{"id":260650,"date":"2026-07-13T17:47:05","date_gmt":"2026-07-13T14:47:05","guid":{"rendered":"https:\/\/1kitap1.com\/en\/forty-years-of-algebraic-groups-jie-du\/"},"modified":"2026-07-13T17:47:05","modified_gmt":"2026-07-13T14:47:05","slug":"forty-years-of-algebraic-groups-jie-du","status":"publish","type":"post","link":"https:\/\/1kitap1.com\/en\/forty-years-of-algebraic-groups-jie-du\/","title":{"rendered":"Forty Years Of Algebraic Groups – Jie Du"},"content":{"rendered":"
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Referring to the multiplication formula in [8, Thm 8.1] (i,h,j) \u2208T ordered by \u22642 (aj,iEh+1,h)(0) (i,h,j) \u2208T ordered by \u22641 (ai, jEh,h+1)(0) = A(0) + B \u2208M(m|n)\u00b1 j\u2208Zm+n, B\u227aA 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- \ufb01ned as in (3.7) is the (super) subalgebra generated by Eh = Eh,h+1(0), Fh = Eh+1,h(0), = O(\u00b1ei), for all 1 \u2264h < m + n and 1 \u2264i \u2264m + n.<\/p>\n

Theorem 3.7 ([8, Thm 6.3, Thm 8.4]). There is an F-algebra isomorphism \u03b7: U\u03c5(glm|n) \u2212\u2192A(m|n) (3.15) sending Eh,h+1, Eh+1,h and K\u00b11 to Eh, Fh and K\u00b11 i , respectively. Thus the quantum enveloping superalgebra U\u03c5(glm|n) is isomorphic to the superalge- bra A(m|n). In particular, U\u03c5(glm|n) can be regarded as the F-superalgebra with the basis {A( j) | A \u2208M(m|n)\u00b1, j \u2208Zm+n}. In Section 2, the \u03c5-Schur superalgebra S v(m|n, r) is isomorphic to the image of U\u03c5(glm|n) over the representation \u03c1r. This relationship can be de- scribed more precisely as follows.<\/p>\n

Proposition 3.8 ([8, Cor 6.4]). There is an F-superalgebra homomor- phism \u03b7r : U\u03c5(glm|n) \u2212\u2192S v(m|n, r) sending Eh,h+1, Eh+1,h and K\u00b11 to Eh,h+1(0, r), Eh+1,h(0, r) and O(\u00b1ei, r), re- spectively. Haixia Gu and Zhongguo Zhou 3.2 The canonical bases for U\u03c5(glm|n) From now on, we can identify U\u03c5(glm|n) with A(m|n). Based on the multi- plication formulas over A(m|n), Du and the \ufb01rst author constructed a basis of the positive part (or negative part) of U\u03c5(glm|n) whose elements are sta- ble under an involution called \u201cbar involution\u201d(see [9]).<\/p>\n

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 U\u03c5(glm|1), this basis can deduce the bases of its \ufb01nite dimensional simple polynomial modules. The \u201cbar involution\u201d \u00af : U\u03c5(glm|n) \u2192U\u03c5(glm|n) is de\ufb01ned by \u00af\u03c5 = \u03c5\u22121, \u00afEh = Eh, \u00afFh = Fh, i = K\u2213 i , (3.16) where 1 \u2264h \u2264m + n \u22121 and 1 \u2264i \u2264m + n.<\/p>\n

For any A = (ai, j) \u2208M(m|n)\u00b1 and j \u2208Zm+n, let MA, j := (i,h,j) \u2208T ordered by \u22642 (aj,iEh+1,h)(0) \u00b7 O( j) \u00b7 (i,h,j) \u2208T ordered by \u22641 (ai, jEh,h+1)(0).<\/p>\n

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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.<\/p>\n

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\u2019s 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.<\/p>\n

Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Control Number: 2022040240 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. East China Normal University Scientific Reports \u2014 Vol. 16 FORTY YEARS OF ALGEBRAIC GROUPS, ALGEBRAIC GEOMETRY, AND REPRESENTATION THEORY IN CHINA In Memory of the Centenary Year of Xihua Cao\u2019s Birth Copyright \u00a9 2023 by World Scientific Publishing Co.<\/p>\n

Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA.<\/p>\n

In this case permission to photocopy is not required from the publisher.<\/p>\n<\/blockquote>\n

This is a short excerpt from the opening of “” by Unknown, quoted for review and introduction purposes. All rights belong to the copyright holders.<\/em><\/p>\n

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