k-Modules Over Linear Spaces by n-Linear Maps Admitting a Multiplicative Basis
Tóm tắt
We study the structure of certain k-modules 𝕍 over linear spaces 𝕎 with restrictions neither on the dimensions of 𝕍 and 𝕎 nor on the base field 𝔽. A basis
$\mathfrak {B} = \{v_{i}\}_{i\in I}$
of 𝕍 is called multiplicative with respect to the basis
$\mathfrak {B}^{\prime } = \{w_{j}\}_{j \in J}$
of 𝕎 if for any
$\sigma \in S_{n}, i_{1},\dots ,i_{k} \in I$
and
$j_{k + 1},\dots , j_{n} \in J$
we have
$[v_{i_{1}},\dots , v_{i_{k}}, w_{j_{k + 1}}, \dots , w_{j_{n}}]_{\sigma } \in \mathbb {F}v_{r_{\sigma }}$
for some rσ ∈ I. We show that if 𝕍 admits a multiplicative basis then it decomposes as the direct sum
$\mathbb {V} = \bigoplus _{\alpha } V_{\alpha }$
of well described k-submodules Vα each one admitting a multiplicative basis. Also the minimality of 𝕍 is characterized in terms of the multiplicative basis and it is shown that the above direct sum is by means of the family of its minimal k-submodules, admitting each one a multiplicative basis. Finally we study an application of k-modules with a multiplicative basis over an arbitrary n-ary algebra with multiplicative basis.
Tài liệu tham khảo
Abdesselam, B.: The twisted Heisenberg algebra e h,w(h(4)). J. Math. Phys. 38(12), 6045–6060 (1997)
Adashev, J.Q., Khudoyberdiyev, A.K.h., Omirov, B.A.: Classifications of some classes of Zinbiel algebras. J. Gen. Lie Theory Appl. 4, 10 (2010). Art. ID S090601
Albeverio, S., Ayupov, S.h.A., Omirov, B.A., Khudoyberdiyev, A.K.h.: n-Dimensional filiform Leibniz algebras of length (n − 1) and their derivations. J. Algebra 319(6), 2471–2488 (2008)
Bai, R., Wu, Y.: Constructions of 3-Lie algebras. Linear Multilinear Algebra 63(11), 2171–2186 (2015)
Balogh, Z.: Further results on a filtered multiplicative basis of group algebras. Math. Commun. 12(2), 229–238 (2007)
Bautista, R., Gabriel, P., Roiter, A.V., Salmeron, L.: Representation-finite algebras and multiplicative basis. Invent. Math. 81, 217–285 (1985)
Beites, P., Kaygorodov, I., Popov, Yu.: Generalized derivations of multiplicative n-ary Hom-O color algebras. Bulletin of the Malaysian Mathematical Sciences Society. https://doi.org/10.1007/s40840-017-0486-8 (2017)
Bovdi, V.: On a filtered multiplicative basis of group algebras. Arch. Math. (Basel) 74(2), 81–88 (2000)
Bovdi, V.: On a filtered multiplicative bases of group algebras. II. Algebr. Represent. Theory 6(3), 353–368 (2003)
Bovdi, V., Grishkov, A., Siciliano, S.: Filtered multiplicative bases of restricted enveloping algebras. Algebr. Represent. Theory 14(4), 601–608 (2011)
Bovdi, V., Grishkov, A., Siciliano, S.: On filtered multiplicative bases of some associative algebras. Algebr. Represent. Theory. 18(2), 297–306 (2015)
Calderón, A.: On split Lie algebras with a symmetric root system. Proc. Indian Acad. Sci. Math. Sci. 118(3), 351–356 (2008)
Calderón, A.J., Navarro, F.J., Sánchez, J.M.: n-Algebras admitting a multiplicative basis. J. Algebra Appl. 16(11), 11 (2018). 1850025
