数理解析研究所講究録
第 1795 巻 2012 年 179-188

179

A SUPER ANALOG OF THE KHOVANOV-LAUDA-ROUQUIER ALGEBRASSHUNSUKE TSUCHIOKA

1. INTRODUCTION
In the conference, I reported ajoint work [KKT] with Masaki Kashiwara (RIMS) and Seok-Jin
Kang (SNU) that proposes a super analog of the Khovanov-Lauda-Rouquier algebras which wecall quiver Hecke superalgebras. Our main results [KKT, Theorem 4.4,Theorem 5.4] establish
a “Morita superequivalence” (see [KKT, \S 2.4]) between the cyclotomic quotient of the quiver
Hecke superalgebras and the cyclotomic quotient of the affine Hecke-Clifford superalgebrasl andits degeneration.
If you are interested in our work, I believe the best way to grasp the synopsis is reading the
introduction of [KKT] since our motivations and results of the work is best summarized in it.
Acknowledgements The author would like to thank professor Reiho Sakamoto for giving mea chance to talk in the conference ”Topics in Combinatorial Representation Theory” in October
2011 at RIMS Kyoto University.

2. KLR ALGEBRAS AND THE SYMMETRIC GROUPS
Recently, Khovanov-Lauda and Rouquier independently introduced a remarkable family ofalgebras (the KLR algebras, the quiver Hecke algebras) that categorifies the negative half of thequantized enveloping algebras associated with symmetrizable Kac-Moody Lie algebras [KLl,KL2, Rou] (see Definition 2.1 and Theorem 2.4). An application of the KLR algebras is thegradation of the symmetric group algebras BKl,(see Theorem 2.5) which quantizes Ariki
categorification of the Kostant Z-form of the basic -module( Proj $(F_{p}6_{n})$ ).
The story is also valid for its q-analog, the Iwahori-Heck algebra of type A.
Definition 2.1 ([KLl, KL2, Rou]). Let be a field and let I be a finite set. Take a matrix$Q=(Q_{ij}(u, v))\in Mat_{I}(k[u, v])$ such that $Q_{ii}(u, v)=0,$ $Q_{ij}(u, v)=Q_{ji}(v, u)$for all $i,j\in I$ .
(a) The Khovanov-Lauda-Rouquier algebra ($KLR$ algebra, for short) $R_{m}(k;Q)$ for $n\geq 0$ is ak-algebra generated by $\{x_{p}, \tau_{a}, e_{\nu}|1\leq p\leq n, 1\leq a<n, \nu\in I^{n}\}$ with the following defining$1\leq p,$ $q\leq n,$ $1\leq b<a\leq n-1$ .
relations for all
$[$

$s$

$Ro_{\text{へ}}u]$$V(\Lambda_{0})^{Z}\cong\oplus_{n>0}K_{0}$$\epsilon 1_{p}$

$k$

...
.

$\mu.\nu\in I^{n},$$e_{\mu}e_{\nu}=\delta_{\mu\nu}e_{\mu},$$1= \sum_{\mu\in I^{n}}e_{\mu},$$\tau_{a}^{2}e_{\nu}=Q_{\nu_{a},\nu_{a+1}}(x_{a}, x_{a+1})e_{\nu},$$x_{p}x_{q}=x_{q}x_{p},$ $x_{p}e_{\mu}=e_{\mu}x_{p},$$\tau_{a}e_{\mu}=e_{s_{a}(\mu)}\tau_{a},$$(\tau_{a}x_{a+1}-x_{a}\tau_{a})e_{\nu}=(x_{a+1}\tau_{a}-\tau_{a}x_{a})e_{\mu}=\delta_{\nu_{a},\nu_{a+1}}e_{\nu}$,..

$\tau_{a}\tau_{b}=\tau_{b}\tau_{a}if|a-b|>1$$\tau_{a}x_{p}=x_{p}\tau_{a}$if,$p\neq a,$ $a+1$,$(\tau_{b+1}\tau_{b}\tau_{b+1}-\tau_{b}\tau_{b+1}\tau_{b})e_{\nu}=\delta_{\nu_{b},\nu_{b+2}}((x_{b+2}-x_{b})^{-1}(Q_{\nu_{b},\nu_{b+1}}(x_{b+2}, x_{b+1})-Q_{\nu_{b},\nu_{b+1}}(x_{b}, x_{b+1})))e_{\nu}$(b) For $\beta=\sum_{i\in I}\beta_{i}\cdot i\in N[I]$ with $n=$ ht, we define $R_{\beta}(k;Q)=R_{n}(k;Q)e_{\beta}$$( \beta)=\{(i_{j})_{j=1}^{n}\in I^{n}|\sum_{j=1}^{n}i_{j}=\beta\}$ .
wheoe
Seq
and
Seq
$(\beta)$$e_{\beta}= \sum_{\nu\in}$$:= \sum_{i\in I}\beta_{i}$$(\beta)^{e_{\nu}}$

Date: March 3, 2012.
2000 Mathematics Subject Classification. Primary $81R50$ , Secondary $20C08$ .

Key words and phrases. categorification, super representation theory, spin representations of symmetric groups,
Sergeev superalgebras, Hecke-Clifford superalgebras, symmetric groups, Iwahori-Hecke algebras, graded representation theory, quantum groups, Khovanov-Lauda-Rouquier algebras.
The research was supported by Research Fellowships for Young Scientists 23.8363, Japan Society for the
Promotion of Science.
lThey can be regarded as a superanalog of the Ariki-Koike algebras.

.

180
SHUNSUKE TSUCHIOKA(c) Forand$\lambda=\sum_{i\in l}\lambda_{i}\cdot i\in N[I]$$\beta\in N[I]$with $n=$ ht$(\beta)$, we define$R_{n}^{\lambda}( k;Q)=R_{n}(k;Q)/R_{n}(k;Q)(\sum_{\nu\in I^{n}}x_{1}^{\lambda_{h_{\nu_{1}}}}e_{\nu})R_{n}(k;Q)$,.

$R_{\beta}^{\lambda}( k;Q)=R_{\beta}(k;Q)/R_{\beta}(k;Q)(\sum_{\nu\in Seq(\beta)}x_{1}^{\lambda_{h_{\nu_{1}}}}e_{\nu})R_{\beta}(k;Q)$

As a consequence of PBW theorem for KLR algebras, we see that $\{e_{\beta}| ht(\beta)=n\}$ exhausts
all the primitive central idempotents of $R_{n}(k;Q)$ . Thus, $R_{n}(k;Q)=\oplus\beta\in N[I|R_{\beta}(k;Q)$ is a$h(\beta\ovalbox{\tt\small REJECT} n$decomposition into indecomposable factors. It is not difficult to see that bothare finite dimensional k-algebras.

