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2 A parameterization of the canonical basis of affine modified quantized enveloping algebras Minghui Zhao Department of Mathematical Sciences, Tsinghua University CRA 2012, Bielefeld Page 2 of 22

3 Quantized enveloping algebra? A = (A,,, P, P ): a Cartan datum associated with a generalized Cartan matrix A;? U : the quantized enveloping algebra associated to the above Cartan datum generated by Ei, Fi and Kµ, i and µ P over Q(v);? f : the Lusztig s algebra associated to the above Cartan datum generated by θi, i ;? + : f U (x 7 x+ ): the algebra homomorphism with image U + ; Page 3 of 22? : f U (x 7 x ): the algebra homomorphism with image U ;? B: the canonical basis of f introduced by Lusztig.

4 Modified quantized enveloping algebra? U : the modified quantized enveloping algebra;? U = λ P U 1λ, where U 1λ = U/ P µ P U (Kµ v hλ,µi );? {b+ b0 1λ b, b0 B} is a basis of the Q(v)-space U 1λ ;? B = {b λ b0 b, b0 B, λ P }: the canonical basis of U ; Page 4 of 22? b λ b0 = b+ b0 1λ + lower terms.

5 Aim? Give a parameterization of the canonical basis of affine modified quantized enveloping algebras using root categories. Page 5 of 22

6 Quivers and path algebra? Q = (, H, s, t): a quiver;? : the set of vertices;? H: the set of arrows;? s, t : H : s(i j) = i, t(i j) = j; Page 6 of 22

7 Ringel-Hall algebras? Fq : a finite field with q elements;? Λ = Fq (Q): the path algebra of Q over Fq ;? L, M, N : three modules in mod-λ; L? gm N : the number of Λ-submodules W of L such that W ' N and L/W ' M in mod-λ; Page 7 of 22

8 Ringel-Hall algebras? Hq (Λ): the Ringel-Hall algebra of Λ over Q(vq ) (vq = q);? Basis: {u[m ] [M ] P} (P: the set of isomorphism classes of finite dimension Λ-modules);? Product: u[m ] u[n ] = vqhdimm,dimn i X L gm N u[l]. [L] P Page 8 of 22? Cq (Λ) := hu[si ] i: The composition algebra;

9 Ringel-Hall algebras? C (Q): the generic composition algebra of quiver Q over Q(v);? Theorem 1 (Green-Ringel) Let Q be a connected quiver without oriented cycles. Then the correspondence ui 7 θi induces an algebra isomorphism C (Q) f. Page 9 of 22

10 PBW-type basis of f : Finite type? H (Q) = C (Q);? Example: AuslanderõReiten quiver of Q = A3 ; (111) : $ u uu uu u u uu (011) : uu uu u uu uu (001) $ (110) : uu uu u uu uu (010) Page 10 of 22 $ (100)

11 PBW-type basis of f : Finite type? Theorem 2 (Lusztig, Ringel) (1) The set {hm i [M ] P} is a A-basis of CA (Q); (2) There exists a canonical basis E[M ] = hm i + X [M 0 ] [M ] with u[m 0 ][M ] v 1 Q[v 1 ]; u[m 0 ][M ] hm 0 i, Page 11 of 22 (3) {E[M ] [M ] P} = B.

12 PBW-type basis of f : Affine type AR-quiver of Q = {1 2} (12) G ; ww ww w w (01)... GG GG G# < zz zz z z zz (23)... O... O (33) O (33) O (22) O (22) O (11) (11)... (32) G < zz zz z z zz... GG GG G# (21) (10) ; ww ww w w Page 12 of 22

13 PBW-type basis of f : Affine type? P rep (resp. P rei): the iso-classes of indecomposable preprojective (resp. preinjective) modules;? Ti (i = 1,..., s): the non-homogeneous tubes with the period ri ;? NPf rep := {a : P rep N a is support-finite};? NPf rei := {b : P rei N b is support-finite};? Πai : the set of aperiodic ri -tuples of partitions;? M := {c = (ac, bc, πc, wc ) ac NPf rep, bc NPf rei, (π1c, π2c,..., πsc ) Πa1 Πa2 Πas, wc = (w1 wt ) is a partition}. Page 13 of 22 πc = w2

14 PBW-type basis of f : Affine type? For each c M there is an element in C (Q): ec := hm (ac )i Eπ1c Eπ2c Eπsc Swc δ hm (bc )i. c? Theorem 3 (Lin-Xiao-Zhang) (1) The set {e c M} is a Q(v)-basis of C (Q); (2) There exists a canonical basis E c = ec + X c0 c with uc0 c v 1 Q[v 1 ]; 0 uc0 c ec, Page 14 of 22 (3) {E c c M} = B.

15 Root category? Q: a connected tame quiver without oriented cycles;? A = kq: the path algebra of Q;? R(Q) = Db (A)/T 2 : the root category;? mod-a can be embedded into R(Q) as a full subcategory and R(Q) is divided into two parts: mod-a and T (mod-a). Page 15 of 22

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17 Root category? There is a bijection between the isomorphism classes of the indecomposable objects in root categories R = Db (A)/T 2 and the roots of the corresponding Lie algebra (Happle);? The root categories of finite dimensional hereditary algebras give a realization of all symmetrizable Kac-Moody algebra in a global way (PengXiao). Page 17 of 22

18 PBW-type basis of U 1λ? Ti (i = 1,..., 2s): all the non-homogeneous tubes in R(Q);? P: the indecomposable iso-classes in R(Q) excluding those in homogeneous tubes and non-homogeneous tubes;? M : the set of quadruplets c = (dc, πc, wc, wc 0 ), where dc NPf, πc = (π1c, π2c,..., π2sc ) Πa1 Πa2 Πa2s, wc = (w1 w2 wt ) and wc 0 = (w10 w20 wt0 ) are partitions. Page 18 of 22? f Q0 is another quiver such that R(Q) ' R(Q0 ), then they give the same M.

19 PBW-type basis of U 1λ? Consider the root category R corresponding to the quantized enveloping algebra U.? Fix Q such that R(Q) ' R.? There is a bijection ψq : M M M.? e c 1λ := ec1 + ec2 1λ, where (c1, c2 ) = ψq (c ), c M. Page 19 of 22? P BWQ (U 1λ ) := {e c 1λ c M } is a PBW-type basis of U 1λ.

20 A bar invariant basis of U 1λ? : an order on M = M M by c c 0 c1 c01 and c2 c02 but c 6= c 0.? A bar invariant basis BQ (U 1λ ) of U 1λ from P BWQ (U 1λ ): X 0 E c 1λ = e c 1λ + u c 0 c e c 1λ, c 0 c with u c 0 c v 1 Q[v 1 ]. Page 20 of 22? Theorem 4 BQ (U 1λ ) = {E c 1λ c M } = {b+ b0 1λ b, b0 B}.

21 A parameterization of the canonical basis of U 1λ? Theorem 5 There is a bijection ΨQ : M B λ given by c 7 e c 1λ 7 E c 1λ = E c1 + E c2 1λ 7 E c1 λ E c2, Page 21 of 22 where B λ is the canonical basis of U 1λ and (c1, c2 ) = ψq (c ).

22 Thank Youœ œ Page 22 of 22