Abstract
In this chapter, we will be primarily interested in the question of which associative algebras, viewed either as Lie algebras under the Lie product or as Jordan algebras under the Jordan product, are zero product determined. We will see that the results in the Jordan case are similar to those in the associative case, while the Lie case is quite different. For example, a unital associative algebra (over a field of characteristic not 2) that is generated by idempotents is zpd when considered as a Jordan algebra, but not necessarily when considered as a Lie algebra. On the other hand, it is easy to find examples of associative algebras that are zpd as Lie algebras, but not as associative algebras.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
J. Alaminos, M. Brešar, J. Extremera, A.R. Villena, Zero Lie product determined Banach algebras. Studia Math. 239, 189–199 (2017)
G. An, J. Li, J. He, Zero Jordan product determined algebras. Linear Algebra Appl. 475, 90–93 (2015)
M. Brešar, Introduction to Noncommutative Algebra. Universitext (Springer, Berlin, 2014)
M. Brešar, P. Šemrl, On bilinear maps on matrices with applications to commutativity preservers. J. Algebra 301, 803–837 (2006)
M. Brešar, M. Grašič, J. Sanchez, Zero product determined matrix algebras. Linear Algebra Appl. 430, 1486–1498 (2009)
M. Brešar, X. Guo, G. Liu, R. Lü, K. Zhao, Zero product determined Lie algebras. European J. Math. 5, 424–453 (2019)
H. Ghahramani, On derivations and Jordan derivations through zero products. Oper. Matrices 8, 759–771 (2014)
M. Grašič, Zero product determined classical Lie algebras. Linear Multilinear Algebra 58, 1007–1022 (2010)
M. Grašič, Zero product determined Jordan algebras, I. Linear Multilinear Algebra 59, 741–757 (2011)
M. Grašič, Zero product determined Jordan algebras, II. Algebra Colloq. 22, 109–118 (2015)
W. Hu, Z. Xiao, A characterization of algebras generated by idempotents. J. Pure Appl. Algebra 225, 106693 (2021)
M.T. Kosan, T.-K. Lee, Y. Zhou, Bilinear forms on matrix algebras vanishing on zero products of xy and yx. Linear Algebra Appl. 453, 110–124 (2014)
X. Ma, G. Ding, L. Wang, Square-zero determined matrix algebras. Linear Multilinear Algebra 59, 1311–1317 (2011)
P. Moravec, Unramified Brauer groups of finite and infinite groups. Amer. J. Math. 134, 1679–1704 (2012)
D. Wang, X. Yu, Z. Chen, A class of zero product determined Lie algebras. J. Algebra 331, 145–151 (2011)
D. Wang, X. Yu, Z. Chen, Maps determined by action on square-zero elements. Comm. Algebra 40, 4255–4262 (2012)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Brešar, M. (2021). Zero Lie/Jordan Product Determined Algebras. In: Zero Product Determined Algebras. Frontiers in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-80242-4_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-80242-4_3
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-80241-7
Online ISBN: 978-3-030-80242-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)