Forty Years Of Algebraic Groups – Jie Du

Referring to the multiplication formula in [8, Thm 8.1] (i,h,j) ∈T ordered by ≤2 (aj,iEh+1,h)(0) (i,h,j) ∈T ordered by ≤1 (ai, jEh,h+1)(0) = A(0) + B ∈M(m|n)± j∈Zm+n, B≺A gB,A, jB( j), (3.14) the following theorem yields. Theorem 3.6 ([8, Thm 8.1]). The (super) subspace A(m|n) of S (m|n) de- fined as in (3.7) is the (super) subalgebra generated by Eh = Eh,h+1(0), Fh = Eh+1,h(0), = O(±ei), for all 1 ≤h < m + n and 1 ≤i ≤m + n.

Theorem 3.7 ([8, Thm 6.3, Thm 8.4]). There is an F-algebra isomorphism η: Uυ(glm|n) −→A(m|n) (3.15) sending Eh,h+1, Eh+1,h and K±1 to Eh, Fh and K±1 i , respectively. Thus the quantum enveloping superalgebra Uυ(glm|n) is isomorphic to the superalge- bra A(m|n). In particular, Uυ(glm|n) can be regarded as the F-superalgebra with the basis {A( j) | A ∈M(m|n)±, j ∈Zm+n}. In Section 2, the υ-Schur superalgebra S v(m|n, r) is isomorphic to the image of Uυ(glm|n) over the representation ρr. This relationship can be de- scribed more precisely as follows.

Proposition 3.8 ([8, Cor 6.4]). There is an F-superalgebra homomor- phism ηr : Uυ(glm|n) −→S v(m|n, r) sending Eh,h+1, Eh+1,h and K±1 to Eh,h+1(0, r), Eh+1,h(0, r) and O(±ei, r), re- spectively. Haixia Gu and Zhongguo Zhou 3.2 The canonical bases for Uυ(glm|n) From now on, we can identify Uυ(glm|n) with A(m|n). Based on the multi- plication formulas over A(m|n), Du and the first author constructed a basis of the positive part (or negative part) of Uυ(glm|n) whose elements are sta- ble under an involution called “bar involution”(see [9]).

And this basis is called canonical bases in non-super case, and is equivalent to the pseudo- canonical bases described in [3]. In the special case Uυ(glm|1), this basis can deduce the bases of its finite dimensional simple polynomial modules. The “bar involution” ¯ : Uυ(glm|n) →Uυ(glm|n) is defined by ¯υ = υ−1, ¯Eh = Eh, ¯Fh = Fh, i = K∓ i , (3.16) where 1 ≤h ≤m + n −1 and 1 ≤i ≤m + n.

For any A = (ai, j) ∈M(m|n)± and j ∈Zm+n, let MA, j := (i,h,j) ∈T ordered by ≤2 (aj,iEh+1,h)(0) · O( j) · (i,h,j) ∈T ordered by ≤1 (ai, jEh,h+1)(0).

This book series reports valuable research results and progress in scientific and related areas. Mainly contributed by the distinguished professors of the East China Normal University, it will cover a number of research areas in pure mathematics, financial mathematics, applied physics, computer science, environmental science, geography, estuarine and coastal science, education information technology, etc.

Subseries of Symposia and Topic Studies Published Vol. 16 Forty Years of Algebraic Groups, Algebraic Geometry, and Representation Theory in China: In Memory of the Centenary Year of Xihua Cao’s Birth edited by Jie Du (University of New South Wales, Australia), Jianpan Wang (East China Normal University, China) and Lei Lin (East China Normal University, China) Subseries on Educational Information Technology Published Vol. 15 Achieving Greater Educational Impact through Data Intelligence: Practice, Challenges and Expectations of Education by Bian Wu (East China Normal University, China), Yiling Hu (East China Normal University, China) and Xiaoqing Gu (East China Normal University, China) More information on this series can also be found at https://www.worldscientific.com/series/ecnusr (Continued at end of book) Published by World Scientific Publishing Co.

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