*{\mathcal {L}}$.

• (d) ${\mathrm {G}}={}^*{\mathrm {H}}$ where $\{{\mathrm {H}}\}={\mathcal {L}}\cap {\mathcal {L}}^*$.

• (e) ${\mathrm {G}}\,\hat{L}_{{\mathrm {G}}}\ne 0$.

Proof

We have

• (a) $\Rightarrow$ (b) by Proposition 27(v);

• (b) $\Rightarrow$ (d) as $|{\mathcal {L}}\cap {\mathcal {L}}^*|=1$;

• (d) $\Rightarrow$ (c) applying ${\mathrm {F}}\mapsto {}^*{\mathrm {F}}$;

• (c) $\Rightarrow$ (b) applying ${\mathrm {F}}\mapsto {\mathrm {F}}^*$;

• (b) Rightarrow (a) again as $|{\mathcal {L}}\cap {\mathcal {L}}^*|=1$.

Finally, (b)$\Leftrightarrow$(e) follows from strong regularity of ${\mathcal {J}}$ and Proposition 26(i). $\square$

Let ${\mathcal {J}}$ be a strongly regular two-sided cell in ${\mathscr {C}}$ and ${\mathcal {L}}$ a left cell in ${\mathcal {J}}$. To simplify computational expressions, we assume that ${\mathcal {J}}$ is a maximal two-sided cell. In the cell 2-representation ${\mathbf{C}}_{{\mathcal {L}}}$ as defined in Sect. 3.1, we have indecomposable projective objects $P_{{\mathrm {F}}}$, indecomposable injective objects $I_{{\mathrm {F}}}$ and simple objects $L_{{\mathrm {F}}}$, where ${\mathrm {F}}\in {\mathcal {L}}$. Set ${\mathrm {G}}={\mathrm {G}}_{{\mathcal {L}}}$.

Lemma 29

For any ${\mathrm {F}}\in {\mathcal {L}}$ we have ${\mathrm {F}}{\mathrm {G}}\cong {\mathbf{m}}_{{\mathrm {G}}}{\mathrm {F}}$.

Proof

Strong regularity of ${\mathcal {J}}$ implies that ${\mathrm {G}}{\mathrm {G}}=m{\mathrm {G}}$ for some non-negative integer m. Applying this to $L_{{\mathrm {G}}}$, using exactness of ${\mathrm {G}}$ and a character argument, we see that

$$\begin{aligned} m=[P_{{\mathrm {G}}}:L_{{\mathrm {G}}}]=\dim {\mathrm {End}}(P_{{\mathrm {G}}}). \end{aligned}$$

At the same time, $\dim {\mathrm {End}}(P_{{\mathrm {G}}})= \dim {\mathrm {Hom}}({\mathrm {G}}\,L_{{\mathrm {G}}},{\mathrm {G}}\,L_{{\mathrm {G}}})$ equals ${\mathbf{m}}_{{\mathrm {G}}}$ by adjunction. Thus $m={\mathbf{m}}_{{\mathrm {G}}}$.

Strong regularity of ${\mathcal {J}}$ again yields ${\mathrm {F}}{\mathrm {G}}\cong k{\mathrm {F}}$ for some non-negative integer k. Moreover, k is, in fact, positive, since ${\mathrm {F}}{\mathrm {G}}\, L_{{\mathrm {G}}}\ne 0$. Now, computing ${\mathrm {F}}{\mathrm {G}}{\mathrm {G}}$ in two different ways using associativity, and then dividing by k, we obtain $k={\mathbf{m}}_{{\mathrm {G}}}$. $\square$

Proposition 30

Let ${\mathrm {F}}\in {\mathcal {L}}$ and ${\mathrm {H}}\in {\mathcal {J}}$.

• (i) The projective object $P_{{\mathrm {F}}}$ is injective.

• (ii) The object ${\mathrm {H}}\, L_{{\mathrm {F}}}$, when non-zero, has a non-zero projective-injective summand.

• (iii) We have ${\mathrm {F}}^*L_{{\mathrm {F}}}=I_{{\mathrm {G}}}$.

• (iv) The object ${\mathrm {H}}\, L_{{\mathrm {F}}}$ is both projective and injective.

• (v) The object ${\mathrm {H}}\, L_{{\mathrm {F}}}$ is indecomposable or zero.