$R_{n}^{\lambda}(k;Q)$

and$R_{\beta}^{\lambda}(k;Q)$

Definition 2.2 ([KLl, KL2, Rou]). Let $A=(a_{ij})_{i,g\in I}$ be a symmetrizable generalized Cartan, i. e., a uniquesuch thatmatrix with the symmetrizationfor
all $j\in I$ and $gcd(d_{i})_{i\in I}=1$ . Take $Q^{A}=(Q_{ij}^{A}(u, v))\in Mat_{I}(k[u, v])$ subject to$d=(d_{i})_{i\in I}$$d_{i}a_{ij}=d_{j}a_{ji}$$d\in Z_{\geq 1}^{I}$$i,$$Q_{ii}^{A}(u, v)=0$

for all $i,j\in I$ where,$Q_{ij}^{A}(u, v)=Q_{ji}^{A}(v, u)$,$t_{i,j,-a_{1j},0}=t_{j,i,0,-a}.j\neq 0$$Q_{ij}^{A}(u, v)= \sum_{p,q\geq 0 ,pd.+qd_{j}=-d_{i}a:j}t_{ijpq}u^{p}v^{q}$.

For $n\geq 0$ and $\beta\in N[I]$ with $ht(\beta)=n$ , all of $R_{n}(k;Q^{A}),$$1\leq p\leq n,$ $1\leq a<n$ .
are Z-graded via the assignment where$R_{\beta}(k;Q^{A}),$$\lambda,$$R_{n}^{\lambda}(k;Q^{A}),$$R_{\beta}^{\lambda}(k;Q^{A})$$\nu\in I^{n},$$\deg(e_{\nu})=$

O,

deg

$(x_{p}e_{\nu})=2d_{\nu_{p}}$,$\deg(\tau_{a}e_{\nu})=-d_{\nu_{a}}a_{\nu_{a},\nu_{a+l}}$.

Definition 2.3. Let a graded algebra. We denote by $Proj_{gr}(R)$ the category of finitely generated left graded projective R-modules and degree $presen$)$ing$ R-homomorphisms.
$R$

The grading shift autoequivalence
structure on $K_{0}(Proj_{gr}(R))$ via$\{-1\rangle$$v=[\langle-1\rangle]$: $Proj_{gr}(R)arrow^{\sim}Proj_{gr}(R)$ affords a $Z[v, v^{-1}]$ -module.

Theorem 2.4 ([KLl, KL2, Rou]). Let be a symmetrizable generalized Cartan matrix and let
Then, the following categorification results hold (though we don’t explain how to-module structure in $(b))$ .
define an algebra stru cture in $(a)$ nor how to define a$A$$d=Z[v, v^{-1}]$ .

$U_{v}^{d}(A)$(a) as an d-algebra, we $have\oplus_{n\geq 0}K_{0}(Proj_{gr}(R_{n}(k;Q^{A})))\cong U_{v}^{-,d}(A)$ .
(b) as a $U_{v}^{d}(A)$ -module, we $have\oplus_{n\geq 0}K_{0}(Proj_{gr}(R_{n}^{\lambda}(k;Q^{A})))\cong V(\lambda)^{d}$ .

resp. $U_{v}^{-,d}(A),V(\lambda)^{d})$ is the Lusztig’s d-lattice of $U_{v}(A)$ resp.
Here
.
with
and we identify$U_{v}^{d}(A)$

$($

$($

$\lambda\in \mathcal{P}^{+}$

Recall that$\sum_{i\in I}\lambda(h_{i})\cdot i\in N[I]$$A_{\ell-1}^{(1)}=(2\delta_{ij}-\delta_{i+1,j}-\delta_{i-1,j})_{i,j\in Z/\ell Z}$

Theorem 2.5 ([BKl, Rou]). Let
have

$k$

for

$\ell\geq 2$

and$\hat{\epsilon 1}_{\ell}=\mathfrak{g}(A_{\ell-1}^{(1)})$.

be a field of characteristic $p>0$ . Then, as a k-algebra we

where(though we don’t explain how to choose signs).
$k6_{n}\cong R_{n^{0}}^{\Lambda}(k;Q^{A_{p-1}^{(1)}})$$U_{v}^{-}(A),V(\lambda))$$Q_{ij}^{A_{p-1}^{(1)}}(u, v)=\pm(u-v)^{-2\delta_{lj}+\delta.+1,j}+\delta_{l-1,j}$

for $i\neq j\in Z/pZ$

You can find related topics to Theorem 2.5 in a well-written survey paper [Kll] which can beseen as an update of [K12].

3. SUPER REPRESENTATIONS
We briefly recall our conventions and notations for superalgebras and supermodules following [BK2, \S 2-b] (see also the references therein). Although they are different from [KKT, \S 2],
we review [BK2, \S 2-b] in order to cite [BK2, Tsu]. In this section, we always assume that in ourfield we have $2\neq 0$ .
$k$

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A SUPER ANALOG OF THE KHOVANOV-LAUDA-ROUQUIER ALGEBRAS

3.1. Superspaces. By a vector superspace, we mean a $Z/2Z$-graded vector spaceover and denote the parity of a homogeneous vector $v\in V$ by. Given two vector
superspaces
and $W$ , an k-linear map: $Varrow W$ is called homogeneous if there exists$p\in Z/2Z$ such thatfor $i\in Z/2Z$ . In this case we call the parity of and denote$V=V_{\overline{0}}\oplus V_{\overline{1}}$$k$$\overline{v}\in Z/2Z$

$V$

$f$

$f(V_{i})\subseteq W_{p+i}$it by$\overline{f}$

$f$

$p$

.

3.2. Superalgebras. A superalgebra is a vector superspace which is an unital associative
k-algebra such thatfor $j\in Z/2Z$ . By an A-supermodule, we mean a vector
superspace $M$ which is a left A-module such thatfor $i,j\in Z/2Z$ .
$A$$A_{i}A_{j}\subseteq A_{i+j}$$i,$$A_{i}M_{j}\subseteq M_{i+j}$3.3. Super categories. In the rest of the paper, we only deal with finite-dimensional Asupermodules. Given two A-supermodules: $Varrow W$ is
and $W$ , an A-homomorphism
an k-linear map such that $f(av)=(-1)^{\overline{\Gamma}a}af(v)$ for $a\in A$ and $v\in V$ . We denote the set of Ahomomorphisms from to $W$ by $Hom_{A}(V, W)$ . By this, we can form a superadditive category
A-smod whose hom-set is a vector superspace in a way that is compatible with composition.
However, we adapt a slightly different definition of isomorphisms from the categorical one.
$V$

$f$

$V$

3.4. Parity change functors. Two A-supermodules
and $W$ are called evenly isomorphic(and denoted by $V\simeq W$ ) if there exists an even A-homomorphism: $Varrow W$ which is anspace
k-vector
isomorphism. They are called isomorphic (and denoted by $V\cong W$ ) if $V\simeq W$$V\simeq\Pi
W$
or. Here for an A-supermodule $M,$ $\Pi M$ is an A-supermodule defined by the samebut the opposite grading underlying vector superspacefor $i\in Z/2Z$ and a new
action given as follows from the old one $a\cdot newm=(-1)^{\overline{a}}a\cdot 0|dm$.
$V$

