# Roiter A. V.

### Singular locally scalar representations of quivers in Hilbert spaces and separating functions

Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 796–809

We consider locally scalar representations of extended Dynkin graphs in Hilbert spaces. The relation between these representations and the function ρ( *n* ) = 1 + ( *n* − 1 ) / ( *n* + 1 ) is established. We construct a family of separating functions that generalize the function ρ and play a similar role in a broader class of graphs.

### The Norm of a Relation, Separating Functions, and Representations of Marked Quivers

Ukr. Mat. Zh. - 2002. - 54, № 6. - pp. 808-840

We consider numerical functions that characterize Dynkin schemes, Coxeter graphs, and tame marked quivers.

### Finitely Represented $K$-Marked Quivers

Belousov K. I., Nazarova L. A., Roiter A. V.

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 550-555

We present necessary and sufficient conditions for the finite representability of *K*-marked quivers.

### Finitely representable dyadic sets

Ukr. Mat. Zh. - 2000. - 52, № 10. - pp. 1363-1396

A criterion of finite representability of dyadic sets is presented.

### Finitely represented dyadic sets and their multielementary representations

Belousov K. I., Nazarova L. A., Roiter A. V.

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1465–1477

We obtain the direct reduction of representations of a dyadic set *S* such that |Ind *C(S)*| < ∞ to the bipartite case.

### Elementary and multielementary representations of vectroids

Belousov K. I., Nazarova L. A., Roiter A. V., Sergeychuk V. V.

Ukr. Mat. Zh. - 1995. - 47, № 11. - pp. 1451–1477

We prove that every finitely represented vectroid is determined, up to an isomorphism, by its completed biordered set. Elementary and multielementary representations of such vectroids (which play a central role for biinvolutive posets) are described.

### Existence of a multiplicative basis for a finitely spaced module over an aggregate

Roiter A. V., Sergeychuk V. V.

Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 567–579

It is proved that a finitely spaced module over $k$-category admits a multiplicative basis (such a module gives rise to a matrix problem in which the allowed column transformations are determined by a module structure, the row transformations are arbitrary, and the number of canonical matrices is finite).

### Tame and wild subspace problems

Gabriel Р., Nazarova L. A., Roiter A. V., Sergeychuk V. V., Vossieck D.

Ukr. Mat. Zh. - 1993. - 45, № 3. - pp. 313–352

Assume that $B$ is a finite-dimensional algebra over an algebraically closed field $k$, $B_d = \text{Spec} k[B_d]$ is the affine algebraic scheme whose $R$-points are the $B ⊗_k k[B_d]$-module structures on $R^d$, and $M_d$ is a canonical $B ⊗_k k[B_d]$-module supported by $k[Bd^]d$. Further, say that an affine subscheme $Ν$ of $B_d$ isclass true if the functor $F_{gn} ∶ X → M_d ⊗_{k[B]} X$ induces an injection between the sets of isomorphism classes of indecomposable finite-dimensional modules over $k[Ν]$ and $B$. If $B_d$ contains a class-true plane for some $d$, then the schemes $B_e$ contain class-true subschemes of arbitrary dimensions. Otherwise, each $B_d$ contains a finite number of classtrue puncture straight lines $L(d, i)$ such that for eachn, almost each indecomposable $B$-module of dimensionn is isomorphic to some $F_{L(d, i)} (X)$; furthermore, $F_{L(d, i)} (X)$ is not isomorphic to $F_{L(l, j)} (Y)$ if $(d, i) ≠ (l, j)$ and $X ≠ 0$. The proof uses a reduction to subspace problems, for which an inductive algorithm permits us to prove corresponding statements.

### Representations of finite p-groups over the ring of formal power series with integral p-adic coefficients

Gudivok P. M., Oros V. M., Roiter A. V.

Ukr. Mat. Zh. - 1992. - 44, № 6. - pp. 753–765

### Integral p-adic representations and representations over a ring of residue classes

Ukr. Mat. Zh. - 1967. - 19, № 2. - pp. 125–126

### Divisibility in the category of representations over a complete local Dedekind ring

Ukr. Mat. Zh. - 1965. - 17, № 4. - pp. 124-129

### $E$-systems of representations

Ukr. Mat. Zh. - 1965. - 17, № 2. - pp. 88-96

### On the categary of representations

Ukr. Mat. Zh. - 1963. - 15, № 4. - pp. 448-453

### Whole-number representations of a symmetrical group of third degree

Ukr. Mat. Zh. - 1962. - 14, № 3. - pp. 271-288

The authors discuss whole-number representations to a symmetrica! group of the third degree. It is shown that there exists only a finite number, i. e. ten, prime representations of this group. The dimensions of the prime representations do not exceed the order of the group.

It is further shown that the factoring of any representation into a direct sum of primes is univalent.

Thus the first example has been given of a complete description of whole-number representations of a non-commutative group.