TY - JOUR
T1 - Verbally prime T-ideals and graded division algebras
AU - Aljadeff, Eli
AU - Karasik, Yaakov
N1 - Publisher Copyright:
© 2018 Elsevier Inc.
PY - 2018/7/9
Y1 - 2018/7/9
N2 - Let F be an algebraically closed field of characteristic zero and let G be a finite group. We consider graded Verbally prime T-ideals in the free G-graded algebra. It turns out that equivalent definitions in the ordinary case (i.e. ungraded) extend to nonequivalent definitions in the graded case, namely verbally prime G-graded T-ideals and strongly verbally prime T-ideals. At first, following Kemer's ideas, we classify G-graded verbally prime T-ideals. The main bulk of the paper is devoted to the stronger notion. We classify G-graded strongly verbally prime T-ideals which are T-ideal of affine G-graded algebras or equivalently G-graded T-ideals that contain a Capelli polynomial. It turns out that these are precisely the T-ideal of G-graded identities of finite dimensional G-graded, central over F (i.e. Z(A)e=F) which admit a G-graded division algebra twisted form over a field k which contains F or equivalently over a field k which contains enough roots of unity (e.g. a primitive n-root of unity where n=ord(G)).
AB - Let F be an algebraically closed field of characteristic zero and let G be a finite group. We consider graded Verbally prime T-ideals in the free G-graded algebra. It turns out that equivalent definitions in the ordinary case (i.e. ungraded) extend to nonequivalent definitions in the graded case, namely verbally prime G-graded T-ideals and strongly verbally prime T-ideals. At first, following Kemer's ideas, we classify G-graded verbally prime T-ideals. The main bulk of the paper is devoted to the stronger notion. We classify G-graded strongly verbally prime T-ideals which are T-ideal of affine G-graded algebras or equivalently G-graded T-ideals that contain a Capelli polynomial. It turns out that these are precisely the T-ideal of G-graded identities of finite dimensional G-graded, central over F (i.e. Z(A)e=F) which admit a G-graded division algebra twisted form over a field k which contains F or equivalently over a field k which contains enough roots of unity (e.g. a primitive n-root of unity where n=ord(G)).
KW - Graded algebras
KW - Graded division algebras
KW - Polynomial identities
KW - Verbally prime
UR - http://www.scopus.com/inward/record.url?scp=85047093051&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2018.05.004
DO - 10.1016/j.aim.2018.05.004
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AN - SCOPUS:85047093051
SN - 0001-8708
VL - 332
SP - 142
EP - 175
JO - Advances in Mathematics
JF - Advances in Mathematics
ER -