$f$

$(\Pi M)_{i}=M_{i+\overline{1}}$3.5. Types of simple supermodules. We denote the isomorphism class of an A-supermodule$M$ by $[M]$ and denote the set of isomorphism classes of irreducible A-supermodules by lrr(A-smod).
Let us assume that is irreducible. We say that is type if $V\simeq\Pi V$ otherwise type M.
$V$

$V$

$Q$

3.6. Super tensor products. Given two superalgebras and $B,$
with multiplication
defined by
for $a_{i}\in A,$ $b_{j}\in B$ is again a superalgebra2.
Let be an A-supermodule and let $W$ be a B-supermodule. Their tensor product $V\otimes W$ is an-supermodule by the action given by
for $a\in A,$$B,$ $v\in V,$ $w\in W$ .
$W$
Let us assume thatandare both irreducible. If and $W$ are bothof type , then there exists a unique (up to odd isomorphism) irreducible-supermodule$X$ of typesuch that $V\otimes W\simeq X\oplus\Pi X$ as-supermodules. We denote $X$ by $VOW$ .
Otherwise $V\otimes W$ is irreducible but we also write it as $VOW$ . Note that $VOW$ is defined only
up to isomorphism in general and $VOW$ is of typeif and only if and $W$ are of the same$A$

$A\otimes B$$(a_{1}\otimes b_{1})(a_{2}\otimes b_{2})=(-1)^{\overline{b_{1}}\overline{a_{2}}}(a_{1}a_{2})\otimes(b_{1}b_{2})$$V$$A\otimes B$$(a\otimes b)(v\otimes w)=(-1)^{\overline{\triangleright}\overline{v}}(av)\otimes(bw)$$b\in$

$V$

$V$

$Q$

$A\otimes B$$M$$A\otimes B$

$V$

$M$

type.

3.7. Grothendieck groups. For a superalgebra$A$, we define the Grothendieck group$K_{0}$(A-smod)

to be the quotient of the -module freely generated by all finite-dimensional A-supermodules
by the Z-submodule generated by$V_{1}-V_{2}+V_{3}$ for every short exact sequence $0arrow V_{1}arrow V_{2}arrow V_{3}arrow 0$ in.
$M-\Pi M$ for every A-supermodule $M$ .
Hereis the abelian subcategory of A-smod whose objects are the same but morphisms
are consisting of even A-homomorphisms. Clearly, ( -smod) is a free Z-module with basis( -smod). The importance of the operation
lies in the fact that it gives an isomorphism..

$\mathbb{Z}$$A-smod_{\overline{0}}$$A-smod_{\overline{0}}$$K_{0}$

$lrr$

$A$

(3.1)

$A$

$O$

(

$K_{0}(A-smod )\otimes_{Z}K_{0}(B-smod )arrow^{\sim}K_{0}$ $A\otimes B$

for two superalgebras

$A$

and

$B$

-smod),$[V]\otimes[W]\mapsto[VOW]$.

that in general we have $|A\otimes B|\not\cong|A|\otimes|B|$ where for a superalgebra $C$ we denote by $|C|$ the underlying
unital associative algebra.
$2_{Note}$

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SHUNSUKE TSUCHIOKA

3.8. Projective supermodules. Let be a superalgebra. A projective A-supermodule is, bydefinition, a projective object in A-smod and it is equivalent to saying that it is a projective
since there are canonical isomorphisms
object in$A$$A-smod_{\overline{0}}$$Hom_{A}$-smod $(V,$ $W)_{\overline{0}}\cong Hom_{A-smod_{\overline{0}}}(V,$ $W)$ ,$Hom_{A}$-smod $(V,$ $W)_{\overline{1}}\cong Hom_{A-smod}\sigma(V,$ $\prod W)(\cong Hom_{A-smod_{\overline{0}}}(\prod V,$ $W))$

We denote by Proj $(A)$ the full subcategory of A-smod consisting of all the projective A-supermodules.

3.9. Cartan pairings. Let us assume further that is finite-dimensional. Then, as in theusual finite-dimensional algebras, every A-supermodule $X$ has a (unique up to even isomor. If $X$ is irreducible, thenphism) projective coveris (evenly) isomorphic
into a projective indecomposable A-supermodule. From this, we easily see $M\cong N$ if and only$A$

$P_{X}$$P_{X}$$A-smod_{\overline{0}}$(Proj $(A)$ ) is identified with
if $P_{M}\cong P_{N}$ for $M,$ $N\in$ lrr( -smod). Thus,$Hom_{Z}$ (( -smod), Z) through the non-degenerate canonical pairing$A$$K_{0}$$(,$$K_{0}$$K_{0}$( $A$ -smod)’$def=$$A$$\}_{A}:K_{0}$( Proj $(A)$ ) $\cross K_{0}(A-smod )arrow Z$ ,$([P_{M}], [N])\mapsto\{\begin{array}{ll}\dim Hom_{A}(P_{M}, N) if type M=M,- \dim Hom_{A}(P_{M}, N) if type M=Q,\end{array}$for all $M\in lrr$ ( -smod) and $N\in$ A-smod. Note that the left hand side is nothing but thecomposition multiplicity $[N : M]$ . We also reserve the symbol$A$$\omega_{A}:K_{0}$(Proj $(A)$ ) $arrow K_{0}$ ( -smod)$A$for the natural Cartan map.

3.10. Clifford superalgebras. The Clifford superalgebra is defined as
for $n\geq 0$
is a 2-dimensional superalgebra generated by the odd generator
with $C^{2}=1$ .
where$\sqrt{-1}\in k$ ,is split-simple if and only if
is a split-simple superalgebra, but
Assumethenis even. We denote by
the Clifford module, i.e., a $2^{[(n+1)/2]}$ -dimensional irreducible$C_{n}=C_{1}^{\otimes n}$$C$$C_{1}$$|C_{n}|$$C_{n}$$U_{n}=C_{1}^{On}$$n$$C_{n}$-supermodule (of type

$Q$

iff

$n$

is odd) characterized by

$1rr$

(

$C_{n}$-smod) $=\{[U_{n}]\}$ noting (3.1).

