Introduction

We start our discussion of PBW-algebras by recalling their definition. Let $K$ be a field. Given a set of variables $x=\{x_1, \ldots, x_n\},$ we write ${\left\langle {x}\right\rangle}:=\langle x_{1},\ldots, x_{n} \rangle$ for the free monoid on $x$. That is, the elements of $\langle x \rangle$ are the words in the finite alphabet $x$, multiplication means concatenation of words, and the identity element is the empty word. The free associative $K$-algebra generated by $x_{1},\dots, x_{n}$ is the corresponding monoid algebra

\[K \langle {x}\rangle:= K \langle x_{1},\dots, x_{n} \rangle.\]

We consider quotients of type $A = K\langle x_1, \dots, x_n \rangle/J$, for some $n$ and some two-sided ideal $J$ of $K\langle x_1, \dots, x_n \rangle$. In case $J$ is given by a finite set of two-sided generators $g_1, \dots, g_r$, we say that $A$ is generated by $x_1, \dots, x_n$, subject to the relations $g_1 = 0, \dots, g_r = 0$, and write

\[A = K\langle x_1, \dots , x_n \mid g_k=0, \ 1\leq k \leq r \rangle.\]

The relations considered in this chapter are "commutation relations", written as

\[x_jx_i = c_{ij}x_ix_j+d_{ij}.\]

Working with Gröbner bases requires that we take monomial orderings into account (see the section on Gröbner bases in the commutative algebra chapter for monomial orderings). In our context here, we use the following notation. A standard monomial in $K \langle x \rangle$ is a word of type $x^\alpha=x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}},$ where $\alpha=(\alpha_1,\dots,\alpha_n)\in\mathbb N^n$. A standard polynomial in $K \langle x \rangle$ is a $K$-linear combination of standard monomials. Each global monomial ordering $>$ on $K[x]$ gives rise to a (total) well-ordering on the set of standard monomials. Abusing our notation, we denote the induced ordering again by $>$, and refer to it as a global monomial ordering on $A$. Given $>$, it makes sense to speak of the leading monomial $\text{LM}_>(f)$ of a standard polynomial $0\neq f \in K \langle x \rangle.$ The notion of a global elimination ordering with respect to a subset of $\{ x_{1},\ldots, x_{n} \}$ carries over from $K[x]$ to $A$.

Definition. $\;$ Let $A$ be a $K$-algebra of type

\[A = K\langle x_1, \dots , x_n \mid x_jx_i = c_{ij}x_ix_j+d_{ij}, \ 1\leq i<j \leq n \rangle,\]

where the $c_{ij}\in K$ are nonzero scalars, and the $d_{ij}\in K\langle x_1, \dots , x_n\rangle$ are standard polynomials. Then $A$ is called a PBW-algebra if the following two conditions hold:

(1) $\;$ There exists a global monomial ordering $>$ on $A$ such that

\[d_{ij}=0\ \text{ or }\ x_ix_j> \text{LM}_>(d_{ij})\ \text{ for all }\ 1\leq i<j \leq n.\]

(2) $\;$ The standard monomials in $K \langle x \rangle$ represent a $K$-basis for $A$. We then refer to this basis as a PBW-basis for $A$.

Every ordering as in (1) is called admissible for the given relations or simply admissible for $A$.

Given a PBW-algebra $A$ as above, we sometimes abuse our notation by denoting the class of a standard monomial $x^{\alpha}$ in $A$ also by $x^{\alpha}$, and refer to this class as a standard monomial in $A$. As these monomials form a $K$-basis for $A$, every element $0\neq f\in A$ has a unique representation

\[f=\sum c_{\alpha}x^{\alpha}, \; \text{ with nonzero coefficients } \; c_{\alpha}\in K.\]

We refer to this representation as the standard representation of $f$, with coefficients $c_{\alpha}$, and exponents $\alpha$.

Proposition. $\;$ Let $A$ be a PBW-algebra. Then:

• $A$ is an integral domain,
• $A$ is (left and right) Noetherian.

Given any associative $K$-algebra $A = (A, +, \cdot)$, its opposite algebra is defined by setting $A^{\text{op}} = (A, +, \ast)$, where $f\ast g:=g\cdot f$ for all $f, g\in A.$ If

\[A = K\langle x_1, \dots , x_n \mid x_jx_i = c_{ij} x_ix_j+d_{ij}, \ 1\leq i<j \leq n \rangle\]

is a PBW-algebra, then $A^{\text{op}}$ is again a PBW-algebra in a natural way. Indeed, consider the automorphism ${\text{op}}$ of $K\langle x_1, \dots , x_n\rangle$ which sends a word $x_{i_1}\cdots x_{i_r}$ to the "opposite word" $x^{\text{op}}:=x_{i_r}\cdots x_{i_1}$. Apply this automorphism to the relations of $A$ to obtain the "opposite relations"

\[x_ix_j = c_{ij}x_jx_i+d_{ij}^{\text{\;\!op}}.\]

Also, if $\alpha=(\alpha_1, \ldots, \alpha_n)\in \mathbb{N}^n$, then set $\alpha^{\text{op}} =(\alpha_n, \ldots, \alpha_1)\in \mathbb{N}^n$, and if $>$ is an admissible monomial ordering for $A$, then define the "opposite ordering" $ >^{\text{op}}$ by setting

\[\alpha >^{\text{op}} \beta \;\Leftrightarrow\; \alpha^{\text{op}} > \beta^{\text{op}}.\]

Finally, reverse the order of the variables: set $x ^{\text{op}}=\{x_n, \ldots, x_1\}$, and consider the free associative $K$-algebra $K \langle x^\text{op}\rangle = K \langle x_{n},\dots, x_{1} \rangle.$ Altogether, we obtain a PBW-algebra which can be naturally identified with $A^{\text{op}}$:

\[A ^{\text{op}} = K\langle x_n, \dots , x_1 \mid x_ix_j = c_{ij}x_jx_i+d_{ij}^{\text{\;\!op}}, \ 1\leq i<j \leq n \rangle,\]

with admissible ordering $>^{\text{op}}$.

When implementing functionality for PBW-algebras, taking the opposite algebra into account often allows one to focus on left ideals, left modules, and left Gröbner bases: Given a PBW-algebra $A$, right Gröbner bases in $A$ are found by computing left Gröbner bases in $A^{\text{op}}$. Here, Gröbner bases are considered with respect to an admissible ordering $>$ for $A$ and the opposite ordering $>^{\text{op}}$ for $A^{\text{op}}$, respectively. Each left Gröbner basis of a two-sided ideal $I$ is also a right Gröbner basis of $I$. Moreover, there is an algorithm which, starting from a left Gröbner basis of $I$, computes a two-sided Gröbner basis of $I$ (see, for example, [DL06]).