Calderón, A.J., Navarro, F.J., Sánchez, J.M.: Modules over linear spaces admitting a multiplicative basis. Linear Multilinear Algebra 65(1), 156–165 (2017)
Calderón, A.J., Navarro, F.J.: Arbitrary algebras with a multiplicative basis. Linear Algebra Appl. 498(1), 106–116 (2016)
Chu, Y.J., Huang, F., Zheng, Z.J.: A commutant of ß γ-system associated to the highest weight module v 4 of \(sl(2,\mathbb {C})\). J. Math. Phys. 51(9,092301), 32 (2010)
De la Mora, C., Wojciechowski, P.J.: Multiplicative bases in matrix algebras. Linear Algebra Appl. 419(2-3), 287–298 (2006)
Ding, L., Jia, X., Zhang W.: On infinite-dimensional 3-Lie algebras. J. Math. Phys. 55, 041704 (2014)
Dimitrov, I., Futorny, V., Penkov, I.: A reduction theorem for highest weight modules over toroidal Lie algebras. Comm. Math. Phys. 250(1), 47–63 (2004)
Dzhumadil’daev, A.S.: Representations of vector product n-Lie algebras. Comm. Algebra 32(9), 3315–3326 (2004)
Filippov, V.T.: n-Lie algebras. Sib. Mat. Zh. 26(6), 126–140 (1985)
Grantcharov, D., Jung, J.H., Kang, S.J., Kim, M.: Highest weight modules over quantum queer superalgebra u q(q(n)). Comm. Math. Phys. 296(3), 827–860 (2010)
Iohara, K.: Unitarizable highest weight modules of the N = 2 super Virasoro algebras: untwisted sectors. Lett. Math. Phys. 91(3), 289–305 (2010)
Kaygorodov, I., Popov, Y.: Generalized derivations of (color) n-ary algebras. Linear Multilinear Algebra 64(6), 1086–1106 (2016)
Kupisch, H., Waschbusch, J.: On multiplicative basis in quasi-Frobenius algebras. Math. Z. 186, 401–405 (1984)
Liu, D., Gao, S., Zhu, L.: Classification of irreducible weight modules over W-algebra W(2, 2). J. Math. Phys. 49(1), 6 (2008). 113503
Makhlouf, A.: Hom-alternative algebras and Hom-Jordan algebras. Int. Electron. J. Algebra 8, 177–190 (2010)
Michor, P.W., Vinogradov, A.M.: Lie n-ary Associative algebras. Geometrical structures for physical theories, II (Vietri, 1996). Rend. Sem. Mat. Univ. Politec. Torino 54(4), 373-392 (1996)
Pozhidaev, A.P.: Monomial n-Lie algebras. Algebra and Logic 37(5), 307–322 (1998)
Pozhidaev, A.P.: Simple factor algebras and subalgebras of algebras of Jacobians. Siberian Math. J. 39(3), 512–517 (1998)
Pozhidaev, A.P.: On simple n-Lie algebras. Algebra and Logic 38(3), 181–192 (1999)
Pozhidaev, A.P.: Two classes of central simple n-Lie algebras. Siberian Math. J. 40(6), 1112–1118 (1999)
Ren, B., Ji Meng, D.: Some two steps nilpotent Lie algebras I. Linear Algebra Appl. 338, 77–98 (2001)
Roiter, A.V., Sergeichuk, V.V.: Existence of a multiplicative basis for finitely spaced module over an aggregate. Ukrainian Math. J. 46(5), 604–617 (1995)
Sagle, A.A.: On simple extended Lie algebras over fields of characteristic zero. Pacific J. Math. 15(2), 621–648 (1965)
Takemura, K.: The decomposition of level-1 irreducible highest-weight modules with respect to the level-0 actions of the quantum affine algebra. J. Phys. A 31 5, 1467–1485 (1998)
Zapletal, A.: Difference equations and highest-weight modules of U q[sl(n)]. J. Phys A 31(47), 9593–9600 (1998)