Nonlinear Operator Superalgebras and BFV-BRST Operators for Lagrangian Description of Mixed-symmetry HS Fields in AdS Spaces

We study the properties of nonlinear superalgebras $\mathcal{A}$ and algebras ${\mathcal{A}}_b$ arising from a one-to-one correspondence between the sets of relations that extract AdS-group irreducible representations $D(E_0,s_1,s_2)$ in AdS${}_d$-spaces and the sets of operators that form $\mathcal{A}$ and ${\mathcal{A}}_b$, respectively, for fermionic, $s_i=n_i+1\mathrm{/}2$, and bosonic, $s_i=n_i$, $n_i\in {\mathbb{N}}_0$, $i=1,2$, HS fields characterized by a Young tableaux with two rows. We consider a method of constructing the Verma modules $V_{\mathcal{A}}$, $V_{{\mathcal{A}}_b}$ for $\mathcal{A}$, ${\mathcal{A}}_b$ and establish a possibility of their Fock-space realizations in terms of formal power series in oscillator operators which serve to realize an additive conversion of the above (super)algebra ($\mathcal{A}$) ${\mathcal{A}}_b$, containing a set of 2nd-class constraints, into a converted (super)algebra ${\mathcal{A}}_{bc}$ = ${\mathcal{A}}_b$ + ${\mathcal{A}}_b^{\mathrm{\prime }}$ (${\mathcal{A}}_c$ = $\mathcal{A}$ + ${\mathcal{A}}^{\mathrm{\prime }}$), containing a set of 1st-class constraints only. For the algebra ${\mathcal{A}}_{bc}$, we construct an exact nilpotent BFV–BRST operator $Q^{\mathrm{\prime }}$ having nonvanishing terms of 3rd degree in the powers of ghost coordinates and use $Q^{\mathrm{\prime }}$ to construct a gauge-invariant Lagrangian formulation (LF) for HS fields with a given mass $m$ (energy $E_0(m)$) and generalized spin $\mathbf{s}$=$(s_1,s_2)$. LFs with off-shell algebraic constraints are examined as well.