arXiv subjects

A. A. Provorov

Publications and source records attributed to A. A. Provorov.

4 recordsLinked to original sources

arXiv2026

Split Casimir Operator of the Lie Algebra so(2r) in Spinor Representations, Colour Factors and Yang-Baxter Equation

In this paper, we derive characteristic identities for the split Casimir operator of the Lie algebra $so(2r)$ in tensor products of spinor representations of the same and opposite chiralities. Using these identities, we explicitly construct projectors onto invariant subspaces of this operator and compute their traces. The results obtained allow us to derive explicit expressions for the colour factors of ladder Feynman diagrams in gauge theories with gauge group $Spin(2r)$. In addition, we obtain a new form of a solution to the Yang-Baxter equation that is invariant under the action of the Lie algebra $so(2r)$ in spinor representations.

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arXiv2023

Split Casimir operator for simple Lie algebras in the cube of $\mathsf{ad}$-representation and Vogel parameters

We constructed characteristic identities for the 3-split (polarized) Casimir operators of simple Lie algebras in the adjoint representations $\mathsf{ad}$ and deduced a certain class of subrepresentations in $\mathsf{ad}^{\otimes 3}$. The projectors onto invariant subspaces for these subrepresentations were directly constructed from the characteristic identities for the 3-split Casimir operators. For all simple Lie algebras, universal expressions for the traces of higher powers of the 3-split Casimir operators were found and dimensions of the subrepresentations in $\mathsf{ad}^{\otimes 3}$ were calculated. All our formulas are in agreement with the universal description of (irreducible) subrepresentations in $\mathsf{ad}^{\otimes 3}$ for simple Lie algebras in terms of the Vogel parameters.

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arXiv2022

The split Casimir operator and solutions of the Yang-Baxter equation for the $osp(M|N)$ and $s\ell(M|N)$ Lie superalgebras, higher Casimir operators, and the Vogel parameters

We find the characteristic identities for the split Casimir operator in the defining and adjoint representations of the $osp(M|N)$ and $s\ell(M|N)$ Lie superalgebras. These identities are used to build the projectors onto invariant subspaces of the representation $T^{\otimes 2}$ of the $osp(M|N)$ and $s\ell(M|N)$ Lie superalgebras in the cases when $T$ is the defining and adjoint representations. For defining representations, the $osp(M|N)$- and $s\ell(M|N)$-invariant solutions of the Yang-Baxter equation are expressed as rational functions of the split Casimir operator. For the adjoint representation, the characteristic identities and invariant projectors obtained are considered from the viewpoint of a universal description of Lie superalgebras by means of the Vogel parametrization. We also construct a universal generating function for higher Casimir operators of the $osp(M|N)$ and $s\ell(M|N)$ Lie superalgebras in the adjoint representation.

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arXiv2021

Projectors on invariant subspaces of representations $\operatorname{ad}^{\otimes 2}$ of Lie algebras $so(N)$ and $sp(2r)$ and Vogel parametrization

Explicit formulae for the projectors onto invariant subspaces of the $\operatorname{ad}^{\otimes 2}$ representation of the Lie algebras $so(N)$ and $sp(2r)$ have been found by means of the split Casimir operator. These projectors have also been considered from the viewpoint of the universal complex simple Lie algebra description by using the Vogel parametrisation.

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