3.11. Morita superequivalences. We must clarify our meaning of the terminology Morita
superequivalence. Again we emphasize that our meaning of Morita superequivalence in thisarticle is similar to [K12, BK2, Wan] and different from that of [KKT. \S 2.4].
Two superalgebras and $B$ are called Morita superequivalent of type if there exist superadid, $FoG\simeq$ id
ditive functors $F:$ A-smod $arrow B$ -smod and $G:$ B-smod $arrow A$-smod such thatlrr( $B$-smod), $G|_{1rr(B-smod )}$ : lrr $(B-smod )arrow^{\sim}$ lrr( -smod)
and both $F|_{1rr(A-smod )}$ : lrr(A-smod)
are type preserving. We say that and $B$ are called Morita superequivalent of type if there: $B-smod arrow A$ -smod such thatexist superadditive functors $F$ : $A-smod arrow B$-smod and$GoF\simeq$ id$FoG\simeq$ id
and induces type reversing bijections
$M$

$A$

$G\circ F\simeq$$A$$arrow^{\sim}$

$Q$

$A$

$G$

$\oplus\Pi$$\oplus\Pi,${ $[V]\in 1$ rr(A-smod){ $[V]\in$ lrr(B-smod)

$|$

$|$

type $V=M$ }type $V=M$ }$arrow^{\sim}${ $[W]\in 1$ rr(B-smod) type $W=Q$ },{ $[W]\in$ lrr(A-smod) type $W=Q$ }.
$|$$arrow^{\sim}$$|$

We say that and are called Morita superequivalent if they are either Morita superequivalentof typeor type Q.
$A$

$B$

$M$

$e^{2}=e$
Example 3.1. Let be a superalgebra and $e\in A$ a full even idempotent, i.e.,
and $A=AeA^{d}=^{ef} \{\sum_{i=1}^{n}a_{i}eb_{i}|a_{i}, b_{i}\in A, n\geq 0\}$ . Then, and $eAe$ are Morita superequivalentof type M.
$A$$e\in A_{\overline{0}},$

$A$

183
A SUPER ANALOG OF THE KHOVANOV-LAUDA-ROUQUIER ALGEBRAS$A_{2}^{(2)}$$0\Leftarrow 0-\cdot\cdot-$$D_{\ell+1}^{(2)}$$\alpha_{0}\alpha_{1}\circ\in\circ$$\alpha_{0}$$A_{2\ell}^{(2)}$$0$$\Rightarrow 0$$\alpha\ell-\iota$$\alpha_{1}$$\alpha_{l}$$b_{\infty}$$\alpha_{0}0\Leftarrow\alpha_{1}\alpha_{\ell-1}0-\cdots-0\Leftarrow\alpha_{\ell}o$$\alpha_{0}0\Leftarrow\alpha_{1}\alpha 2\alpha 0-0-0-3\ldots$

FIGURE 1. Dynkin diagrams of type

$A_{2\ell}^{(2)},$$D_{\ell+1}^{(2)}$and.

$b_{\infty}$

Example 3.2. Let and $B$ superalgebras and suppose there exists a superalgebra isomorphism
for some $n\geq 0$ . Then, and $B$ are Morita equivalent of type (resp. type M)
if is odd (resp. is even) via$A$$A\otimes C_{n}arrow^{\sim}B$
$n$

$A$

$Q$

$n$

$F:A-smod arrow B$-smod,$V\mapsto Hom_{C_{n}}(U_{n}, V)$$G:B-smod arrow A$ -smod,$W\mapsto W\otimes U_{n}$,.

4. PARTIAL CATEGORIFICATIONS USING $HECKE-CLIFFORD$ SUPERALGEBRASFrom now on, we reserve a non-zero quantum parameterand set $\xi=q-q^{-1}$ for
convenience. Let us define the affine Hecke-Clifford superalgebra [JN, \S 3]. Although Jones andNazarov introduced it under the name of affine Sergeev algebra, we call it affine Heckc-Clifford
superalgebra following [BK2, \S 2-d].
$q\in k^{\cross}$

Definition 4.1 ([JN]). Let $n\geq 0$ be an integer. The affine Hecke-Clifford superalgebm
is
by
even
genemiors
generators
odd
andwith thedefinedfollowing relations.
(1) $X_{i}X_{i}^{-1}=X_{i}^{-1}X_{i}=1,$ $X_{i}X_{j}=X_{i}X_{j}$ for all $1\leq i,j\leq n$ .
$\mathcal{H}_{n}$$X_{1}^{\pm 1},$$\cdots,$$X_{n}^{\pm 1},$

$T_{1},$$\cdots,$ $T_{n-1}$$C_{1},$$\cdots,$$C_{n}$(2) $C_{i}^{2}=1,$ $C_{i}C_{j}+C_{j}C_{i}=0$ for all $1\leq i\neq j\leq n$ .

(3) $T_{i}^{2}=\xi T_{i}+1,$ $T_{i}T_{j}=T_{j}T_{i},$ $T_{k}T_{k+1}T_{k}=T_{k+1}T_{k}T_{k+1}$ for all $1\leq k\leq n-2$ and $1\leq i,j\leq$$n-1$ with $|i-j|\geq 2$ .

(4)
for all $1\leq i\neq j\leq n$ .
(5) $T_{i}C_{i}=C_{i+1}T_{i},$ $(T_{i}+\xi C_{i}C_{i+1})X_{i}T_{i}=X_{i+1}$ for all $1\leq i\leq n-1$ .
(6) $T_{i}C_{j}=C_{j}T_{i},$for all $1\leq i\leq n-1$ and $1\leq j\leq n$ with $j\neq i,$ $i+1$ .
$C_{i}X_{i}^{\pm 1}=X_{i}^{\mp 1}C_{i},$ $C_{i}X_{j}^{\pm 1}=X_{j}^{\pm 1}C_{i}$$T_{i}X_{j}^{\pm 1}=X_{j}^{\pm 1}T_{i}$Definition 4.2 ([BK2, Tsu]). Let.
(a) Repa fgw of$k$be a field whose characteristic different from 2 and take$q\in k^{\cross}$-supermodule $M$ such that the set-smod consisting ofof eigenvalues of $X_{j}+X_{j}^{-1}$ is a subset of $\{q(i)|i\in Z\}$ for all $1\leq j\leq n^{3}$ where $q(i)=$$\mathcal{H}_{n}\iota s$$\mathcal{H}_{n}$$2\cdot(q^{2i+1}+q^{-(2i+1)})/(q+q^{-1})$$\mathcal{H}_{\mu}$.

(b) Put I be the set of vertices of Dynkin diagmm $X$ (see Figure 1) where$A_{2\ell}^{(2)}$$X=$$\{$ $D_{\ell+1}^{(2)}$$b_{\infty}$

is a primitive $(2\ell+1)$ -the root of unity for some(if
is a primitive $2(\ell+1)$ -the root of unity for some).
(if otherwise and moreove we have(if$q^{2}$$q^{2}$)$\ell\geq 1$)$q^{4}\neq 1$

We define for a dominant integml weight
algebrawhere$\mathcal{H}_{n}=\{f^{\lambda}\rangle$$\ell\geq 1$$\lambda\in \mathcal{P}^{+}$

of

$X$

a finite-dimensional quotient superand$g^{\lambda}= \prod_{i\in I}(X_{1}^{2}-q(i)X_{1}+1)^{\lambda(h_{i})}$$f^{\lambda}=\{\begin{array}{ll}g^{\lambda}/((X_{1}-1)^{\lambda(h_{0})}(X_{1}-1)^{\lambda(h_{\ell})}) (if X=D_{\ell+1}^{(2)})g^{\lambda}/(X_{1}-1)^{\lambda(h_{0})} (if X=A_{2\ell}^{(2)}, b_{\infty})\end{array}$$3_{It}$4.4].

