Lie groups, Lie algebras and their representation theories are important parts of mathematical physics. They play a crucial character in symmetries. As a generalization of Lie algebra, Lie superalgebras are from the comprehending and description for supersymmetry of mathematical physics. Unlike the semisimple Lie algebras, understanding the representation theory of Lie superalgebras is a difficult problem. Lie superalgebras graded by root supersystems are Lie superalgebras of great significance. In recent years, the representations of types B(m,n), C(n), D(m,n), P(n) and Q(n)-graded Lie superalgebras coordinatized by quantum tori have been studied. In this paper, we construct fermionic-bosonic representations for a class of A(M-1,N-1)-graded Lie superalgebras coordinatized by quantum tori with nontrivial central extensions. At first, we introduce the background of the research on the graded Lie superalgebras and present some basics on it. Then, a set of bases for A(M-1,N-1)-graded Lie superalgebras and the multiplication operations among them are given specifically to present the construction of the vector space. By using the tensor product of fermionic and bosonic module, the operators and their operation relations are derived. Finally, we obtain a brief and pretty representation theorem of A(M-1,N-1)-graded Lie superalgebras with nontrivial central extensions.
Published in | American Journal of Applied Mathematics (Volume 9, Issue 2) |
DOI | 10.11648/j.ajam.20210902.12 |
Page(s) | 44-51 |
Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
Copyright |
Copyright © The Author(s), 2021. Published by Science Publishing Group |
Graded Lie Superalgebras, Quantum Tori, Fermionic-Bosonic Representations, Nontrivial Central Extensions
[1] | Varadarajan V S. (2004). Supersymmetry for mathematicians. New York: American Mathematical Society. |
[2] | Berman S, Moody R V. (1992). Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy. Inventiones mathematicae, 108 (1): 323-347. |
[3] | Benkart G, Zelmanov E. (1996). Lie algebras graded by finite root systems and intersection matrix algebras. Inventiones mathematicae, 126 (1): 1-45. |
[4] | Neher E. (1996). Lie algebras graded by 3-graded root systems and Jordan pairs covered by grids. American Journal of Mathematics. 118 (2): 439-491. |
[5] | Allison B N, Benkart G, Gao Y. (2000). Central extensions of Lie algebras graded by finite root systems. Mathematische Annalen. 316 (3): 499-527. |
[6] | Kac V G. (1977). Lie superalgebras. Advances in Math, 26 (1): 8-96. |
[7] | P Benkart G, Elduque A. (2002). Lie superalgebras graded by the root systems C(n), D(m,n), D(2,1;α), F(4), G(3). Canad Math Bull, 45 (4): 509-524. 0. |
[8] | Benkart G, Elduque A. (2002). Lie superalgebras graded by the root system A(m,n). |
[9] | Benkart G, Elduque A. (2003). Lie superalgebras graded by the root system B(m,n). Selecta Mathematica, New Series, 9: 313-360. |
[10] | Gao Y. (2002). Fermionic and bosonic representations of the extended affine Lie algebra gl_{N}(C_{q}) Canadian Mathematical Bulletin, 45 (4): 623-633. |
[11] | Lau M. (2005). Bosonic and fermionic representations of Lie algebra central extensions. Advances in Mathematics, 194 (2): 225-245. |
[12] | Chen H, Gao Y. (2007). BC_{N}-graded Lie algebras arising from fermionic representations. J Algebra, 308 (2): 1740-1752. |
[13] | Chen H, Gao Y. (2006). B(0,N)-graded Lie superalgebras coordinatized by quantum tori. Sci China Ser A, 49: 545-566 2. |
[14] | Cheng J. (2016). Generalized B(m,n), C(n), D(m,n)-graded Lie superalgebras arising from fermionic-bosonic representations. Frontiers of Mathematics in China, 11 (6): 1451-1470. |
[15] | Cheng J, Gao Y. Generalized P(N)-graded Lie superalgebras. submitted. |
[16] | Cheng J. (2016). Q(N)-graded Lie superalgebras arising from fermionic-bosonic representations. Pacific Journal of Mathematics, 283 (1): 63-74. |
[17] | Manin Y I. (1991). Topics in noncommutative geometry. M. B. Porter Lectures. Princeton: Princeton University Press. |
[18] | Berman S, Gao Y, Krylyuk Y. (1996). Quantum tori and the structure of elliptic quasi-simple Lie algebras. J Funct Anal, 135 (2): 339-389. |
