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PRESENTING INFINITESIMAL q-SCHUR ALGEBRAS
- Part of
-
- Journal:
- Nagoya Mathematical Journal , First View
- Published online by Cambridge University Press:
- 13 January 2025, pp. 1-22
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- Article
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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.
be a commutative ring containing a primitive 
over 
form an ascending chain of subalgebras of the 
, which are useful in studying representations of the Frobenius kernel of the associated quantum linear group. Let 
be the quantized enveloping algebra of 
. There is a natural surjective algebra homomorphism 
to 
. The map 
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 
as a quotient of 
. The map 
, where 
is the modified quantum algebra associated with 
. We also give a generating set for the kernel of 
-modules over a field of characteristic 