is equivalent to require only the set of eigenvalues of $X_{1}+X_{1}^{-1}$ is a subset of $\{q(i)|i\in Z\}$ by [BK2, Lemma

184
SHUNSUKE TSUCHIOKA

Remark 4.3. In the setting of Definition 4.2 (b), forョ

$\lambda\in \mathcal{P}^{+},$$M\in \mathcal{H}_{n}$-smod we have $M\in$ Rep$\mathcal{H}_{n}\Leftrightarrow$$f^{\lambda}M=0$

Theorem 4.4 ([BK2, Tsu]). Let be an algebmically closed field whose characterestic different
have the following.
and $X$ as in Definition 4.2 $(b)$ . Then,from 2 and take) is isomorphic to
as gmded Z-Hopf algebm.
(a) the gmded dual of $K(\infty)=\oplus_{n\geq 0}K_{0}$ (Rep-smod)
is isomorphic to thewhich-module structure(
has a left(b)
-module of highest weight .
integrable highest weight.
) is isomorphic to Kashiwara’s crystal associated with(c) $B(\infty)=U_{n\geq 0}$ lrr(Rep$B(\lambda)=U_{n>0}$
theintecrystal
withtsomorphic
associated-smod) isto Kashiwam’s(d)lrr(-module of highest weight .
gmble-smod) are two integml lattices
(
) and
(e)
( Proj. Moreover,is minimum latticecontaining the trivial representationof.
in the sense thatHereis $the\pm$ -part of the Kostant Z-form of the universal enveloping algebm of $g(X)$ andis the -subalgebm of the universal enveloping algebm of $g(X)$ genemted by the Chevalley
genemtors.
$k$$u$) $e$$q\in k^{x}$$U_{Z}^{+}$$\mathcal{H}_{n}$$K(\lambda)_{\mathbb{Q}}=\oplus_{n>0}\mathbb{Q}\otimes K_{0}$$\mathcal{H}_{n}^{\lambda}$$U_{\mathbb{Q}}$$\lambda$$U_{\mathbb{Q}}$$U_{v}^{-}(\mathfrak{g}(X))$$\mathcal{H}_{n}$$\mathcal{H}_{n}^{\lambda}$$U_{v}(\mathfrak{g}\overline{(}X))$$\lambda$$K(\lambda)\oplus_{n\geq 0}K_{0}$$(\mathcal{H})_{n}^{\lambda}$$K(\lambda)^{*}=\oplus_{n\geq 0}K_{0}$$[1_{\lambda}]$$ofK(\lambda)_{Q}$$\mathcal{H}_{n}^{\lambda}$$K(\lambda)^{*}$$\mathcal{H}_{0}^{\lambda}=k$$K(\lambda)^{*}=U_{Z}^{-}[1_{\lambda}]$$U_{Z}^{\pm}$$U_{\mathbb{Q}}$$\mathbb{Q}$

Remark 4.5. Since A-smod is not necessarily an abelian category for a superalgebra , Theorem
4.4 cannot be seen as a categorification result in the usual sense (see for example [KMS]). For
example, in the identification Theorem 4.4 (b) neither the action of Chevalley generators nor
are “exact” functors, of course. We just can assign for each simple module identified up toparity change (which is aof the Grothendieck groups (see 3.7)) a well-defined destination
in a “module-theoretic“ way.
$A$

$e_{i}$$f_{i}$$ba\backslash ^{\backslash }i_{b^{v}}$Remark 4.6. Under the identification (b) and (e) of Theorem 4.4, the Cartan pairing on
coincides with the Shapovalov form [BK2, Tsu]. It is expected but not proved so far4 that thedecomposition ofcomes from the block decomposition ofcoincides withthe weight space decomposition of the corresponding integrable highest weight module.
$K(\lambda)_{\mathbb{Q}}$$\{\mathcal{H}_{n}^{\lambda}|n\geq 0\}$$K(\lambda)_{\mathbb{Q}}$5. AN EXPECTATION AND TWO COUNTEREXAMPLES
Considering both Theorem 2.4 and Theorem 4.4, it is reasonable to expect that in the settingof Definition 4.2 (b),andhas a “good relation” as Theorem 2.5. However, we
believe that this expectation never achieved because of the following two facts.
$R_{n}^{\lambda}(X;Q^{X})$$\mathcal{H}_{n}^{\lambda}$

5.1. $X=D_{2}^{(2)}$ case. Let $q=\exp(2\pi\sqrt{-1}/8)\in k$ and let char $k=0$ . In virtue of Theo)(resp.
rem 2.4 and Theorem 4.4, the family of (super)algebras-module).
categorifies-module (resp.
However, there is no Morita equivalence between
nor Morita suand
perequivalence of type between
whatever superalgebra structure weand. This is because we have
imposeon and for any choice of parameters$\{R_{n^{0}}^{\Lambda}(k;Q^{X})\}_{n\geq 0}$$\{\mathcal{H}_{n^{0}}^{\Lambda}(q)\}_{n\geq 0}$$U(\mathfrak{g}(X))$$U_{v}(\mathfrak{g}(X))$$V(\Lambda_{0})$$|\mathcal{H}_{4}^{\Lambda_{O}}(X)|$$M$$\mathcal{H}_{4}^{\Lambda_{0}}(X)$$R_{4}^{\Lambda_{O}}(k;Q^{X})$$R_{4}^{\Lambda_{0}}(k;Q^{X})$$Q^{X}$$R_{4}^{\Lambda_{0}}(k;Q^{X})$$\dim Z(|\mathcal{H}_{4}^{\Lambda_{0}}(q)|)=4\neq 5=\dim Z(|R_{4}^{\Lambda_{0}}(k;Q^{X})|)$.

-smod) consists of 2 irreducible superBecause lrr $(Mod_{gr}(R_{4}^{\Lambda_{0}}(k;Q^{X})))=2$ and lrr(
get Morita superequivand
modules of type , there is no possibility thatappropriately.
alence of type by defining a superalgebra structure on$\mathcal{H}_{4}^{\Lambda_{0}}(q)$

$\#$

$M$

$\mathcal{H}_{4}^{\Lambda_{0}}(X)$$Q$$4_{For}$the degenerate case, some partial results are known $[RuQ$ .

$R_{4}^{\Lambda_{0}}(k;Q^{X})$$R_{4}^{\Lambda_{0}}(k;Q^{X})$185
A SUPER ANALOG OF THE KHOVANOV-LAUDA-ROUQUIER ALGEBRAS

5.2.

and degenerate case. Let us briefly recall the affine Sergeev superalgebra
introduced by Nazarov in his study of spin Young symmetrizers for the symmetric groups [Naz].
$X=A_{2}^{(2)}$$\overline{\mathcal{H}}_{n}$Definition 5.1.

