IJPAM: Volume 81, No. 1 (2012)

COMPUTING CONSTANTS IN SOME WEIGHT SUBSPACES
OF FREE ASSOCIATIVE COMPLEX ALGEBRA

Milena Sosic
Department of Mathematics
University of Rijeka
2, Radmile Matejcic, Rijeka 51000, CROATIA


Abstract. Let ${{\mathcal{N}}=\{i_{1},i_{2},\dots, i_{N}\}}$ be a fixed subset of nonnegative integers and let $q_{ij}$, ${i,j\in \mathcal{N}}$ be given complex numbers. We consider a free unital associative complex algebra ${\mathcal{B}}$ generated by $N$ generators ${\{e_{i}\}_{i\in{\mathcal{N}}}}$ (each of degree one) together with $N$ linear operators ${\partial_{i}\colon {\mathcal{B}} \to {\mathcal{B}}}$, ${i\in \mathcal{N}}$ that act as twisted derivations on ${\mathcal{B}}$. The algebra ${\mathcal{B}}$ is graded by total degree. More generally ${\mathcal{B}}$ could be considered as multigraded. Then it has a direct sum decomposition into multigraded (weight) subspaces ${\mathcal{B}}_{Q}$, where $Q$ runs over multisets (over ${\mathcal{N}}$). An element $C$ in ${\mathcal{B}}$ is called a constant if it is annihilated by all operators ${\partial_{i}}$. Then the fundamental problem is to describe the space ${\mathcal{C}}$ of all constants in algebra ${\mathcal{B}}$. The space ${\mathcal{C}}$ also inherits the direct sum decomposition into multigraded subspaces ${{\mathcal{C}}_{Q}={\mathcal{B}}_{Q}\cap{\mathcal{C}}}$. Thus it is enough to determine the finite dimensional spaces ${\mathcal{C}}_{Q}$.

Received: August 14, 2012

AMS Subject Classification: 05Exx

Key Words and Phrases: q-algebras, noncommutative polynomial algebras, twisted derivations

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 81
Issue: 1