Proof

By adjunction, $L_{{\mathrm {G}}}$ injects into ${\mathrm {F}}^*L_{{\mathrm {F}}}$. Consider an injective object I and let $L_{{\mathrm {K}}}$ be a simple quotient of I, where ${\mathrm {K}}\in {\mathcal {L}}$. Then $L_{{\mathrm {G}}}$ is a subquotient of the injective object ${\mathrm {K}}^*\, I$. As ${\mathcal {J}}$ is strongly regular, it follows from Proposition 26(i) that ${\mathrm {G}}$ annihilates all simples except for $L_{{\mathrm {G}}}$. Hence $L_{{\mathrm {G}}}$ appears in the top of the injective object ${\mathrm {G}}{\mathrm {K}}^*\, I$. Applying ${\mathrm {F}}$, we obtain that $P_{{\mathrm {F}}}$ is a quotient (and hence a direct summand) of the injective object ${\mathrm {F}}{\mathrm {G}}{\mathrm {K}}^*\, I$ and is therefore injective. This proves claim (i).

Assume that ${\mathrm {H}}\, L_{{\mathrm {F}}}\ne 0$. The argument in the previous paragraph shows that ${\mathrm {H}}'\, L_{{\mathrm {F}}}$ has a non-zero projective-injective summand for some ${\mathrm {H}}'$ in the same left cell as ${\mathrm {H}}$. Claim (ii) is now deduced by multiplying the latter on the left with elements in ${\mathcal {J}}$ and using strong regularity of ${\mathcal {J}}$.

Using adjunction and strong regularity of ${\mathcal {J}}$, we see that ${\mathrm {F}}^*L_{{\mathrm {F}}}$ has simple socle $L_{{\mathrm {G}}}$. Therefore claim (iii) follows from claim (ii). Claim (iv) is implied by claim (iii) as any inequality ${\mathrm {H}}\, L_{{\mathrm {F}}}\ne 0$ means, by strong regularity of ${\mathcal {J}}$, that ${\mathrm {H}}$ and ${\mathrm {F}}^*$ are in the same left cell and hence ${\mathrm {H}}\, L_{{\mathrm {F}}}$ is a direct summand of the projective-injective object ${\mathrm {K}}{\mathrm {F}}^*L_{{\mathrm {F}}}$ for some ${\mathrm {K}}$.

It remains to prove claim (v). By strong regularity of ${\mathcal {J}}$, there is a unique element $\tilde{{\mathrm {G}}}$ in the right cell of ${\mathrm {F}}$ such that $\tilde{{\mathrm {G}}}\, L_{{\mathrm {F}}}\ne 0$. This implies that $\tilde{{\mathrm {G}}}\, \hat{L}_{{\mathrm {F}}}\ne 0$ and thus $\tilde{{\mathrm {G}}}\, \hat{L}_{\tilde{{\mathrm {G}}}}\ne 0$ by Proposition 26(i), as $\tilde{{\mathrm {G}}}$ and ${\mathrm {F}}$ are in the same right cell. In particular, by Proposition 28, $\tilde{{\mathrm {G}}}$ is the Duflo involution in its left cell.

Using claim (iv) and strong regularity of ${\mathcal {J}}$, we deduce $\tilde{{\mathrm {G}}}\, L_{{\mathrm {F}}}\cong k P_{{\mathrm {F}}}$ for some non-negative integer k. We compute ${\mathrm {F}}^*\tilde{{\mathrm {G}}}\, L_{{\mathrm {F}}}$ in two different ways. On the one hand,

$$\begin{aligned} {\mathrm {F}}^*\tilde{{\mathrm {G}}}\, L_{{\mathrm {F}}}\cong k{\mathrm {F}}^*\,P_{{\mathrm {F}}}\cong k{\mathrm {F}}^*{\mathrm {F}}\,L_{{\mathrm {F}}}\cong k{\mathbf{m}}_{{\mathrm {F}}}P_{{\mathrm {G}}^*}, \end{aligned}$$

where the last isomorphism follows from the isomorphism ${\mathrm {F}}^*{\mathrm {F}}\cong ({}^*{\mathrm {F}}{\mathrm {F}})^*$ and the fact that $\{{\mathrm {G}}^*\}={\mathcal {L}}\cap {\mathcal {L}}^*$. On the other hand, ${\mathrm {F}}^*$ is in the same left cell as $\tilde{{\mathrm {G}}}^*$ and hence in the same left cell as $\tilde{{\mathrm {G}}}$ as the latter is the Duflo involution. Therefore, Lemma 29 and claim (iii) give

$$\begin{aligned} {\mathrm {F}}^*\tilde{{\mathrm {G}}}\, L_{{\mathrm {F}}}\cong {\mathbf{m}}_{\tilde{{\mathrm {G}}}}P_{{\mathrm {G}}^*}. \end{aligned}$$