(i) The spin symmetric group superalgebm
and the following relations$k6_{\overline{n}}$is defined by odd genemtors$\{t_{i}|1\leq i\leq n-1\}$$t_{a}^{2}=1$,$t_{a}t_{b}=-t_{b}t_{a}if|a-b|>1$ ,$t_{c}t_{c+1}t_{c}=t_{c+1}t_{c}t_{c+1}$.

(ii) The Sergeev supemlgebm is defined as(for super tensor product, see \S 3.6)
whereis the Clifford supemlgebm (see \S 3.10).
(iii) The affine Sergeev superalgebrais the k-superalgebm genemted by the even genemtors,
and the odd genemtors
with the following relations.
(i) $x_{i}x_{j}=x_{j}x_{i}$ for all $1\leq i,j\leq n$ ,(ii) $C_{i}^{2}=1,$ $C_{i}C_{j}+C_{j}C_{i}=0$ for all $1\leq i\neq j\leq n$ ,(iii) $t_{i}^{2}=1,$ $t_{i}t_{i+1}t_{i}=t_{i+1}t_{i}t_{i+1},$ $t_{i}t_{j}=t_{j}t_{i}(|i-j|\geq 2)$ ,(iv),(v) $C_{i}x_{j}=x_{j}C_{i}$ for all $1\leq i\neq j\leq n$ ,(vi) $C_{i}x_{i}=-x_{i}C_{i}$ for all $1\leq i\leq n$ ,(vii) $t_{i}x_{i}=x_{i+1}t_{i}-1-C_{i}C_{i+1},$ $t_{i}x_{i+1}=x_{i}t_{i}+1-C_{i}C_{i+1}$ for all $1\leq i\leq n-1$ ,(viii)
if $j\neq i,$ $i+1$ .
$y_{n}=k6_{\overline{n}}\otimes C_{n}$$C_{n}$$\overline{\mathcal{H}}_{n}$$x_{1},$$\ldots,$ $x_{n},$$t_{1},$$\ldots,$$t_{n-1}$$C_{1},$$\ldots$$C_{n}$$t_{i}C_{j}=C_{s_{i}(j)}t_{i}$$t_{i}x_{j}=x_{j}t_{i}$is an affinization of the Sergeev superalgebra
as its finite-dimensionalandhas
quotientsince there is a non-trivial superisomorphism$\overline{\mathcal{H}}_{n}$$y_{n}$$\mathcal{Y}_{n}\cong\overline{\mathcal{H}}_{n}^{\Lambda_{0}}$(5.1)$y_{n}$$\mathcal{H}_{n}$$:=\overline{\mathcal{H}}_{n}/\{x_{1}\rangle$$1\otimes C_{j}\mapsto 1\otimes C_{j}$$k6_{\overline{n}}\otimes C_{n}\frac{\sim_{\backslash }}{\prime}k6_{\overline{n}}\ltimes C_{n}$,$t_{i} \otimes 1\mapsto\frac{1}{\sqrt{-2}}s_{i}\otimes(\dot{C}_{i}-C_{i+1})$.

due to Sergeev and Yamaguchi [Ser, Yam]. Note thatis Morita superequivalent tosee
Example 3.2),
Modular representation theory of
was considerably developed in [BK2] using the methodof Grojnowski [Gro]. A consequence of [BK2] is that theof finite-dimensional integral-supermodules partially categorifies(resp.
when char $k=0$ (resp.
char $k=2P+1$ for) as Theorem 4.4.
Assume char $k=3$ and put $X=A_{2}^{(2)}$ (see Figure 1). Take a block subsuperalgebra $B$ ofwhich categorifies
where $\nu=8\alpha_{0}+3\alpha_{1}$ . Although $R_{\nu}(k;Q^{X})$ categorifies$1rr(Mod_{gr}(R_{\nu}(k;Q^{X})))$ and lrr(B-smod) correspond to different perfect basis at
the specialization $v=1$ .
Let us explain in detail. By [BK2] (see also [K12, part II]), we have$k6_{\overline{n}}($$\mathcal{Y}_{n}$$7?_{n}$$(:ats_{\text{ノ}}gory$$\overline{\mathcal{H}}_{n}$$U^{-}(\mathfrak{g}(b_{\infty}))$$U^{-}(\mathfrak{g}(A_{2l}^{(2)}))$$\ell\geq 1$$\overline{\mathcal{H}}_{11}$$U^{-}(\mathfrak{g}(X))_{-\nu}$$U_{v}^{-}(\mathfrak{g}(X))_{-\nu},$(5.2)$\bigoplus_{n\geq 0}K_{0}(\overline{\mathcal{H}}_{n}^{\Lambda_{0}}-smod )_{\mathbb{C}}\cong V(\Lambda_{0})$,$n\geq u_{0}|rr(\overline{\mathcal{H}}_{n}^{\Lambda_{0}}-smod )\cong RP_{3}\cong B(\Lambda_{0})$where the left isomorphism is as U(g(X))-modules and the right isomorphism is as
crystals. In virtue of (5.1) and Example 3.2, the same Lie-theoretic descriptions hold when we
replacewith.
Recall $RP$3 is the set of a113-restricted 3-strict partitions. A partitionis3-restricted 3-strict if the following conditions are satisfied [Kan, K12. LT].
$U_{v}(\mathfrak{g}(X))-$$\overline{\mathcal{H}}_{n}^{\Lambda_{0}}$$k6_{\overline{n}}$$\lambda=$..

$e\lambda_{k}=\lambda_{k+1}$

implies$\lambda_{k}-\lambda_{k+1}<3$$\lambda_{k}-\lambda_{k+1}\leq 3$

For each
irreducibles of$\lambda\in$

RP3

if
if

$\lambda_{k}\in 3Z$$\lambda_{k}\in 3Z$,$\lambda_{k}\not\in 3Z$.

,, we denote by. Note thatis of type$\cong B(\Lambda_{0})$$V_{\lambda}^{spin}$$k6_{|\lambda|}^{-}$$(\lambda_{1}, \cdots , \lambda_{r})$the corresponding isomorphism class ofis odd.
if and only if$V_{\lambda}^{sp\dot{\ovalbox{\tt\small REJECT}}n}$$Q$$\gamma_{1}(\lambda)$$:= \sum_{k\geq 1}L\frac{1+\lambda_{k}}{3}\rfloor$

