Free bicommutative superalgebras
Drensky V. Ismailov N. Mustafa M. Zhakhayev B.
15 August 2024Academic Press Inc.
Journal of Algebra
2024#652158 - 187 pp.
We introduce the variety Bsup of bicommutative superalgebras over an arbitrary field of characteristic different from 2. The variety consists of all nonassociative Z2-graded algebras satisfying the polynomial super-identities of super- left- and right-commutativity x(yz)=(−1)x‾y‾y(xz) and (xy)z=(−1)y‾z‾(xz)y, where u‾∈{0,1} is the parity of the homogeneous element u. We present an explicit construction of the free bicommutative superalgebras, find their bases as vector spaces and show that they share many properties typical for ordinary bicommutative algebras and super-commutative associative superalgebras. In particular, in the case of free algebras of finite rank we compute the Hilbert series and find explicitly its coefficients. As a consequence we give a formula for the codimension sequence. We establish an analogue of the classical Hilbert Basissatz for two-sided ideals. We see that the Gröbner-Shirshov bases of these ideals are finite, the Gelfand-Kirillov dimensions of finitely generated bicommutative superalgebras are nonnegative integers and the Hilbert series of finitely generated graded bicommutative superalgebras are rational functions. Concerning problems studied in the theory of varieties of algebraic systems, we prove that the variety of bicommutative superalgebras satisfies the Specht property. In the case of characteristic 0 we compute the sequence of cocharacters.
Bicommutative superalgebras , Cocharacters , Codimensions , Free bicommutative superalgebras , Gröbner-Shirshov basis , Hilbert Basissatz , Hilbert series , Specht property
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Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, 1113, Bulgaria
Astana IT University, Astana, 010000, Kazakhstan
SDU University, Kaskelen, Kazakhstan
Department of Mathematics, School of Sciences and Humanities, Nazarbayev University, 53 Qabanbay Batyr Avenue, Astana, 010000, Kazakhstan
SDU University, Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Institute of Mathematics and Informatics
Astana IT University
SDU University
Department of Mathematics
SDU University
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