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Identities for the special linear Lie algebra with the Pauli and Cartan gradings
Israel Journal of Mathematics ( IF 1 ) Pub Date : 2021-01-15 , DOI: 10.1007/s11856-021-2093-5
Claudemir Fidelis , Diogo Diniz , Franciélia Limeira de Sousa

Let $$\mathbb{K}$$ K be a field of characteristic zero. We study the graded identities of the special linear Lie algebra with the Pauli and Cartan gradings. Given a prime number p we provide a finite basis for the graded identities of $$s{l_p}\left(\mathbb{K}\right)$$ s l p ( K ) with the Pauli grading by the group ℤ p × ℤ p and compute its graded codimensions. We also prove that $${{\mathop{\rm var}} ^{{\mathbb{Z}_p} \times {\mathbb{Z}_p}}}\left( {s{l_p}\left(\mathbb{K} \right)} \right)$$ var ℤ p × ℤ p ( s l p ( K ) ) is a minimal variety and satisfies the Specht property. As a by-product we determine a basis for the identities of certain graded Lie algebras with a grading in which every homogeneous subspace has dimension ≤ 1. For $$s{l_m}\left(\mathbb{K}\right)$$ s l m ( K ) with the Cartan grading a finite basis for the graded identities is determined, moreover a basis for the subspace of the multilinear polynomials in the relatively free algebra $$L\left\langle {{X_G}} \right\rangle /{T_G}\left( {s{l_m}\left( \right)} \right)$$ L 〈 X G 〉 / T G ( s l m ( K ) ) , as a vector space, is exhibited. As a consequence we compute the graded codimensions for m = 2 and provide bases for the graded identities and for the subspace of the multilinear polynomials in the relatively free algebra of certain Lie subalgebras of $${M_m}{\left(\mathbb{K}\right)^{\left( - \right)}}$$ M m ( K ) ( − ) with the Cartan grading.

中文翻译:

具有泡利和嘉当分级的特殊线性李代数的恒等式

令 $$\mathbb{K}$$ K 为特征为零的域。我们用泡利和嘉当分级研究特殊线性李代数的分级恒等式。给定一个素数 p,我们为 $$s{l_p}\left(\mathbb{K}\right)$$ slp ( K ) 的分级恒等式提供了一个有限基,其中的泡利分级由群 ℤ p × ℤ p并计算其分级的codimensions。我们还证明 $${{\mathop{\rm var}} ^{{\mathbb{Z}_p} \times {\mathbb{Z}_p}}}\left( {s{l_p}\left(\ mathbb{K} \right)} \right)$$ var ℤ p × ℤ p ( slp ( K ) ) 是一个极小变体并且满足 Specht 性质。作为副产品,我们确定了某些分级李代数的恒等式的基础,其中每个齐次子空间的维数都≤ 1。对于具有嘉当分级的 $$s{l_m}\left(\mathbb{K}\right)$$ slm ( K ),确定分级恒等式的有限基,此外还确定了相对自由的代数 $$L\left\langle {{X_G}} \right\rangle /{T_G}\left( {s{l_m}\left( \right)} \right)$$ L 〈 XG 〉 / TG ( slm ( K ) ) 作为向量空间被展示出来。因此,我们计算 m = 2 的分级余维,并为分级恒等式和在 $${M_m}{\left(\mathbb{K }\right)^{\left( - \right)}}$$ M m ( K ) ( − ) 与 Cartan 分级。而且是相对自由代数中多重线性多项式的子空间的基 $$L\left\langle {{X_G}} \right\rangle /{T_G}\left( {s{l_m}\left( \right)} \right)$$ L 〈 XG 〉 / TG ( slm ( K ) ) 作为向量空间被展示。因此,我们计算 m = 2 的分级余维,并为分级恒等式和在 $${M_m}{\left(\mathbb{K }\right)^{\left( - \right)}}$$ M m ( K ) ( − ) 与 Cartan 分级。而且是相对自由代数中多重线性多项式的子空间的基 $$L\left\langle {{X_G}} \right\rangle /{T_G}\left( {s{l_m}\left( \right)} \right)$$ L 〈 XG 〉 / TG ( slm ( K ) ) 作为向量空间被展示。因此,我们计算 m = 2 的分级余维,并为分级恒等式和在 $${M_m}{\left(\mathbb{K }\right)^{\left( - \right)}}$$ M m ( K ) ( − ) 与 Cartan 分级。
更新日期:2021-01-15
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