TY - JOUR

T1 - On solvability of the first hochschild cohomology of a finite-dimensional algebra

AU - EISELE, FLORIAN

AU - RAEDSCHELDERS, THEO

N1 - Funding Information:
Received by the editors April 26, 2019, and, in revised form, October 6, 2019. 2010 Mathematics Subject Classification. Primary 16E40, 16G10, 16G60. Key words and phrases. Hochschild cohomology, finite-dimensional algebras, Lie algebras, representation type. The second author was supported by an EPSRC postdoctoral fellowship EP/R005214/1.
Publisher Copyright:
© 2020 American Mathematical Society. All rights reserved.

PY - 2020

Y1 - 2020

N2 - For an arbitrary finite-dimensional algebra A A , we introduce a general approach to determining when its first Hochschild cohomology H H 1 ( A ) \mathrm {HH}^1(A) , considered as a Lie algebra, is solvable. If A A is, moreover, of tame or finite representation type, we are able to describe H H 1 ( A ) \mathrm {HH}^1(A) as the direct sum of a solvable Lie algebra and a sum of copies of s l 2 \mathfrak {sl}_2 . We proceed to determine the exact number of such copies, and give an explicit formula for this number in terms of certain chains of Kronecker subquivers of the quiver of A A . As a corollary, we obtain a precise answer to a question posed by Chaparro, Schroll, and Solotar.

AB - For an arbitrary finite-dimensional algebra A A , we introduce a general approach to determining when its first Hochschild cohomology H H 1 ( A ) \mathrm {HH}^1(A) , considered as a Lie algebra, is solvable. If A A is, moreover, of tame or finite representation type, we are able to describe H H 1 ( A ) \mathrm {HH}^1(A) as the direct sum of a solvable Lie algebra and a sum of copies of s l 2 \mathfrak {sl}_2 . We proceed to determine the exact number of such copies, and give an explicit formula for this number in terms of certain chains of Kronecker subquivers of the quiver of A A . As a corollary, we obtain a precise answer to a question posed by Chaparro, Schroll, and Solotar.

KW - Finite-dimensional algebras

KW - Hochschild cohomology

KW - Lie algebras

KW - Representation type

UR - http://www.scopus.com/inward/record.url?scp=85094866324&partnerID=8YFLogxK

U2 - 10.1090/TRAN/8064

DO - 10.1090/TRAN/8064

M3 - Article

AN - SCOPUS:85094866324

SN - 0002-9947

VL - 373

SP - 7607

EP - 7638

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 11

ER -