186
SHUNSUKE TSUCHIOKAOn the other hand, by [KK, LV] we have,$n\geq u_{0}|rr(Mod_{gr}(R_{n^{0}}^{\Lambda}(k;Q^{X}))\cong B(\Lambda_{0})$$\bigoplus_{n\geq 0}K_{0}(Mod_{gr}(R_{n^{0}}^{\Lambda}(k;Q^{X})))_{\mathbb{C}}\cong V(\Lambda_{0})$where the left isomorphism is as $U_{v}(g(X))$ -modules and the right isomorphism is as $U_{v}(g(X))-$, we denote by
the corresponding isomorphism classcrystals. For eachRP3.
of irreducibles ofIf both lrr $(Mod_{gr}(R_{\nu}(k;Q^{X})))$ and lrr(B-smod) correspond (after the specialization $v=1$ ) the, then we must havesame perfect basis in the sense of $[BeKa]$ on $U(9(X))$ -module$\cong B(\Lambda_{0})$$\lambda\in$$V_{\lambda}^{KLR}$$R_{n^{0}}^{\Lambda}(k;Q^{X})$$V(\Lambda_{0})$$\dim V_{\lambda}^{spin}/\dim V_{\lambda}^{KLR}=2^{[(1+\gamma_{1}(\lambda))/2]}$

for any
RP3 (see [K12, Lemma 22.3.8]). A computer calculation shows that for $\lambda=(6,4,1)$ ,we have $\dim V_{\lambda}^{KLR}=648$ while it is known that $\dim V_{\lambda}^{spin}=2880$ . It may be interesting to pointout that in history this dimension $\dim V_{\lambda}^{spin}=2880$ was first miscalculated as $\dim V_{\lambda}^{spin}=2592$
in [MY]. If it were correct, observing such a direct discrepancy between the KLR algebras andthe spin symmetric groups must become more difficult.
$\lambda\in$

6. QUIVER HECKE SUPERALGEBRASDefinition 6.1 ([KKT, \S 3.1]). Let$k$be a field such that $2\neq 0$ and let I be a finite set withparity decomposition $I=I_{odd}uI_{even}$ . For $i\in I$ , we denote the parity of by par $(i)\in Z/2Z$ , i. e.,$i$otherwise . Take $Q=(Q_{ij}(u, v))$ such that
$Q_{ij}\in k\{w, v\}/\{uv-(-1)$ par(i)parfor all $i,j\in I$ ,$Q_{ij}(u, v)=0$ for all $i,j\in I$ with $i=j$ ,$Q_{ij}(u, v)=Q_{ji}(v, u)$ for all $i,j\in I$ ,$Q_{ij}(u, v)=Q_{ij}(-u, v)$ for all $i\in I_{odd},j\in I$ .
(a) The quiver Hecke supemlgebra5 $R_{n}(k;Q)$ is the k-supemlgebm generated by$p\leq n,$ $1\leq a<n.\nu\in I^{n}\}$ unth parity
par
par
par
$1\leqp,$$q\leqn,$$1\leq b<a\leq n-1$ .
with the following defining relations6 for all

par

..
.

$(i)=\overline{1}$if$i\in I_{odd}$$\overline{0}$$(j)_{vu\rangle}$$\bullet$$\{x_{p},$$\overline{e(\nu)}=\overline{0},$$\overline{x_{p}e(\nu)}=$$(\nu_{p}),$$\overline{\tau_{a}e(\nu)}=$$\tau_{a},$$e_{\nu}|1\leq$$(\nu_{a+1})$$(\nu_{a})$$\mu,$ $\nu\in I^{n},$$x_{p}x_{q}e_{\nu}=(-1)^{par(\nu_{p})par(\nu_{q})}x_{q}x_{p}e_{\nu}$$e_{\mu}e_{\nu}=\delta_{\mu\nu}e_{\mu},$$1= \sum_{\mu\in I^{n}}e_{\mu},$$x_{p}e_{\nu}=e_{\nu}x_{p},$ $\tau_{a}\tau_{b}e_{\nu}=(-1)$

par$\tau_{a}^{2}e_{\nu}=Q_{\nu_{a},\nu_{a+1}}(x_{a}, x_{a+1})e_{\nu},$$(\tau_{a}x_{a+1}-(-1)$

par$(\nu_{a})$$(\nu_{a})$par $(\nu_{a+1})$ par$(\nu_{b})$$\tau_{a}e_{\mu}=e_{s_{a}(\mu)}\tau_{a},$,par $(\nu_{b+1})_{\tau_{b}\tau_{a}e_{\nu}if}|a-b|>1$ ,$\tau_{a}x_{p}e_{\nu}=(-1)^{par(\nu_{p})}$par $(\nu_{a+1})_{X_{a}\tau_{a})e_{\nu}=}(x_{a+1}\tau_{a}-(-1)$ par$(\nu_{a})$

par

par

$(\nu_{a})$par $(\nu_{a+1})_{X_{p}\tau_{a}e_{\nu}ifp}\neq a,$ $a+1$ ,,$(\nu_{a+1})_{\mathcal{T}_{a}X_{a})e_{\nu}=}\delta_{\nu_{a},\nu_{a+1}}e_{\nu}$$(7b+1^{T}b^{7}b+1-\tau_{b}\tau_{b+1}\tau_{b})e_{\nu}=$$\{\begin{array}{ll}\frac{Q_{\nu_{b}.\nu_{b+1}}(x_{b+2},x_{b+1})-Q_{\nu_{b}.\nu_{b+1}}(x_{b},x_{b+1})}{x_{b+2}-x_{b}}e_{\nu} if \nu_{b}=\nu_{b+2}\in I_{even},(-1)^{par(\nu_{b})}(x_{b+2}-x_{b})\frac{Q_{\nu_{b}.\nu_{b+1}}(x_{b+2},x_{b+1})-Q_{\nu_{b^{y}b+1}}.(x_{b},x_{b+1})}{x_{b+2}^{2}-x_{b}^{2}}e_{\nu} if \nu_{b}=\nu_{b+2}\in I_{odd},0 otherwise\end{array}$(b) For $\beta=\sum_{i\in I}\beta_{i}\cdot i\in N[I]$ with $n=$ ht,
where$(\beta)$$:= \sum_{i\in I}\beta_{i}$, we define $R_{\beta}(k;Q)=R_{n}(k;Q)e_{\beta}$$e_{\beta}= \sum_{\nu\in Seq(\beta)}e_{\nu}$$5_{Because}$ whenthe quiver Hecke superalgebra is the same as the Khovanov-Lauda-Rouquier algebra,
the notation $R_{n}(k_{iQ)}$ for the quiver Hecke superalgebra is justified.
6 Whenbelongs to the commutative ring
, and hence we can define
is odd,$I_{odd}=\emptyset$$\nu_{b}$$k[x_{b}^{2}, x_{b+1}]$$Q_{\nu_{b}.\nu_{b+1}}(x_{b}, x_{b+1})$$\frac{Q_{\nu_{b^{\nu}b+1}}.(x_{b+2},x_{b+1})-Q_{\nu_{b},\nu_{b+1}}(x_{b},x_{b+1})}{x_{b+2}-x_{b}^{l}}$as an element of$k[x_{b}^{2}, x_{b+1}, x_{b+2}^{2}]$.