[19] | Feingold A J, Frenkel I B. (1985). Classical affine algebras. Adv Math, 56 (2): 117-172. |
APA Style
Lingyu Yu. (2021). The Fermionic-bosonic Representations of A(M-1,N-1)-graded Lie Superalgebras. American Journal of Applied Mathematics, 9(2), 44-51. https://doi.org/10.11648/j.ajam.20210902.12
ACS Style
Lingyu Yu. The Fermionic-bosonic Representations of A(M-1,N-1)-graded Lie Superalgebras. Am. J. Appl. Math. 2021, 9(2), 44-51. doi: 10.11648/j.ajam.20210902.12
AMA Style
Lingyu Yu. The Fermionic-bosonic Representations of A(M-1,N-1)-graded Lie Superalgebras. Am J Appl Math. 2021;9(2):44-51. doi: 10.11648/j.ajam.20210902.12
@article{10.11648/j.ajam.20210902.12, author = {Lingyu Yu}, title = {The Fermionic-bosonic Representations of A(M-1,N-1)-graded Lie Superalgebras}, journal = {American Journal of Applied Mathematics}, volume = {9}, number = {2}, pages = {44-51}, doi = {10.11648/j.ajam.20210902.12}, url = {https://doi.org/10.11648/j.ajam.20210902.12}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20210902.12}, abstract = {Lie groups, Lie algebras and their representation theories are important parts of mathematical physics. They play a crucial character in symmetries. As a generalization of Lie algebra, Lie superalgebras are from the comprehending and description for supersymmetry of mathematical physics. Unlike the semisimple Lie algebras, understanding the representation theory of Lie superalgebras is a difficult problem. Lie superalgebras graded by root supersystems are Lie superalgebras of great significance. In recent years, the representations of types B(m,n), C(n), D(m,n), P(n) and Q(n)-graded Lie superalgebras coordinatized by quantum tori have been studied. In this paper, we construct fermionic-bosonic representations for a class of A(M-1,N-1)-graded Lie superalgebras coordinatized by quantum tori with nontrivial central extensions. At first, we introduce the background of the research on the graded Lie superalgebras and present some basics on it. Then, a set of bases for A(M-1,N-1)-graded Lie superalgebras and the multiplication operations among them are given specifically to present the construction of the vector space. By using the tensor product of fermionic and bosonic module, the operators and their operation relations are derived. Finally, we obtain a brief and pretty representation theorem of A(M-1,N-1)-graded Lie superalgebras with nontrivial central extensions.}, year = {2021} }
TY - JOUR T1 - The Fermionic-bosonic Representations of A(M-1,N-1)-graded Lie Superalgebras AU - Lingyu Yu Y1 - 2021/04/10 PY - 2021 N1 - https://doi.org/10.11648/j.ajam.20210902.12 DO - 10.11648/j.ajam.20210902.12 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 44 EP - 51 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20210902.12 AB - Lie groups, Lie algebras and their representation theories are important parts of mathematical physics. They play a crucial character in symmetries. As a generalization of Lie algebra, Lie superalgebras are from the comprehending and description for supersymmetry of mathematical physics. Unlike the semisimple Lie algebras, understanding the representation theory of Lie superalgebras is a difficult problem. Lie superalgebras graded by root supersystems are Lie superalgebras of great significance. In recent years, the representations of types B(m,n), C(n), D(m,n), P(n) and Q(n)-graded Lie superalgebras coordinatized by quantum tori have been studied. In this paper, we construct fermionic-bosonic representations for a class of A(M-1,N-1)-graded Lie superalgebras coordinatized by quantum tori with nontrivial central extensions. At first, we introduce the background of the research on the graded Lie superalgebras and present some basics on it. Then, a set of bases for A(M-1,N-1)-graded Lie superalgebras and the multiplication operations among them are given specifically to present the construction of the vector space. By using the tensor product of fermionic and bosonic module, the operators and their operation relations are derived. Finally, we obtain a brief and pretty representation theorem of A(M-1,N-1)-graded Lie superalgebras with nontrivial central extensions. VL - 9 IS - 2 ER -