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PRESENTING INFINITESIMAL q-SCHUR ALGEBRAS

Part of: Linear algebraic groups and related topics Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  13 January 2025

QIANG FU Open the ORCID record for QIANG FU [Opens in a new window]  and
CHENGQUAN SUN Open the ORCID record for CHENGQUAN SUN [Opens in a new window]
QIANG FU*
Affiliation:
School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Ministry of Education) Tongji University Shanghai, 200092 China
CHENGQUAN SUN
Affiliation:
School of Mathematical Sciences Tongji University Shanghai, 200092 China [email protected]
*
[email protected] [email protected]
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Abstract

Let be a commutative ring containing a primitive $l'$th root $\varepsilon $ of $1$. The infinitesimal q-Schur algebras over form an ascending chain of subalgebras of the q-Schur algebra , which are useful in studying representations of the Frobenius kernel of the associated quantum linear group. Let be the quantized enveloping algebra of $\mathfrak {gl}_n$ over . There is a natural surjective algebra homomorphism $\zeta _{d}$ from to . The map $\zeta _{d}$ restricts to a surjective algebra homomorphism $\zeta _{d,r}$ from to , where is a certain Hopf subalgebra of , which is closely related to Frobenius–Lusztig kernels of . We give the extra defining relations needed to define the infinitesimal q-Schur algebra as a quotient of . The map $\zeta _{d,r}$ induces a surjective algebra homomorphism , where is the modified quantum algebra associated with . We also give a generating set for the kernel of $\dot {\zeta }_{d,r}$. These results can be used to give a classification of irreducible -modules over a field of characteristic p.

Keywords

infinitesimal q-Schur algebraFrobenius–Lusztig kernelsthe quantized enveloping algebra of gln

MSC classification

Primary: 20G43: Schur and $q$-Schur algebras 17B37: Quantum groups (quantized enveloping algebras) and related deformations
Type
Article
Information
Nagoya Mathematical Journal , First View , pp. 1 - 22
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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