187
A SUPER ANALOG OF THE KHOVANOV-LAUDA-ROUQUIER ALGEBRAS$A_{2}^{(2)}$$D_{\ell+1}^{(2)}$$\alpha_{0}\alpha_{1}\not\in 0$$\alpha_{0}()\Leftarrow\alpha_{1}\alpha_{\ell-1}0-\cdots-0\Rightarrow\alpha\ell(j$$A_{2\ell}^{(2)}$$b_{\infty}$$\alpha_{0}\Leftarrow\alpha_{1}\alpha_{\ell-1}0-\cdots-0\Leftarrow\alpha po$$\alpha 0\Leftarrow\alpha_{1}\alpha_{2}\alpha 30-0-0-\cdots$

FIGURE 2. Dynkin diagrams of typeindicates an odd vertex.
(c) For$\lambda=\sum_{i\in I}\lambda_{i}\cdot i\in N[I]$$A_{2\ell}^{(2)},$

and$D_{\ell+1}^{(2)}$and $\beta\in N[I]$ with $n=$ ht$(\beta)$$b_{\infty}$

with parity. Here

, we define$R_{n}^{\lambda}( k;Q)=R_{\eta}(k;Q)/R_{n}(k;Q)(\sum_{\nu\in I^{n}}x_{1}^{\lambda_{h_{\nu_{1}}}}e_{\nu})R_{\eta}(k;Q)$,.

$R_{\beta}^{\lambda}( k;Q)=R_{\beta}(k;Q)/R_{\beta}(k;Q)(\sum_{\nu\in Seq(\beta)}x_{1}^{\lambda_{h_{\nu_{1}}}}e_{\nu})R_{\beta}(k;Q)$

Definition 6.2 ([BKM, KKT]). A genemlized Cartan matrix $(GCM)$ with parity is a $GCM$
with the parity decomposition $I=I_{even}uI_{odd}$ such that $a_{ij}\in 2Z$ for all $i\in I_{odd}$
and $j\in I$ .

$A=(a_{ij})_{i,j\in I}$

Definition 6.3 ([KKT, \S 3.6]). Let $A=(a_{ij})_{i,j\in I}$ be a $symmetr\cdot\iota zableGCM$ with parity. Take
the symmetrization. For $j\in I$ , let
be the set of $(r, s)$ where and are integers
satisfying the following conditions. Note thatwhen $i=j$ .
(i) $0\leq r\leq-a_{ij},$ $0\leq s\leq-a_{ji}$ and $d_{i}r+d_{j}s=-d_{i}a_{ij}$ ,(ii) $r\in 2Z$ if $i\in I_{odd}$ and $s\in 2Z$ if $j\in I_{odd}$ .
Take a sequenceinsuch that $t_{i,j,r,s}=t_{j,i,s,r}$ and $t_{i,j,-a_{i,j},0}\neq 0$ and put$z\}/\langle zw-(-1)^{par(i)par(j)}wz\rangle$ .
$d=(d_{i})_{i\in I}$$i,$$S_{ij}$$r$$s$$S_{i,j}=\emptyset$$(t_{i,j,r,s})_{(r,s)\in S_{ij}}$$k$$Q_{i,j}^{A}(u, v)= \sum_{(r,s)\in S_{ij}}t_{i,j,r,s}u^{r}v^{s}\in k_{A}\langle w,$

For $n\geq 0$ and $\beta\in N[I]$ with $ht(\beta)=n$ , all of $R_{n}(k;Q^{A}),$$1\leq p\leq n,$ $1\leq a<n$ .
are $(Z\cross Z/2Z)$ -graded via the assignment where$R_{\beta}(k;Q^{A}),$$\lambda,$$R_{n}^{\lambda}(k;Q^{A}),$$R_{\beta}^{\lambda}(k;Q^{A})$$\nu\in I^{n},$$\deg(e_{\nu})=(0, \overline{0})$,$\deg(x_{p}e_{\nu})=(2d_{\nu_{P}}$, par$(\nu_{p}))$,$\deg(\tau_{a}e_{\nu})=$($-d_{\nu_{a}}a_{\nu_{a},\nu_{a+1}}$, par$(\nu_{a})$par$(\nu_{a+1})$).

Theorem 6.4 ([KKT, Corollary 4.8,Theorem 3.13]). Let be an algebmically closed field whosechamctenstic different from 2 and takeand $X\in Mat_{I}(Z)$ as in Definition 4.2 $(b)$ andmake $X$ a $GCM$ with panty as in Figure 2. Then,andare Morita superequivalent(see \S 3.11) for all.
where we identify
and$k$$q\in k^{\cross}$$R_{n}^{\lambda}(X;Q^{X})$$\mathcal{H}_{n}^{\lambda}$$\lambda\in \mathcal{P}^{+}$$\lambda\in \mathcal{P}^{+}$$\sum_{i\in I}\lambda(h_{i})\cdot i\in N[I]$

Remark 6.5. Actually, in [KKT, Theorem 4.4] we also treat other blocks of-smod thanRep
where Dynkin diagram without parity of typeappear (in addition towith parity).
$\mathcal{H}_{n}$$\mathcal{H}_{n}$$b_{\infty},$$A_{2\ell}^{(2)},$$a_{\infty},$$A_{\ell}^{(1)},$$C_{\ell}^{(1)}$$c_{\infty},$$D_{\ell+1}^{(2)}$$a_{\infty}$...

$A_{1}^{(1)}$$A_{\ell}^{(1)}\alpha_{1}\alpha 2\alpha\ell 0/^{\prime\alpha 0}-0-\cdot\cdot-0/^{O}.\backslash _{\backslash _{\backslash }}$$-0-0-0-\alpha_{-1}\alpha 0\alpha_{1}\ldots$$\alpha 0o\Leftrightarrow\alpha_{1}\circ$$C_{\ell}^{(1)}$$c_{\infty}$$\alpha_{0}0\Rightarrow\alpha_{1}\alpha 2\alpha_{3}0-0-0-\cdots$$0\Rightarrow 0-\cdots-$$\alpha_{0}$$\alpha_{1}$$0$$\alpha_{\ell-1}$$\Leftarrow 0$$\alpha\ell$Remark 6.6. We believe thatwhile theyhas simpler representation theory thanare Morita superequivalent. For example, we conjectured that all the simple supermodules ofare of type M. This “type phenomenon” are verified in [HW. \S 6.5]. Moreover,$R_{n}^{\lambda}(k;Q^{X})$$R_{n}^{\lambda}(k;Q^{X})$$\mathcal{H}_{n}^{\lambda}$$M$

Hill and Wang claims that $R_{n}(k;Q^{A})$ categorifies the half of quantum Kac-Moody superalgebra
introduced by Benkart-Kang-Melville [BKM].

188
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$D_{\ell}^{(2)}$

INSTITUTE FOR THE PHYSICS AND MATHEMATICS OF THE UNIVERSE, UNIVERSITY OF TOKYO, KASHIWANO-HA
5-1-5, KASHIWA CITY, CHIBA 277-8582, JAPANE-mail address: tshunQkur $ins$ . kyoto-u. ac. jp