From Proposition 1, we obtain ${\mathbf{m}}_{\tilde{{\mathrm {G}}}}={\mathbf{m}}_{{\mathrm {F}}}$ which yields $k=1$ and proves claim (v) in the case ${\mathrm {H}}=\tilde{{\mathrm {G}}}$.

Now, in the general case, assume ${\mathrm {H}}\, L_{{\mathrm {F}}}\cong k P_{{\mathrm {K}}}$ for some ${\mathrm {K}}\in {\mathcal {L}}$ and a positive integer k (note that ${\mathrm {K}}$ and ${\mathrm {H}}$ are then in the same right cell). Then ${}^*{\mathrm {H}}{\mathrm {H}}\, L_{{\mathrm {F}}}\ne 0$ by adjunction and hence ${}^*{\mathrm {H}}{\mathrm {H}}\cong {\mathbf{m}}_{{\mathrm {H}}}\tilde{{\mathrm {G}}}$. Let us now compute the dimension of ${\mathrm {End}}({\mathrm {H}}\, L_{{\mathrm {F}}})$. On the one hand, by adjunction and the fact that $\tilde{{\mathrm {G}}}\, L_{{\mathrm {F}}}\cong P_{{\mathrm {F}}}$, proved in the previous paragraph, this dimension equals ${\mathbf{m}}_{{\mathrm {H}}}$. On the other hand, it equals $k^2\dim {\mathrm {End}}(P_{{\mathrm {K}}})$ which, in turn, by adjunction, equals $k^2{\mathbf{m}}_{{\mathrm {K}}}$. Proposition 1 implies ${\mathbf{m}}_{{\mathrm {K}}}={\mathbf{m}}_{{\mathrm {H}}}$ and hence $k=1$, completing the proof. $\square$

Proposition 30 shows that indecomposable 1-morphisms in ${\mathcal {J}}$ act, under ${\mathbf{C}}_{{\mathcal {L}}}$, as indecomposable projective functors for some self-injective algebra. In analogy to [17, Theorem 43], this implies the following.

Theorem 31

Let ${\mathscr {C}}$ be a weakly fiat 2-category and ${\mathcal {J}}$ a strongly regular two-sided cell in ${\mathscr {C}}$.

• (i) For any left cell ${\mathcal {L}}$ in ${\mathcal {J}}$, the cell 2-representation ${\mathbf{C}}_{{\mathcal {L}}}$ is strongly simple in the sense of [17, Section 6.2].

• (ii) If ${\mathcal {L}}$ and ${\mathcal {L}}'$ are two right cells in ${\mathcal {J}}$, then the cell 2-representations ${\mathbf{C}}_{{\mathcal {L}}}$ and ${\mathbf{C}}_{{\mathcal {L}}'}$ are equivalent.

Using this, and following the arguments in [19], we can generalize [19, Theorem 13] to all weakly fiat 2-categories (in the notation of Sect. 5).

Theorem 32

Let ${\mathscr {C}}={\mathscr {C}}_{{\mathcal {J}}}$ be a skeletal weakly fiat ${\mathcal {J}}$-simple 2-category for a strongly regular ${\mathcal {J}}$. Then ${\mathscr {C}}$ is biequivalent to ${\mathscr {C}}_{A,X}$ for appropriate self-injective A and $X\subset Z(A)$.

Similarly, we obtain the following generalization of [21, Theorem 8].

Theorem 33

Let ${\mathscr {C}}$ be a weakly fiat 2-category such that all two-sided cells in ${\mathscr {C}}$ are strongly regular. Then any simple transitive 2-representation of ${\mathscr {C}}$ is equivalent to a cell 2-representation.