On the representations of certain graded tensor products

In this article we obtain a tensor product theorem for a certain class of associative algebras in the braided category of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -graded vector spaces over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a finite abelian group. The tensor product theorem describes the irreducible modules of the tensor product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -graded associative algebras in this class as the tensor product of the irreducibles of each of its constituents. Applying this general result to the universal enveloping algebras of basic classical Lie superalgebras or the queer Lie superalgebra or even certain Lie color algebras, we get an extension of a result of Yousofzadeh [J. Algebra 606 (2022), pp. 19–29] to this larger class of algebras. Along the way, we also extend to the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -graded setting the results of Lemire [Proc. Amer. Math. Soc. 22 (1969), pp. 192–197] which relates the irreducibility of a representation of a semi-simple Lie algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {L}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to the irreducibility of its weight spaces considered as modules for the centralizer of the Cartan subalgebra in the universal enveloping algebra of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {L}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We prove it in a more general setting of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -graded associative algebras with a <italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -compatible </italic> grading by another abelian group. In the case of the universal enveloping algebras of Lie superalgebras and Lie color algebras belonging to the class mentioned earlier, these results give us the analogues of Lemire’s results.