Strong conjugation actions induced by flags of exact Borel subalgebras |
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Authors: | Email author" target="_blank">Yuehui?ZhangEmail author |
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Institution: | Department of Mathematics, Shanghai Jiaotong University, Shanghai 200030, China |
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Abstract: | Let A be a quasi-hereditary algebra with a strong exact Borel subalgebra. It is proved that for any standard semisimple subalgebra
T there exists an exact Borel subalgebra B of A such that T is a maximal semisimple subalgebra of B. It is shown that the
maximal length of flags of exact Borel subalgebras of A is the difference of the radium and the rank of Grothendic group of
A plus 2. The number of conjugation-classes of exact Borel subalgebras is 1 if and only if A is basic; the number is 2 if
and only if A is semisimple. For all other cases, this number is 0 or no less than 3. Furthermore, it is shown that all the
exact Borel subalgebras are idempotent-conjugate to each other, that is, for any exact Borel subalgebras B and C of A, there
exists an idempotent e of A, and an invertible element u of A, such that eBe = u-1eCeu. |
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Keywords: | quasi-hereditary algebra exact Borel subalgebra strong conjugation idempotent-conjugate Morita equivalence |
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